Research Article Dynamic of a TB-HIV Coinfection...

14
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 429567, 13 pages http://dx.doi.org/10.1155/2013/429567 Research Article Dynamic of a TB-HIV Coinfection Epidemic Model with Latent Age Xiaoyan Wang, Junyuan Yang, and Fengqin Zhang Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi, China Correspondence should be addressed to Junyuan Yang; [email protected] Received 16 November 2012; Accepted 6 January 2013 Academic Editor: Jinde Cao Copyright © 2013 Xiaoyan Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease. Recently, this has happened with human immunodeficiency virus (HIV) and tuberculosis (TB). e density of individuals infected with latent tuberculosis is structured by age since latency. e host population is divided into five subclasses of susceptibles, latent TB, active TB (without HIV), HIV infectives (without TB), and coinfection class (infected by both TB and HIV). e model exhibits three boundary equilibria, namely, disease free equilibrium, TB dominated equilibrium, and HIV dominated equilibrium. We discuss the local or global stabilities of boundary equilibria. We prove the persistence of our model. Our simple model of two synergistic infectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progression of the other disease. We simulate the dynamic behaviors of our model and give medicine explanations. 1. Introduction Coepidemics—the related spread of two or more infectious diseases—have afflicted mankind for centuries. Worldwide, there were an estimated 1.37 million coinfected HIV and TB patients globally in 2007. Around 80 percent of patients live in sub-Saharan Africa. 456000 people died of HIV-associated TB in 2007. HIV/AIDS and tuberculosis (TB) are commonly called the “deadly duo” and referred to as HIV/TB, despite biological differences. HIV is a retrovirus that is transmitted primarily by homosexual and heterosexual contact, needle sharing, and from mother to child. e disease eventually progresses to AIDS as the immune system weakens. HIV can be treated with highly active antiheroical therapy (HAART), but there is presently no cure [1]. Virtually all HIV-infected individuals can transmit the virus to others, and an infected individual’s chance of spreading the virus generally increases as the disease progresses and damages the immune system [2]. Tuberculosis is caused by mycobacterium tuberculosis bacteria and is spread through the air. Some TB infections are “latent,” meaning that a person has the TB-causing bacteria but it is dormant. A person with latent TB is not sick and is not infectious. However, latent TB can progress to “active” TB. “Active” TB infection means that the TB bacteria are multiplying and spreading in the body. A person with active TB in their lungs or throat can transmit the bacteria to others. Symptoms of active TB include a cough that lasts several weeks, weight loss, loss of appetite, fever, night sweats, and coughing up blood. HIV weakens the immune system and so people are more susceptible to catching TB if they are exposed. At least one- third of people living with HIV worldwide are infected with TB and are 20–30 times more likely to develop TB than those without HIV. TB bacteria accelerate the progression of HIV to AIDS. Persons coinfected with TB and HIV may spread the disease not only to the other HIV-infected persons, but also to members of the general population who do not participate in any of the high risk behaviors associated with HIV. People living with HIV and displaying early diagnosis need treatment in time. If TB is present, they should receive TB preventive treatment (IPT). e treatments are not expensive. erefore, it is essential that adequate attention must be paid to study the transmission dynamics of HIV-TB coinfection in the population. Current treatment for HIV is known as highly active antiretroviral therapy (HAART) [3]. Some authors have developed simulation models to investigate HIV-TB co-epidemic dynamics [4–7]. West and

Transcript of Research Article Dynamic of a TB-HIV Coinfection...

Page 1: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 429567 13 pageshttpdxdoiorg1011552013429567

Research ArticleDynamic of a TB-HIV Coinfection Epidemic Model withLatent Age

Xiaoyan Wang Junyuan Yang and Fengqin Zhang

Department of Applied Mathematics Yuncheng University Yuncheng 044000 Shanxi China

Correspondence should be addressed to Junyuan Yang yangjunyuan00126com

Received 16 November 2012 Accepted 6 January 2013

Academic Editor Jinde Cao

Copyright copy 2013 Xiaoyan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease Recently thishas happened with human immunodeficiency virus (HIV) and tuberculosis (TB) The density of individuals infected with latenttuberculosis is structured by age since latency The host population is divided into five subclasses of susceptibles latent TB activeTB (without HIV) HIV infectives (without TB) and coinfection class (infected by both TB and HIV) The model exhibits threeboundary equilibria namely disease free equilibrium TB dominated equilibrium and HIV dominated equilibrium We discussthe local or global stabilities of boundary equilibria We prove the persistence of our model Our simple model of two synergisticinfectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progressionof the other disease We simulate the dynamic behaviors of our model and give medicine explanations

1 Introduction

Coepidemicsmdashthe related spread of two or more infectiousdiseasesmdashhave afflicted mankind for centuries Worldwidethere were an estimated 137 million coinfected HIV and TBpatients globally in 2007 Around 80 percent of patients livein sub-Saharan Africa 456000 people died of HIV-associatedTB in 2007 HIVAIDS and tuberculosis (TB) are commonlycalled the ldquodeadly duordquo and referred to as HIVTB despitebiological differences HIV is a retrovirus that is transmittedprimarily by homosexual and heterosexual contact needlesharing and from mother to child The disease eventuallyprogresses to AIDS as the immune system weakens HIV canbe treated with highly active antiheroical therapy (HAART)but there is presently no cure [1] Virtually all HIV-infectedindividuals can transmit the virus to others and an infectedindividualrsquos chance of spreading the virus generally increasesas the disease progresses and damages the immune system[2] Tuberculosis is caused by mycobacterium tuberculosisbacteria and is spread through the air Some TB infections areldquolatentrdquo meaning that a person has the TB-causing bacteriabut it is dormant A person with latent TB is not sick and isnot infectious However latent TB can progress to ldquoactiverdquoTB ldquoActiverdquo TB infection means that the TB bacteria are

multiplying and spreading in the body A person with activeTB in their lungs or throat can transmit the bacteria to othersSymptoms of active TB include a cough that lasts severalweeks weight loss loss of appetite fever night sweats andcoughing up blood

HIVweakens the immune system and so people are moresusceptible to catching TB if they are exposed At least one-third of people living with HIV worldwide are infected withTB and are 20ndash30 times more likely to develop TB than thosewithout HIV TB bacteria accelerate the progression of HIVtoAIDS Persons coinfectedwith TB andHIVmay spread thedisease not only to the other HIV-infected persons but alsoto members of the general population who do not participatein any of the high risk behaviors associated with HIVPeople living with HIV and displaying early diagnosis needtreatment in time If TB is present they should receive TBpreventive treatment (IPT)The treatments are not expensiveTherefore it is essential that adequate attention must be paidto study the transmission dynamics of HIV-TB coinfection inthe population Current treatment forHIV is known as highlyactive antiretroviral therapy (HAART) [3]

Some authors have developed simulation models toinvestigate HIV-TB co-epidemic dynamics [4ndash7] West and

2 Journal of Applied Mathematics

Λ

119878

12057321198781198682

120583119878

12057311198781198681

120583119869

119869

12057511986811198682

1205831198681

120574(119886)119890(119886119905)

12057211198681

120583119890(119886119905)

120584119869

1205720(119886)119890(119886119905)

12057221198682

119890

1198681

1198682

1205831198682

Figure 1 The flow diagram of the model (1) Two yellow rectangu-lars denote the individuals infected by latent tuberculosis and activetuberculosisThe red rectangular denotes the individuals infected byHIV The orange rectangular denotes the individuals coinfected byTB and HIV

Thompson [8] developed models which reflect the trans-mission dynamics of both TB and HIV and discussed themagnitude and duration of the effect that the HIV epidemicmay have on TB Naresh and Tripathi [9] presented amodel for HIV-TB coinfection with constant recruitmentof susceptibles and found that TB will be eradicated fromthe population if more than 90 percent TB infectives arerecovered due to effective treatment Naresh et al [10] studiedthe effect of tuberculosis on the spread of HIV infection in alogistically growing human population They found that asthe number of TB infectives decreases due to recovery thenumber of HIV infectives also decreases and endemic equi-librium tends toTB free equilibrium Long et al [11] discussedtwo synergistic infectious disease epidemics illustrating theimportance of including the effects of each disease on thetransmission and progression of the other disease

However these models may exclude coinfection maygreatly simplify infection dynamics and may include fewdisease states (eg the important distinction between latentand active TBwas absent)They assume thatHIV treatment isunavailable Bifurcation will appear in some models [12 13]We perform the additional latent age analyze the coexistenceequilibria and extend the model to include disease recoveryin this paper At the same time we obtain the persistence ofthe system and global stabilities of the equilibria under someconditions

This paper is organized as follows In Section 2 weintroduce the TB-HIV coinfection model In Section 3 we

introduce the reproduction numbers of TB and HIV 1198771 119877

2

and discuss the existence of the equilibria The values of thedisease-free equilibrium the two boundary equilibria andthe coexistence equilibrium are given explicitly Section 4focuses on local and global stabilities of the equilibria InSection 5 we discuss the persistence of the system in suitableperiod In Section 6 we simulate and illustrate our resultsWegive some biological explanations in this section In Section 7we conclude our results and discuss the defect of our model

2 The Model Formulation

Two diseases mentioned in the introduction are spreading ina population of total size 119873(119905) We classify the total popu-lation into four classes the susceptible 119878(119905) the individualsinfected by the tuberculosis 119890(119886 119905) and 119868

1(119905) denotes the

latent TB class which has no infectious ability and active TBclass who can infect the susceptible class respectively theindividuals infected by HIV 119868

2(119905) the individuals coinfected

by tuberculosis and HIV 119869(119905) The individuals infected withactive TB infect the susceptible and then develop into thelatent individuals at 120573

1 The individuals infected by HIV

infect the susceptible and become the individuals infected byHIV at 120573

2 An individual already infected with TB can be

coinfected with HIV at 120575 and thus become jointly infectedindividuals 119869(119905)

Figure 1 presents a schematic flow diagram of the mathe-matical model as follows

119889119878

119889119905= Λ minus 120583119878 minus 120573

1119878119868

1minus 120573

2119878119868

2

+ int

infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198681+ 120572

21198682

120597119890 (119886 119905)

120597119886+120597119890 (119886 119905)

120597119905= minus (120583 + 120574 (119886) + 120572

0(119886)) 119890 (119886 119905)

119890 (0 119905) = 1205731119878119868

1

1198891198681

119889119905= int

infin

0

120574 (119886) 119890 (119886 119905) 119889119886minus 1205831198681minus 120575119868

11198682minus 120572

11198681

1198891198682

119889119905= 120573

2119878119868

2minus (120583 + 120572

2) 119868

2

119889119869

119889119905= 120575119868

11198682minus (120583 + 120584) 119869

(1)

where 120583 is natural death rate and Λ is birth rate 120574(119886) israte of endogenous reactivation of latent TB We assume thatthe individuals separately infected by TB and HIV are notlethal but the coinfection can lead to the extra death at 120584Specifically we assume that jointly infected individuals donot recover Individuals infected with latent TB active TB orHIV alone may be potentially treated at rates 120572

0(119886) 120572

1 and

1205722 respectively

Assumption 1 Suppose that

(a) Λ 120583 120575 1205721 120572

2 120573

1 120573

2isin (0 +infin)

Journal of Applied Mathematics 3

(b) 120584 isin [0 +infin)(c) 120574(119886) 120572

0(119886) isin 119862

119861119880([0 +infin) 119877) cap 119862

+([0 +infin) 119877) and

for each 119886 ge 0 where 119862119861119880(119877) denotes the space

of bounded and uniformly continuous map from[0 +infin) into 119877

We assume (1) with the initial conditions

119878 (0) = 1198780ge 0 119890 (119886 0) = 119890

0(119886) isin 119871

1

+(0 +infin)

1198681(0) = 119868

10ge 0 119868

2(0) = 119868

20ge 0

119869 (0) = 1198690ge 0

(2)

Set

119883 = 119877 times 119884 times 1198772 119883

+= 119877

+times 119884

+times 119877

+

2

1198830= 119877 times 119884

0times 119877

2 119883

0+= 119883

0cap 119883

+

(3)

with

119884 = 119877 times 1198711(0 +infin) 119884

+= [0 +infin) times 119871

1

+(0 +infin)

1198840= 0 times 119871

1(0 +infin)

(4)

Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882

11(0infin)

for each 119909 = (119878 0119877 119890 119868

1 1198682) isin 119883

0 DefineA 119863(A) sub 119883

0sub

119883 rarr 119883 andF 1198830rarr 119883 by

A119909

=((

(

minus120583119878

minus119890 (0)

minus119889119890 (sdot)

119889119886minus (120583 + 120572

0(sdot) + 120574 (sdot)) 119890 (sdot)

minus1205831198681minus 120572

11198681

minus1205831198682minus 120572

21198682

minus (120583 + 120584) 119869

))

)

if 119909 isin 119863 (A)

(5)

F (119909)

=

((((

(

120583minus 1205731119878119868

1minus 120573

2119878119868

2+ int

infin

0

1205720(119886) 119890 (119886) 119889119886 + 120572

11198681+ 120572

21198682

1205731119878119868

1

01198711

int

infin

0

120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682

1205732119878119868

2

12057511986811198682

))))

)

(6)

Rewrite problem (1) as an abstract Cauchy problem

119889119906119909(119905)

119889119905= A119906

119909(119905) +F (119906

119909(119905)) 119905 ge 0 119906

119909(0) = 119909 isin 119883

0+

(7)

It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583

10038171003817100381710038171003817(120582 minusA)

minus110038171003817100381710038171003817le1

120582 + 120583 (8)

By applying the results in Magal et al [14ndash16] we obtainthe following proposition

Lemma 1 There exists a uniquely determined semiflow119880(119905)

119905ge0on 119883

0+ such that for each 119909 = (119878

0 0 119890

0 11986810

11986820 1198690)119879isin 119883

0+ there exists a unique continuous map 119880 isin

119862([0 +infin)1198830+) which is an integrated solution of the Cauchy

problem (1) that is to say that

int

119905

0

119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0

119880 (119905) 119909 = 119909 +Aint119905

0

119880 (119904) 119909 119889119904 + int

119905

0

F (119880 (119904) 119909) 119889119904 forallt ge 0

(9)

The total population size 119873(119905) is the sum of all individuals inall classes

119873 = 119878 (119905) + int

+infin

0

119890 (119886 119905) 119889119886 + 1198681(119905) + 119868

2(119905) + 119869 (119905) (10)

The total population size satisfies the equation 1198731015840(119905) = Λ minus

120583119873 minus 120584119869 We introduce the notation

120587 (119886) = exp(minusint119886

0

(1205720(120591) + 120574 (120591)) 119889120591) (11)

To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity

119861 = int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 (12)

which gives the probability of treatment since the individualscan leave TB infectious period via treatment

119862 = int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)

which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is

int

infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 + int

infin

0

120574 (a) 120587 (119886) 119890minus120583119886 119889119886

+ 120583int

infin

0

120587 (119886) 119890minus120583119886119889119886 = 1

(14)

It immediately follows that 119861 + 119862 lt 1

3 Equilibria of the Model with Coinfection

We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is

1198771=Λ120573

1119862

120583 (120583 + 1205721) (15)

4 Journal of Applied Mathematics

and the reproduction number of HIV is

1198772=

Λ1205732

120583 (120583 + 1205722) (16)

We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894

1 1198942 119895 as the equilibria of the modelTherefore equilibria

satisfy the following equations

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ int

+infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

12057321199041198942minus (120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(17)

The ODE in the system can be solved to result in

119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)

Substituting for 119894 in the integrals one obtains

int

+infin

0

1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 = 119890 (0) 119861

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862

(19)

With this notation the system for the equilibria becomes

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ 119890 (0) 119861 + 12057211198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

119890 (0) 119862 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

1205732119904119894minus(120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(20)

This system has three boundary equilibria

(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)

The disease-free equilibrium always exists

(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the

TB equilibrium is given by

1199041=Λ

1205831198771

119890 = 119890 (0) 120587 (119886) 119890minus120583119886

1198941=

Λ (1 minus 11198771)

(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862

119890 (0) = 120573111990411198941

(21)

Thus the equilibrium is

1198641= (119904

1 119890 119894

1 0 0) (22)

(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by

1198642= (119904

2 0 0 119894

2 0) (23)

where 1199042= 1119877

2 119894

2= 0 119890

2= 0 119894

2= (Λ120583)(1 minus

11198772) 119895

2= 0

(4) The coexistence equilibrium exists if and only if 1198771gt

1198772 and [1 minus (1119877

2+120583(120583+120572

1)Λ120575)][(Λ120573

1120583120572

11198772)(1 minus

119861) minus 1] gt 0 and it is given by

1198642= (119904 119890 119894

1 1198942 119895) (24)

where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 1198941=

(Λ1205721)(1 minus (1119877

2+120583(120583+120572

1)Λ120575))((Λ120573

1120583120572

11198772)(1 minus

119861) minus 1) 1198942= ((120583 + 120575)120575)(119877

1119877

2minus 1) 119895 = 120575119894

11198942(120583 + 120584)

Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0

4 Stability of Equilibria

In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not

To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894

1 1198942 119895 That

is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894

1 119868

2= 119906 + 119894

2 119869 = 119908 + 119895

Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and

Journal of Applied Mathematics 5

119908These satisfy the following linear eigenvalue problem (here119904 119890 119894

1 1198942 and 119895 are the corresponding equilibria)

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

120582119908 = 1205751198942119911 + 120575119894

1119906 minus (120583 + 120584)119908

(25)

In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing

41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894

1= 119894

2= 119895 = 0 and 119904 = Λ120583

Thus the system above simplifies to the following system

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(26)

From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864

0

Theorem 2 If 1198771lt 1 and 119877

2lt 1 then the disease-free

equilibrium 1198640is locally asymptotically stable If 119877

1gt 1 or

1198772gt 1 then the disease-free equilibrium 119864

0is unstable

Proof To see this from the second to last equation we have120582119906 = [120573

2119904 minus (120583 + 120572

2)]119906 where either 120582 = 120573

2119904 minus (120583 + 120572

2) or

119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877

2minus 1) lt 0 if and only

if 1198772lt 1 Thus if 119877

2gt 1 the disease-free equilibrium 119864

0

is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation

119910 (119886) = 119910 (0) 119890minus(120582+120583)119886

120587 (119886) = 1205731119904119911119890

minus(120582+120583)119886120587 (119886) (27)

Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the

characteristic equation 120582119911 = 1205731119904 int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 119911 minus

(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572

1)(120573

1119904(120583 + 120572

1))

intinfin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 minus 1) or 119911 = 0

We denote the left hand side of the equation above byF

1(120582) = 120582 andF

2(120582) = (120583 + 120572

1)(119877

1(120582) minus 1) where 119877

1(120582) =

(1205731119904(120583 + 120572

1)) int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 and

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(28)

IfF1(120582) = F

2(120582) has a root with Re 120582 ge 0 then Re F

1(120582) ge

0 But |F2(120582)| le (120583 + 120572

1)(119877

1minus 1) lt 0 when 119877

1lt 1 This is a

contradictionThus if both 119877

1lt 1 and 119877

2lt 1 all eigenvalues

have negative real part and the disease-free equilibrium 1198640

is locally asymptotically stable If only 1198771gt 1 then if we

consider F1(120582) = F

2(120582) for 120582 is real we see that F

2(120582) is

a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F

2(0) = (120583 + 120572

1)(119877

1minus 1) gt 0 and F

1(0) = 0

that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864

0is unstableThis concludes the

proof

In what follows we show that the diseases vanish if 1198771lt

1 1198772lt 1

Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864

0is a global attractor

that is lim119905rarrinfin

119890(119886 119905) = 0 1198681rarr 0 119868

2rarr 0 119869 rarr 0 as

119905 rarr infin

Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin

le

Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have

119890 (119886 119905) =

B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886

(29)

SinceB(119905) = 1205731119878119868

1le 120573

1Λ119868

1120583 and

1198681015840

1le 120573

1

Λ

120583int

119905

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1198651(119905) minus 120583119868

1minus 120572

11198681

(30)

where 1198651(119905) = int

infin

119905120574(119886)119890

0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890

minus120583119905119889119886 and

lim119905rarrinfin

1198651(119905) = 0 from the equation for 119868

1then we have the

following inequality

1198681le 119868

1(0) 119890

minus(120583+1205721)119905

+ 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)120591int

119905minus120591

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889120591

+ int

119905

0

119890minus(120583+120572

1)1205911198651(119905 minus 120591) 119889120591

(31)

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

2 Journal of Applied Mathematics

Λ

119878

12057321198781198682

120583119878

12057311198781198681

120583119869

119869

12057511986811198682

1205831198681

120574(119886)119890(119886119905)

12057211198681

120583119890(119886119905)

120584119869

1205720(119886)119890(119886119905)

12057221198682

119890

1198681

1198682

1205831198682

Figure 1 The flow diagram of the model (1) Two yellow rectangu-lars denote the individuals infected by latent tuberculosis and activetuberculosisThe red rectangular denotes the individuals infected byHIV The orange rectangular denotes the individuals coinfected byTB and HIV

Thompson [8] developed models which reflect the trans-mission dynamics of both TB and HIV and discussed themagnitude and duration of the effect that the HIV epidemicmay have on TB Naresh and Tripathi [9] presented amodel for HIV-TB coinfection with constant recruitmentof susceptibles and found that TB will be eradicated fromthe population if more than 90 percent TB infectives arerecovered due to effective treatment Naresh et al [10] studiedthe effect of tuberculosis on the spread of HIV infection in alogistically growing human population They found that asthe number of TB infectives decreases due to recovery thenumber of HIV infectives also decreases and endemic equi-librium tends toTB free equilibrium Long et al [11] discussedtwo synergistic infectious disease epidemics illustrating theimportance of including the effects of each disease on thetransmission and progression of the other disease

However these models may exclude coinfection maygreatly simplify infection dynamics and may include fewdisease states (eg the important distinction between latentand active TBwas absent)They assume thatHIV treatment isunavailable Bifurcation will appear in some models [12 13]We perform the additional latent age analyze the coexistenceequilibria and extend the model to include disease recoveryin this paper At the same time we obtain the persistence ofthe system and global stabilities of the equilibria under someconditions

This paper is organized as follows In Section 2 weintroduce the TB-HIV coinfection model In Section 3 we

introduce the reproduction numbers of TB and HIV 1198771 119877

2

and discuss the existence of the equilibria The values of thedisease-free equilibrium the two boundary equilibria andthe coexistence equilibrium are given explicitly Section 4focuses on local and global stabilities of the equilibria InSection 5 we discuss the persistence of the system in suitableperiod In Section 6 we simulate and illustrate our resultsWegive some biological explanations in this section In Section 7we conclude our results and discuss the defect of our model

2 The Model Formulation

Two diseases mentioned in the introduction are spreading ina population of total size 119873(119905) We classify the total popu-lation into four classes the susceptible 119878(119905) the individualsinfected by the tuberculosis 119890(119886 119905) and 119868

1(119905) denotes the

latent TB class which has no infectious ability and active TBclass who can infect the susceptible class respectively theindividuals infected by HIV 119868

2(119905) the individuals coinfected

by tuberculosis and HIV 119869(119905) The individuals infected withactive TB infect the susceptible and then develop into thelatent individuals at 120573

1 The individuals infected by HIV

infect the susceptible and become the individuals infected byHIV at 120573

2 An individual already infected with TB can be

coinfected with HIV at 120575 and thus become jointly infectedindividuals 119869(119905)

Figure 1 presents a schematic flow diagram of the mathe-matical model as follows

119889119878

119889119905= Λ minus 120583119878 minus 120573

1119878119868

1minus 120573

2119878119868

2

+ int

infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198681+ 120572

21198682

120597119890 (119886 119905)

120597119886+120597119890 (119886 119905)

120597119905= minus (120583 + 120574 (119886) + 120572

0(119886)) 119890 (119886 119905)

119890 (0 119905) = 1205731119878119868

1

1198891198681

119889119905= int

infin

0

120574 (119886) 119890 (119886 119905) 119889119886minus 1205831198681minus 120575119868

11198682minus 120572

11198681

1198891198682

119889119905= 120573

2119878119868

2minus (120583 + 120572

2) 119868

2

119889119869

119889119905= 120575119868

11198682minus (120583 + 120584) 119869

(1)

where 120583 is natural death rate and Λ is birth rate 120574(119886) israte of endogenous reactivation of latent TB We assume thatthe individuals separately infected by TB and HIV are notlethal but the coinfection can lead to the extra death at 120584Specifically we assume that jointly infected individuals donot recover Individuals infected with latent TB active TB orHIV alone may be potentially treated at rates 120572

0(119886) 120572

1 and

1205722 respectively

Assumption 1 Suppose that

(a) Λ 120583 120575 1205721 120572

2 120573

1 120573

2isin (0 +infin)

Journal of Applied Mathematics 3

(b) 120584 isin [0 +infin)(c) 120574(119886) 120572

0(119886) isin 119862

119861119880([0 +infin) 119877) cap 119862

+([0 +infin) 119877) and

for each 119886 ge 0 where 119862119861119880(119877) denotes the space

of bounded and uniformly continuous map from[0 +infin) into 119877

We assume (1) with the initial conditions

119878 (0) = 1198780ge 0 119890 (119886 0) = 119890

0(119886) isin 119871

1

+(0 +infin)

1198681(0) = 119868

10ge 0 119868

2(0) = 119868

20ge 0

119869 (0) = 1198690ge 0

(2)

Set

119883 = 119877 times 119884 times 1198772 119883

+= 119877

+times 119884

+times 119877

+

2

1198830= 119877 times 119884

0times 119877

2 119883

0+= 119883

0cap 119883

+

(3)

with

119884 = 119877 times 1198711(0 +infin) 119884

+= [0 +infin) times 119871

1

+(0 +infin)

1198840= 0 times 119871

1(0 +infin)

(4)

Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882

11(0infin)

for each 119909 = (119878 0119877 119890 119868

1 1198682) isin 119883

0 DefineA 119863(A) sub 119883

0sub

119883 rarr 119883 andF 1198830rarr 119883 by

A119909

=((

(

minus120583119878

minus119890 (0)

minus119889119890 (sdot)

119889119886minus (120583 + 120572

0(sdot) + 120574 (sdot)) 119890 (sdot)

minus1205831198681minus 120572

11198681

minus1205831198682minus 120572

21198682

minus (120583 + 120584) 119869

))

)

if 119909 isin 119863 (A)

(5)

F (119909)

=

((((

(

120583minus 1205731119878119868

1minus 120573

2119878119868

2+ int

infin

0

1205720(119886) 119890 (119886) 119889119886 + 120572

11198681+ 120572

21198682

1205731119878119868

1

01198711

int

infin

0

120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682

1205732119878119868

2

12057511986811198682

))))

)

(6)

Rewrite problem (1) as an abstract Cauchy problem

119889119906119909(119905)

119889119905= A119906

119909(119905) +F (119906

119909(119905)) 119905 ge 0 119906

119909(0) = 119909 isin 119883

0+

(7)

It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583

10038171003817100381710038171003817(120582 minusA)

minus110038171003817100381710038171003817le1

120582 + 120583 (8)

By applying the results in Magal et al [14ndash16] we obtainthe following proposition

Lemma 1 There exists a uniquely determined semiflow119880(119905)

119905ge0on 119883

0+ such that for each 119909 = (119878

0 0 119890

0 11986810

11986820 1198690)119879isin 119883

0+ there exists a unique continuous map 119880 isin

119862([0 +infin)1198830+) which is an integrated solution of the Cauchy

problem (1) that is to say that

int

119905

0

119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0

119880 (119905) 119909 = 119909 +Aint119905

0

119880 (119904) 119909 119889119904 + int

119905

0

F (119880 (119904) 119909) 119889119904 forallt ge 0

(9)

The total population size 119873(119905) is the sum of all individuals inall classes

119873 = 119878 (119905) + int

+infin

0

119890 (119886 119905) 119889119886 + 1198681(119905) + 119868

2(119905) + 119869 (119905) (10)

The total population size satisfies the equation 1198731015840(119905) = Λ minus

120583119873 minus 120584119869 We introduce the notation

120587 (119886) = exp(minusint119886

0

(1205720(120591) + 120574 (120591)) 119889120591) (11)

To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity

119861 = int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 (12)

which gives the probability of treatment since the individualscan leave TB infectious period via treatment

119862 = int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)

which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is

int

infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 + int

infin

0

120574 (a) 120587 (119886) 119890minus120583119886 119889119886

+ 120583int

infin

0

120587 (119886) 119890minus120583119886119889119886 = 1

(14)

It immediately follows that 119861 + 119862 lt 1

3 Equilibria of the Model with Coinfection

We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is

1198771=Λ120573

1119862

120583 (120583 + 1205721) (15)

4 Journal of Applied Mathematics

and the reproduction number of HIV is

1198772=

Λ1205732

120583 (120583 + 1205722) (16)

We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894

1 1198942 119895 as the equilibria of the modelTherefore equilibria

satisfy the following equations

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ int

+infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

12057321199041198942minus (120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(17)

The ODE in the system can be solved to result in

119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)

Substituting for 119894 in the integrals one obtains

int

+infin

0

1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 = 119890 (0) 119861

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862

(19)

With this notation the system for the equilibria becomes

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ 119890 (0) 119861 + 12057211198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

119890 (0) 119862 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

1205732119904119894minus(120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(20)

This system has three boundary equilibria

(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)

The disease-free equilibrium always exists

(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the

TB equilibrium is given by

1199041=Λ

1205831198771

119890 = 119890 (0) 120587 (119886) 119890minus120583119886

1198941=

Λ (1 minus 11198771)

(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862

119890 (0) = 120573111990411198941

(21)

Thus the equilibrium is

1198641= (119904

1 119890 119894

1 0 0) (22)

(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by

1198642= (119904

2 0 0 119894

2 0) (23)

where 1199042= 1119877

2 119894

2= 0 119890

2= 0 119894

2= (Λ120583)(1 minus

11198772) 119895

2= 0

(4) The coexistence equilibrium exists if and only if 1198771gt

1198772 and [1 minus (1119877

2+120583(120583+120572

1)Λ120575)][(Λ120573

1120583120572

11198772)(1 minus

119861) minus 1] gt 0 and it is given by

1198642= (119904 119890 119894

1 1198942 119895) (24)

where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 1198941=

(Λ1205721)(1 minus (1119877

2+120583(120583+120572

1)Λ120575))((Λ120573

1120583120572

11198772)(1 minus

119861) minus 1) 1198942= ((120583 + 120575)120575)(119877

1119877

2minus 1) 119895 = 120575119894

11198942(120583 + 120584)

Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0

4 Stability of Equilibria

In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not

To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894

1 1198942 119895 That

is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894

1 119868

2= 119906 + 119894

2 119869 = 119908 + 119895

Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and

Journal of Applied Mathematics 5

119908These satisfy the following linear eigenvalue problem (here119904 119890 119894

1 1198942 and 119895 are the corresponding equilibria)

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

120582119908 = 1205751198942119911 + 120575119894

1119906 minus (120583 + 120584)119908

(25)

In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing

41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894

1= 119894

2= 119895 = 0 and 119904 = Λ120583

Thus the system above simplifies to the following system

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(26)

From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864

0

Theorem 2 If 1198771lt 1 and 119877

2lt 1 then the disease-free

equilibrium 1198640is locally asymptotically stable If 119877

1gt 1 or

1198772gt 1 then the disease-free equilibrium 119864

0is unstable

Proof To see this from the second to last equation we have120582119906 = [120573

2119904 minus (120583 + 120572

2)]119906 where either 120582 = 120573

2119904 minus (120583 + 120572

2) or

119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877

2minus 1) lt 0 if and only

if 1198772lt 1 Thus if 119877

2gt 1 the disease-free equilibrium 119864

0

is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation

119910 (119886) = 119910 (0) 119890minus(120582+120583)119886

120587 (119886) = 1205731119904119911119890

minus(120582+120583)119886120587 (119886) (27)

Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the

characteristic equation 120582119911 = 1205731119904 int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 119911 minus

(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572

1)(120573

1119904(120583 + 120572

1))

intinfin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 minus 1) or 119911 = 0

We denote the left hand side of the equation above byF

1(120582) = 120582 andF

2(120582) = (120583 + 120572

1)(119877

1(120582) minus 1) where 119877

1(120582) =

(1205731119904(120583 + 120572

1)) int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 and

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(28)

IfF1(120582) = F

2(120582) has a root with Re 120582 ge 0 then Re F

1(120582) ge

0 But |F2(120582)| le (120583 + 120572

1)(119877

1minus 1) lt 0 when 119877

1lt 1 This is a

contradictionThus if both 119877

1lt 1 and 119877

2lt 1 all eigenvalues

have negative real part and the disease-free equilibrium 1198640

is locally asymptotically stable If only 1198771gt 1 then if we

consider F1(120582) = F

2(120582) for 120582 is real we see that F

2(120582) is

a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F

2(0) = (120583 + 120572

1)(119877

1minus 1) gt 0 and F

1(0) = 0

that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864

0is unstableThis concludes the

proof

In what follows we show that the diseases vanish if 1198771lt

1 1198772lt 1

Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864

0is a global attractor

that is lim119905rarrinfin

119890(119886 119905) = 0 1198681rarr 0 119868

2rarr 0 119869 rarr 0 as

119905 rarr infin

Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin

le

Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have

119890 (119886 119905) =

B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886

(29)

SinceB(119905) = 1205731119878119868

1le 120573

1Λ119868

1120583 and

1198681015840

1le 120573

1

Λ

120583int

119905

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1198651(119905) minus 120583119868

1minus 120572

11198681

(30)

where 1198651(119905) = int

infin

119905120574(119886)119890

0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890

minus120583119905119889119886 and

lim119905rarrinfin

1198651(119905) = 0 from the equation for 119868

1then we have the

following inequality

1198681le 119868

1(0) 119890

minus(120583+1205721)119905

+ 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)120591int

119905minus120591

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889120591

+ int

119905

0

119890minus(120583+120572

1)1205911198651(119905 minus 120591) 119889120591

(31)

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 3

(b) 120584 isin [0 +infin)(c) 120574(119886) 120572

0(119886) isin 119862

119861119880([0 +infin) 119877) cap 119862

+([0 +infin) 119877) and

for each 119886 ge 0 where 119862119861119880(119877) denotes the space

of bounded and uniformly continuous map from[0 +infin) into 119877

We assume (1) with the initial conditions

119878 (0) = 1198780ge 0 119890 (119886 0) = 119890

0(119886) isin 119871

1

+(0 +infin)

1198681(0) = 119868

10ge 0 119868

2(0) = 119868

20ge 0

119869 (0) = 1198690ge 0

(2)

Set

119883 = 119877 times 119884 times 1198772 119883

+= 119877

+times 119884

+times 119877

+

2

1198830= 119877 times 119884

0times 119877

2 119883

0+= 119883

0cap 119883

+

(3)

with

119884 = 119877 times 1198711(0 +infin) 119884

+= [0 +infin) times 119871

1

+(0 +infin)

1198840= 0 times 119871

1(0 +infin)

(4)

Let 119863(A) = 119877 times 119885 times 1198772 with 119885 = 0119877 times 119882

11(0infin)

for each 119909 = (119878 0119877 119890 119868

1 1198682) isin 119883

0 DefineA 119863(A) sub 119883

0sub

119883 rarr 119883 andF 1198830rarr 119883 by

A119909

=((

(

minus120583119878

minus119890 (0)

minus119889119890 (sdot)

119889119886minus (120583 + 120572

0(sdot) + 120574 (sdot)) 119890 (sdot)

minus1205831198681minus 120572

11198681

minus1205831198682minus 120572

21198682

minus (120583 + 120584) 119869

))

)

if 119909 isin 119863 (A)

(5)

F (119909)

=

((((

(

120583minus 1205731119878119868

1minus 120573

2119878119868

2+ int

infin

0

1205720(119886) 119890 (119886) 119889119886 + 120572

11198681+ 120572

21198682

1205731119878119868

1

01198711

int

infin

0

120574 (119886) 119890 (119886) 119889119886 minus 12057511986811198682

1205732119878119868

2

12057511986811198682

))))

)

(6)

Rewrite problem (1) as an abstract Cauchy problem

119889119906119909(119905)

119889119905= A119906

119909(119905) +F (119906

119909(119905)) 119905 ge 0 119906

119909(0) = 119909 isin 119883

0+

(7)

It is well known that A is a Hille-Yosida operator Moreprecisely we have (minus120583 +infin) sub 120588(A) and for each 120582 gt minus120583

10038171003817100381710038171003817(120582 minusA)

minus110038171003817100381710038171003817le1

120582 + 120583 (8)

By applying the results in Magal et al [14ndash16] we obtainthe following proposition

Lemma 1 There exists a uniquely determined semiflow119880(119905)

119905ge0on 119883

0+ such that for each 119909 = (119878

0 0 119890

0 11986810

11986820 1198690)119879isin 119883

0+ there exists a unique continuous map 119880 isin

119862([0 +infin)1198830+) which is an integrated solution of the Cauchy

problem (1) that is to say that

int

119905

0

119880 (119904) 119909 119889119904 isin 119863 (A) forall119905 ge 0

119880 (119905) 119909 = 119909 +Aint119905

0

119880 (119904) 119909 119889119904 + int

119905

0

F (119880 (119904) 119909) 119889119904 forallt ge 0

(9)

The total population size 119873(119905) is the sum of all individuals inall classes

119873 = 119878 (119905) + int

+infin

0

119890 (119886 119905) 119889119886 + 1198681(119905) + 119868

2(119905) + 119869 (119905) (10)

The total population size satisfies the equation 1198731015840(119905) = Λ minus

120583119873 minus 120584119869 We introduce the notation

120587 (119886) = exp(minusint119886

0

(1205720(120591) + 120574 (120591)) 119889120591) (11)

To understand the biological meaning of the quantity 120587(119886) wenote that 120587(119886)119890minus120583119886 is the probability to remain infected with TBtime units after infection In addition we define the quantity

119861 = int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 (12)

which gives the probability of treatment since the individualscan leave TB infectious period via treatment

119862 = int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 (13)

which gives the probability of progression since the individualscan leave TB infectious period via progression Since individu-als can only leave the latent TB infected class through treatmentprogression or death the sum of the probabilities of recoveryprogression and death equals one that is

int

infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 + int

infin

0

120574 (a) 120587 (119886) 119890minus120583119886 119889119886

+ 120583int

infin

0

120587 (119886) 119890minus120583119886119889119886 = 1

(14)

It immediately follows that 119861 + 119862 lt 1

3 Equilibria of the Model with Coinfection

We introduce the reproduction numbers of the two diseasesThe reproduction number of TB is

1198771=Λ120573

1119862

120583 (120583 + 1205721) (15)

4 Journal of Applied Mathematics

and the reproduction number of HIV is

1198772=

Λ1205732

120583 (120583 + 1205722) (16)

We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894

1 1198942 119895 as the equilibria of the modelTherefore equilibria

satisfy the following equations

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ int

+infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

12057321199041198942minus (120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(17)

The ODE in the system can be solved to result in

119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)

Substituting for 119894 in the integrals one obtains

int

+infin

0

1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 = 119890 (0) 119861

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862

(19)

With this notation the system for the equilibria becomes

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ 119890 (0) 119861 + 12057211198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

119890 (0) 119862 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

1205732119904119894minus(120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(20)

This system has three boundary equilibria

(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)

The disease-free equilibrium always exists

(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the

TB equilibrium is given by

1199041=Λ

1205831198771

119890 = 119890 (0) 120587 (119886) 119890minus120583119886

1198941=

Λ (1 minus 11198771)

(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862

119890 (0) = 120573111990411198941

(21)

Thus the equilibrium is

1198641= (119904

1 119890 119894

1 0 0) (22)

(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by

1198642= (119904

2 0 0 119894

2 0) (23)

where 1199042= 1119877

2 119894

2= 0 119890

2= 0 119894

2= (Λ120583)(1 minus

11198772) 119895

2= 0

(4) The coexistence equilibrium exists if and only if 1198771gt

1198772 and [1 minus (1119877

2+120583(120583+120572

1)Λ120575)][(Λ120573

1120583120572

11198772)(1 minus

119861) minus 1] gt 0 and it is given by

1198642= (119904 119890 119894

1 1198942 119895) (24)

where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 1198941=

(Λ1205721)(1 minus (1119877

2+120583(120583+120572

1)Λ120575))((Λ120573

1120583120572

11198772)(1 minus

119861) minus 1) 1198942= ((120583 + 120575)120575)(119877

1119877

2minus 1) 119895 = 120575119894

11198942(120583 + 120584)

Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0

4 Stability of Equilibria

In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not

To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894

1 1198942 119895 That

is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894

1 119868

2= 119906 + 119894

2 119869 = 119908 + 119895

Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and

Journal of Applied Mathematics 5

119908These satisfy the following linear eigenvalue problem (here119904 119890 119894

1 1198942 and 119895 are the corresponding equilibria)

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

120582119908 = 1205751198942119911 + 120575119894

1119906 minus (120583 + 120584)119908

(25)

In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing

41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894

1= 119894

2= 119895 = 0 and 119904 = Λ120583

Thus the system above simplifies to the following system

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(26)

From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864

0

Theorem 2 If 1198771lt 1 and 119877

2lt 1 then the disease-free

equilibrium 1198640is locally asymptotically stable If 119877

1gt 1 or

1198772gt 1 then the disease-free equilibrium 119864

0is unstable

Proof To see this from the second to last equation we have120582119906 = [120573

2119904 minus (120583 + 120572

2)]119906 where either 120582 = 120573

2119904 minus (120583 + 120572

2) or

119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877

2minus 1) lt 0 if and only

if 1198772lt 1 Thus if 119877

2gt 1 the disease-free equilibrium 119864

0

is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation

119910 (119886) = 119910 (0) 119890minus(120582+120583)119886

120587 (119886) = 1205731119904119911119890

minus(120582+120583)119886120587 (119886) (27)

Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the

characteristic equation 120582119911 = 1205731119904 int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 119911 minus

(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572

1)(120573

1119904(120583 + 120572

1))

intinfin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 minus 1) or 119911 = 0

We denote the left hand side of the equation above byF

1(120582) = 120582 andF

2(120582) = (120583 + 120572

1)(119877

1(120582) minus 1) where 119877

1(120582) =

(1205731119904(120583 + 120572

1)) int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 and

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(28)

IfF1(120582) = F

2(120582) has a root with Re 120582 ge 0 then Re F

1(120582) ge

0 But |F2(120582)| le (120583 + 120572

1)(119877

1minus 1) lt 0 when 119877

1lt 1 This is a

contradictionThus if both 119877

1lt 1 and 119877

2lt 1 all eigenvalues

have negative real part and the disease-free equilibrium 1198640

is locally asymptotically stable If only 1198771gt 1 then if we

consider F1(120582) = F

2(120582) for 120582 is real we see that F

2(120582) is

a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F

2(0) = (120583 + 120572

1)(119877

1minus 1) gt 0 and F

1(0) = 0

that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864

0is unstableThis concludes the

proof

In what follows we show that the diseases vanish if 1198771lt

1 1198772lt 1

Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864

0is a global attractor

that is lim119905rarrinfin

119890(119886 119905) = 0 1198681rarr 0 119868

2rarr 0 119869 rarr 0 as

119905 rarr infin

Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin

le

Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have

119890 (119886 119905) =

B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886

(29)

SinceB(119905) = 1205731119878119868

1le 120573

1Λ119868

1120583 and

1198681015840

1le 120573

1

Λ

120583int

119905

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1198651(119905) minus 120583119868

1minus 120572

11198681

(30)

where 1198651(119905) = int

infin

119905120574(119886)119890

0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890

minus120583119905119889119886 and

lim119905rarrinfin

1198651(119905) = 0 from the equation for 119868

1then we have the

following inequality

1198681le 119868

1(0) 119890

minus(120583+1205721)119905

+ 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)120591int

119905minus120591

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889120591

+ int

119905

0

119890minus(120583+120572

1)1205911198651(119905 minus 120591) 119889120591

(31)

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Stochastic AnalysisInternational Journal of

Page 4: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

4 Journal of Applied Mathematics

and the reproduction number of HIV is

1198772=

Λ1205732

120583 (120583 + 1205722) (16)

We note that the coinfection rate 120575 does not affect thereproduction numbers since coinfection does not lead toadditional infections Setting the derivatives with respect totime to zerowe obtain a systemof algebraic equations and oneODE for the equilibria of (1) For convenience we consider119904 119890 119894

1 1198942 119895 as the equilibria of the modelTherefore equilibria

satisfy the following equations

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ int

+infin

0

1205720(119886) 119890 (119886 119905) 119889119886 + 120572

11198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

12057321199041198942minus (120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(17)

The ODE in the system can be solved to result in

119890 (119886) = 119890 (0) 120587 (119886) 119890minus120583119886 (18)

Substituting for 119894 in the integrals one obtains

int

+infin

0

1205720(119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

1205720(119886) 120587 (119886) 119890

minus120583119886119889119886 = 119890 (0) 119861

int

+infin

0

120574 (119886) 119890 (119886) 119889119886 = 119890 (0) int

+infin

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 = 119890 (0) 119862

(19)

With this notation the system for the equilibria becomes

Λ minus 12057311199041198941minus 120573

21199041198942minus 120583119904

+ 119890 (0) 119861 + 12057211198941+ 120572

21198942= 0

119889119890

119889119886= minus120572

0(119886) 119890 minus 120583119890 minus 120574 (119886) 119890

119890 (0) = 12057311199041198941

119890 (0) 119862 minus 1205831198941minus 120575119894

11198942minus 120572

11198941= 0

1205732119904119894minus(120583 + 120572

2) 119894

2= 0

12057511989411198942minus (120583 + 120584) 119895 = 0

(20)

This system has three boundary equilibria

(1) The disease-free equilibrium 1198640= (Λ120583 0 0 0 0)

The disease-free equilibrium always exists

(2) The TB dominated equilibrium exists if and only if1198771gt 1 The steady distribution of infectives in the

TB equilibrium is given by

1199041=Λ

1205831198771

119890 = 119890 (0) 120587 (119886) 119890minus120583119886

1198941=

Λ (1 minus 11198771)

(1 minus 119861) 120583119862 + (1 minus 119861 minus 119862) 1205721119862

119890 (0) = 120573111990411198941

(21)

Thus the equilibrium is

1198641= (119904

1 119890 119894

1 0 0) (22)

(3) The HIV dominated equilibrium exists if and only if1198772gt 1 and is given by

1198642= (119904

2 0 0 119894

2 0) (23)

where 1199042= 1119877

2 119894

2= 0 119890

2= 0 119894

2= (Λ120583)(1 minus

11198772) 119895

2= 0

(4) The coexistence equilibrium exists if and only if 1198771gt

1198772 and [1 minus (1119877

2+120583(120583+120572

1)Λ120575)][(Λ120573

1120583120572

11198772)(1 minus

119861) minus 1] gt 0 and it is given by

1198642= (119904 119890 119894

1 1198942 119895) (24)

where 119904 = Λ1205831198772 119890 = 119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 1198941=

(Λ1205721)(1 minus (1119877

2+120583(120583+120572

1)Λ120575))((Λ120573

1120583120572

11198772)(1 minus

119861) minus 1) 1198942= ((120583 + 120575)120575)(119877

1119877

2minus 1) 119895 = 120575119894

11198942(120583 + 120584)

Notice that the values of the two dominance equilibria donot depend on the coinfection These exact same equilibriaare present even if 120575 = 0

4 Stability of Equilibria

In this section we investigate local and global stabilitiesof equilibria In particular we derive conditions for thestability of the disease-free equilibrium of the TB dominanceequilibrium and of the HIV dominance equilibrium Thestability of equilibria determines whether both diseasess willbe eliminated one of the diseases will be dominated and bothdiseases will persist or not

To investigate the stability of the equilibria we linearizethe model (1) In particular let 119909(119905) 119910(119886 119905) 119911(119905) 119906(119905) and119908(119905) be the perturbations respectively of 119904 119890(119886) 119894

1 1198942 119895 That

is 119878 = 119909 + 119904 119890 = 119910 + 119890 1198681= 119911 + 119894

1 119868

2= 119906 + 119894

2 119869 = 119908 + 119895

Thus the perturbations satisfy a linear system Further weconsider the eigenvalue problem for the linearized systemWe will denote the eigenvector again with 119909 119910(119886) 119911 119906 and

Journal of Applied Mathematics 5

119908These satisfy the following linear eigenvalue problem (here119904 119890 119894

1 1198942 and 119895 are the corresponding equilibria)

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

120582119908 = 1205751198942119911 + 120575119894

1119906 minus (120583 + 120584)119908

(25)

In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing

41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894

1= 119894

2= 119895 = 0 and 119904 = Λ120583

Thus the system above simplifies to the following system

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(26)

From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864

0

Theorem 2 If 1198771lt 1 and 119877

2lt 1 then the disease-free

equilibrium 1198640is locally asymptotically stable If 119877

1gt 1 or

1198772gt 1 then the disease-free equilibrium 119864

0is unstable

Proof To see this from the second to last equation we have120582119906 = [120573

2119904 minus (120583 + 120572

2)]119906 where either 120582 = 120573

2119904 minus (120583 + 120572

2) or

119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877

2minus 1) lt 0 if and only

if 1198772lt 1 Thus if 119877

2gt 1 the disease-free equilibrium 119864

0

is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation

119910 (119886) = 119910 (0) 119890minus(120582+120583)119886

120587 (119886) = 1205731119904119911119890

minus(120582+120583)119886120587 (119886) (27)

Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the

characteristic equation 120582119911 = 1205731119904 int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 119911 minus

(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572

1)(120573

1119904(120583 + 120572

1))

intinfin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 minus 1) or 119911 = 0

We denote the left hand side of the equation above byF

1(120582) = 120582 andF

2(120582) = (120583 + 120572

1)(119877

1(120582) minus 1) where 119877

1(120582) =

(1205731119904(120583 + 120572

1)) int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 and

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(28)

IfF1(120582) = F

2(120582) has a root with Re 120582 ge 0 then Re F

1(120582) ge

0 But |F2(120582)| le (120583 + 120572

1)(119877

1minus 1) lt 0 when 119877

1lt 1 This is a

contradictionThus if both 119877

1lt 1 and 119877

2lt 1 all eigenvalues

have negative real part and the disease-free equilibrium 1198640

is locally asymptotically stable If only 1198771gt 1 then if we

consider F1(120582) = F

2(120582) for 120582 is real we see that F

2(120582) is

a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F

2(0) = (120583 + 120572

1)(119877

1minus 1) gt 0 and F

1(0) = 0

that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864

0is unstableThis concludes the

proof

In what follows we show that the diseases vanish if 1198771lt

1 1198772lt 1

Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864

0is a global attractor

that is lim119905rarrinfin

119890(119886 119905) = 0 1198681rarr 0 119868

2rarr 0 119869 rarr 0 as

119905 rarr infin

Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin

le

Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have

119890 (119886 119905) =

B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886

(29)

SinceB(119905) = 1205731119878119868

1le 120573

1Λ119868

1120583 and

1198681015840

1le 120573

1

Λ

120583int

119905

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1198651(119905) minus 120583119868

1minus 120572

11198681

(30)

where 1198651(119905) = int

infin

119905120574(119886)119890

0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890

minus120583119905119889119886 and

lim119905rarrinfin

1198651(119905) = 0 from the equation for 119868

1then we have the

following inequality

1198681le 119868

1(0) 119890

minus(120583+1205721)119905

+ 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)120591int

119905minus120591

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889120591

+ int

119905

0

119890minus(120583+120572

1)1205911198651(119905 minus 120591) 119889120591

(31)

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Page 5: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 5

119908These satisfy the following linear eigenvalue problem (here119904 119890 119894

1 1198942 and 119895 are the corresponding equilibria)

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

120582119908 = 1205751198942119911 + 120575119894

1119906 minus (120583 + 120584)119908

(25)

In the following we discuss local stability of the equilib-ria through the characteristic equation (25) Since the lastequation has no relation with the other equations in (1) and(25) we just discuss the first four equations of them in thefollowing

41 Stability of the Disease-Free Equilibrium For the disease-free equilibrium we have 119890(0) = 119894

1= 119894

2= 119895 = 0 and 119904 = Λ120583

Thus the system above simplifies to the following system

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120575119894

1119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(26)

From this system we will establish the following resultregarding the local stability of the disease-free equilibrium119864

0

Theorem 2 If 1198771lt 1 and 119877

2lt 1 then the disease-free

equilibrium 1198640is locally asymptotically stable If 119877

1gt 1 or

1198772gt 1 then the disease-free equilibrium 119864

0is unstable

Proof To see this from the second to last equation we have120582119906 = [120573

2119904 minus (120583 + 120572

2)]119906 where either 120582 = 120573

2119904 minus (120583 + 120572

2) or

119906 = 0 This eigenvalue 120582 = (120583 + 1205722)(119877

2minus 1) lt 0 if and only

if 1198772lt 1 Thus if 119877

2gt 1 the disease-free equilibrium 119864

0

is unstable because this eigenvalue is positive Further fromthe second equation we have that the remaining eigenvaluessatisfying the equation

119910 (119886) = 119910 (0) 119890minus(120582+120583)119886

120587 (119886) = 1205731119904119911119890

minus(120582+120583)119886120587 (119886) (27)

Further from the fourth equation we have that the remainingeigenvalues satisfying the equation also referred to as the

characteristic equation 120582119911 = 1205731119904 int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 119911 minus

(120583 + 1205721) 119911 This eigenvalue 120582 = (120583 + 120572

1)(120573

1119904(120583 + 120572

1))

intinfin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 minus 1) or 119911 = 0

We denote the left hand side of the equation above byF

1(120582) = 120582 andF

2(120582) = (120583 + 120572

1)(119877

1(120582) minus 1) where 119877

1(120582) =

(1205731119904(120583 + 120572

1)) int

infin

0120574(119886)120587(119886)119890

minus(120582+120583)119886119889119886 and

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(28)

IfF1(120582) = F

2(120582) has a root with Re 120582 ge 0 then Re F

1(120582) ge

0 But |F2(120582)| le (120583 + 120572

1)(119877

1minus 1) lt 0 when 119877

1lt 1 This is a

contradictionThus if both 119877

1lt 1 and 119877

2lt 1 all eigenvalues

have negative real part and the disease-free equilibrium 1198640

is locally asymptotically stable If only 1198771gt 1 then if we

consider F1(120582) = F

2(120582) for 120582 is real we see that F

2(120582) is

a decreasing function of 120582 approaching zero as 120582 approachesinfinity Since F

2(0) = (120583 + 120572

1)(119877

1minus 1) gt 0 and F

1(0) = 0

that implies that there is a positive eigenvalue 120582lowast gt 0 andthe disease-free equilibrium119864

0is unstableThis concludes the

proof

In what follows we show that the diseases vanish if 1198771lt

1 1198772lt 1

Theorem 3 If 119877119894lt 1 119894 = 1 2 then 119864

0is a global attractor

that is lim119905rarrinfin

119890(119886 119905) = 0 1198681rarr 0 119868

2rarr 0 119869 rarr 0 as

119905 rarr infin

Proof Since 1198731015840(119905) = Λ minus 120583119873 minus 120584119869 le Λ minus 120583119873 then Ninfin

le

Λ120583 Hence 119878infin le Λ120583 Let B(119905) = 119890(0 119905) Integrating thisinequality along the characteristic lines we have

119890 (119886 119905) =

B (119905 minus 119886) 120587 (119886) 119890minus120583119886 119905 lt 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 ge 119886

(29)

SinceB(119905) = 1205731119878119868

1le 120573

1Λ119868

1120583 and

1198681015840

1le 120573

1

Λ

120583int

119905

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1198651(119905) minus 120583119868

1minus 120572

11198681

(30)

where 1198651(119905) = int

infin

119905120574(119886)119890

0(119886 minus 119905)(120587(119886)120587(119886 minus 119905))119890

minus120583119905119889119886 and

lim119905rarrinfin

1198651(119905) = 0 from the equation for 119868

1then we have the

following inequality

1198681le 119868

1(0) 119890

minus(120583+1205721)119905

+ 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)120591int

119905minus120591

0

120574 (119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889120591

+ int

119905

0

119890minus(120583+120572

1)1205911198651(119905 minus 120591) 119889120591

(31)

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

6 Journal of Applied Mathematics

where 120574(119886) is bounded and the integral of 1198651(119905) goes to zero

as 119905 rarr infin Consequently taking a limsup of both sides as119905 rarr infin we obtain

119868infin

1le 119877

1119868infin

1 (32)

Since 1198771lt 1 this inequality can only be satisfied if 119868infin

1rarr 0

as 119905 rarr infinSince B(119905) le 120573

1(Λ120583)119868

1 it is easy to obtain B(119905) rarr 0

as 119905 rarr infinFrom the equation 119868

2then we have the following inequal-

ity

1198681015840

2le 120573

2

Λ

1205831198682minus (120583 + 120572

2) 119868

2 (33)

Solving the inequality we get

1198682le 119868

2(0) 119890

minus(120583+1205722)119905+ 120573

2

Λ

120583int

infin

0

1198682(119905 minus 120591) 119890

minus(120583+1205722)120591119889120591 (34)

Consequently taking a limsup of both sides we obtain

119868infin

2le 119877

2119868infin

2 (35)

If 1198772lt 1 then 119868

2(119905) rarr 0 as 119905 rarr infin

From the fluctuations lemma we can choose sequence 119905119899

such that 119878(119905119899) rarr 119878

infinand 1198781015840(119905

119899) rarr 0 when 119905

119899rarr +infin It

follows from the first equation of (1) that we have 119878infinge Λ120583

Then Λ120583 le 119878infinle 119878

infinle Λ120583 Hence 119878 rarr Λ120583 when 119905 rarr

infin This completes the theorem

42 Stability of the TB Dominated Equilibrium In this sub-section we discuss stabilities of the equilibrium 119864

1and

derive conditions for domination of TB We show that theequilibrium 119864

1can lose stability and dominance of the TB

is possible in the form of sustained oscillation In this case1198942= 0 119895 = 0 119904 = Λ(120583119877

1) 119894

1= Λ(1 minus 1119877

1)((1 minus 119861)120583119862 +

(1 minus 119861 minus 119862)1205721119862) 119890(119886) = 119890(0)120587(119886)119890minus120583119886

The eigenvalue problem takes the form

120582119909 = minus 1205731119904119911 minus 120573

11198941119909 minus 120573

2119904119906 minus 120583119904

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911 + 120573

11198941119909

120582119911 = int

infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198941119906 minus 120572

1119911

120582119906 = 1205732119904119906 minus (120583 + 120572

2) 119906

(36)

From the last equation we have

120582 = (120583 + 1205722) (1198772

1198771

minus 1) or 119906 = 0 (37)

Hence 1198771gt 119877

2the partial characteristic root of (36) is

negative Substituting

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

119910 (0) = 1205731119904119911 + 120573

11198941119909 (38)

in the first and fourth equations and cancelling 119909 119911 we arriveat the following characteristic equation

(120582 + 120583 + 1205721minus 120573

1119904119862 (120582)) (120582 + 120583 + 120573

11198941(1 minus 119861 (120582)))

minus 12057311198941119862 (120582) (120572

1+ 120573

1119904 (119861 (120582) minus 1)) = 0

(39)

Substituting the formula

1 minus 119861 (120582) = 119862 (120582) + (120582 + 120583) 119864 (120582) (40)

where

119864 (120582) = int

infin

0

120587 (119886) 119890minus(120582+120583)119886 (41)

into (39) we obtain the equivalent characteristic equationwith (39) as follows(120582 + 120583 + 120572

1) (1 + 120573

11198941119864 (120582)) + 120573

11198941119862 (120582)

120583 + 1205721

=119862 (120582)

119862 (42)

It is easy to see if (42) has the roots with Re 120582 gt 0 the leftsidemode of (42) is larger than 1 while the right sidemode of(42) is less than 1 which leads to a contradiction Hence thecharacteristic roots of (42) have negative real parts then theTB dominated equilibrium119864

1is locally asymptotically stable

Therefore we are ready to establish the first result

Theorem 4 Let 1198771gt 1 and 119877

1gt 119877

2 Then the TB dominated

equilibrium 1198641is locally asymptotically stable

For all 119909 = (119878 0119877 119890 119868

1 1198682 119869) isin 119883

0 we define 119875

119878 119883

0rarr

1198771198751198681

1198751198682

1198830rarr 119877 and 119875

11986812

1198830rarr 119877 by

119875119878(119909) = 119878 119875

1198681(119909) = 119868

1 119875

1198682(119909) = 119868

2 (43)

Set119872

11986812

= 1198830+ 119872

119868120

= 119909 isin 11987211986812

| 1198751198681

119909 = 0 1198751198682

119909 = 0

120597119872119868120

=

11987211986812

119872119868120

1198721198681

= 120597119872119868120

11987211986810

= 119909 isin 1198721198681

| 1198751198681

119909 = 0 1198751198682

x = 0 12059711987211986810

=

1198721198681

11987211986810

1198721198682

= 12059711987211986810

11987211986820

= 119909 isin 1198721198682

| 1198751198682

119909 = 0 1198751198681

119909 = 0

12059711987211986820

=

1198721198682

11987211986820

119872119878= 120597119872

11986820

(44)

(see Figure 2)

Lemma 5 If 1198771gt 1 lim inf 119868

1(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 119890 0

119877 0

119877) isin 119872

11986810

We assume there is a1199050gt 0 such that 119868

1lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986810

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

1119878 minus 120583119878

119878 (0) = 1198780

(45)

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 7

11987211986820

119872119868120

11987211986810

119872119878

Figure 2 Zoning the area

We solve them as follows

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

1120576)119905)

120583≐ 119878

lowast(119905) (46)

And 119878lowast(119905) rarr Λ120583 as 119905 rarr +infin Since 1198771gt 1 there

exists 120575 gt 0 and 1198791gt 0 which satisfy 120573

1(Λ120583)

int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 gt 1 From the second

and third equations of (1) and integrating them from thecharacteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) =

1205731119878 (119905 minus 119886) 119868

1(119905 minus 119886) 120587 (119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905 119905 lt 119886

(47)

From the fourth equation of (1) we get

1198681015840

1(119905) = 120573

1int

119905

0

120574 (119886) 119878 (119905 minus 119886) 1198681(119905 minus 119886) 120587 (119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

120587 (119886)

120587 (119886 minus 119905)119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(48)

Solving it we have

1198681(119905) ge 120573

1

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(49)

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(50)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 119905

0

(51)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868

1(119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(52)

Assuming that 1199052lt infin it follows that

1198681(1199052) ge 120573

1

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587 (119886) 119890

minus120583119886119889119886 119889119904

ge 1205731

Λ

120583int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 120587 (119886) 119890minus120583119886119889119886 119889119904 120585

(53)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it

follows that there exists an 1205761gt 0 such that 119868

1(119905) ge 120585

for all 119905 isin [1199052 1199052+ 120576

1] which contradicts the definition of

1199052 Therefore 119868

1(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin=

lim inf119905rarr+infin

1198681(119905) ge 120585 gt 0 Using (51) it follows that

1198681infinge 119868

1infin1205731(Λ120583) int

120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)120587(119886)119890

minus120583119886119889119886 119889119904 which

is impossible Thus 1198681(119905) ge 120576

Theorem 6 If 1198771gt 1 system (1) is permanent in 119872

11986810

Moreover there exists 119860

11986810

a compact subset of 11987211986810

which isa global attractor for 119880(119905)

119905ge0in119872

11986810

Proof Suppose that 119909 = (119878(119905) 0 119890 1198681 0 0) isin 119872

11986810

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt 120583 minus (120573

1+ 120583) 119878 (119905) (54)

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

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Page 8: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

8 Journal of Applied Mathematics

Consider the comparison equation

1199061015840(119905) = 120583 minus (120573

1+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(55)

Similarly (55) exists as a positive unique steady state So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

1 for 119905 large enough where 119906lowast =

120583(120583 + 1205731) From the second and third equations of (1) and

Volterrarsquos formulation we have 119890(119886 119905) ge 12057311198981120587(119886)119890

minus120583119886≐ 119898

2

for 119905 large enough

43 Stability of the HIV Dominated Equilibrium In this sub-section we establish that the equilibrium 119864

2is locally stable

whenever it exists In this case 119890(0) = 0 1198941= 0 119895 = 0 119904 =

Λ1205831198772 1198942= (Λ120583)(1 minus 1119877

2) The linear eigenvalue problem

becomes

120582119909 = minus 1205731119904119911 minus 120573

2119904119906 minus 120573

21198942119909 minus 120583119909

+ int

infin

0

1205720(119886) 119910 (119886) 119889119886 + 120572

1119911 + 120572

2119906

1199101015840(119886) = minus 120582119910 minus 120572

0(119886) 119910 minus 120583119910 minus 120574 (119886) 119910

119910 (0) = 1205731119904119911

120582119911 = int

+infin

0

120574 (119886) 119910 (119886) 119889119886 minus 120583119911 minus 1205751198942119911 minus 120572

1119911

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(56)

From the equation for 119910(119886) we have

119910 (119886) = 119910 (0) 120587 (119886) 119890minus(120582+120583)119886

(57)

Substituting 119910(119886) in the equation for the initial condition119910(0) and assuming that 119910(0) = 0 we obtain the followingcharacteristic equation

120582 + 1205751198942= 120573

1119904 int

infin

0

120574 (119886) 120587 (119886) 119890minus(120582+120583)119886

119889119886 minus (120583 + 1205721)

= (120583 + 1205721) (1198771(120582)

1198772

minus 1)

(58)

Denoting the left hand side of the equation above byF1(120582) =

120582+1205751198942 andF

2(120582) = (120583+120572

1)((119877

1(120582)119877

2)minus1) and also noting

that

1198771015840

1(120582) lt 0 lim

119905rarr+infin

1198771(120582) = 0 lim

119905rarrminusinfin

1198771(120582) = +infin

(59)

it is easy to see that F1(120582) is an increasing function with 120582

Note thatF1(0) = 120575119868

2ge 0 ButF

2(0) le (120583 + 120572

1)((119877

1119877

2) minus

1) lt 0 when 1198771lt 119877

2 Therefore the characteristic roots of

this equation have only negative real part Furthermore for119910(0) = 119910(119886) = 0 and 119911 = 0

120582119909 = minus 1205732119904119906 minus 120573

21198942119909 minus 120583119909 + 120572

2119906

120582119906 = 1205732119904119906 + 120573

21198942119909 minus (120583 + 120572

2) 119906

(60)

We express 119909 from the first equation

119909 =(minus120573

2119904 + 120572

2) 119906

120582 + 120583 + 12057321198942

(61)

and substitute it into the second equation In additionassuming that 119906 is nonzero we cancel it and obtain thefollowing characteristic equation

120582 + 120583 + 1205722= 120573

2119904 +12057321198942(minus120573

2119904 + 120572

2)

120582 + 120583 + 12057321198942

(62)

that is

(120582 + 120583) (120582 + 120583 + 1205722+ 120573

21198942minus 120573

2119904) = 0 (63)

Noticing that 1205732119904 = 120583 + 120572

2 we obtain the following

eigenvalues 120582 = minus120583 and 120582 = minus12057321198942 which are both negative

Consequently the equilibrium 1198642is locally asymptotically

stableTherefore we have the following theorem for equilibrium

1198642

Theorem 7 Let 1198772gt 1 Assume that tuberculosis cannot

invade the equilibrium of HIV that is 1198771lt 119877

2 Then the

equilibrium 1198642is locally asymptotically stable

Lemma 8 If 1198772gt 1 then lim inf 119868

2(119905) ge 120576 in119872

11986820

Proof Let 119909 = (1198780 0

119877 0

1

119871 0

119877 1198682 0

119877) isin 119872

11986820

We assume thereis a 119905

0gt 0 such that 119868

2lt 120576 for all 119905 ge 119905

0 From the first equation

of (1) in11987211986820

it is easy to get

119889119878

119889119905ge Λ minus 120576120573

2119878 minus 120583119878

119878 (0) = 1198780

(64)

We solve them

119878 (119905) ge

Λ (1 minus 119890minus(120583+120573

2120576)119905)

120583≐ 119878

lowast(119905) (65)

Then 119878(119905) ge 119878lowast(119905) and 119878(119905) rarr Λ120583 as 119905 rarr +infin From thefourth of equation (1) exist119905

0gt 0 we have

1198681015840

2(119905) ge 120573

2

Λ

1205831198682(119905) minus (120583 + 120572

2) 119868

2(119905) forall119905 ge 119905

0 (66)

Solving it we have

1198682(119905) ge 120573

2

Λ

120583int

119905

0

119890minus(120583+120572

2)(119905minus119904)1198682(119905 minus 119904) 119889119904 (67)

We do lim inf on both side of (67) and denote 1198682infin=

lim inf119905rarr+infin

1198682(119905) gt 0 thus

1198682infinge 119877

21198682infin (68)

which is impossible Thus 1198682(119905) ge 120576

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 9

Theorem 9 If 1198772gt 1 system (1) is permanent in 119872

11986820

Moreover there exists 119860

11986820

a compact subset of 11987211986820

which isa global attractor for 119880(119905)

119905ge0in119872

11986820

Proof Suppose that 119909 = (119878(119905) 0 0 1198682(119905) 0) isin 119872

11986820

be anysolution of (1) From the first equation of (1) we have

1198781015840(119905) gt Λ minus (120573

2+ 120583) 119878 (119905) (69)

Consider the comparison equation

1199061015840(119905) = Λ minus (120573

2+ 120583) 119906 (119905)

119906 (0) = 119878 (0)

(70)

Similarly (70) exists as a positive unique solution So wehave 119878(119905) ge 119906lowast minus 120576

1≐ 119898

2 for 119905 large enough where 119906lowast =

Λ(1205732+ 120583)

44 Stability of Coexistence Equilibrium 119864lowast In this subsec-

tion we establish that the equilibrium 119864lowastis locally sta-

ble whenever it exists In this case 119904 = Λ1205831198772 119890 =

119890(0)120587(119886)119890minus120583119886 119890(0) = 120573

11199041198941 119894

1= (Λ120572

1)((1 minus (1119877

2+ 120583(120583 +

1205721)Λ120575))((Λ120573

1120583120572

11198772)(1minus119861)minus1)) 119894

2= ((120583+120575)120575)(119877

1119877

2minus

1) 119895 = 12057511989411198942(120583 + 120584) The characteristic equations at the

coexistence equilibrium are as follows

(120582 + 120583 + 1205721+ 120575119894

2minus 120573

1119904119862 (120582))

times (120582 + 12057311198941+ 120573

21198942+ 120583 minus 120573

11198941119861 (120582) +

12057321198942120583

120582)

= (1205731119904119861 (120582) minus 120573

1119904) (120573

11198941minus120575120573

211989411198942

120582)

(71)

Theorem 10 If 1198771gt 119877

2 [1 minus (1119877

2+ (120583(120583 + 120572

1)

Λ120575))][((1205821205731120583120572

11198772)(1 minus 119861)) minus 1] gt 0 and the characteristic

equation (71) has only negative real parts roots then thecoexistence equilibrium 119864

lowastis locally asymptotically stable

5 Persistence of the System

In this section we consider persistence of the system in119872

119868120

when 120584 = 0 It is easy to check if the system (1)is dissipative and the dynamical system is asymptoticallysmooth 119872

119868120

11987211986810

11987211986820

and 119872119904are positively invariant

Define

120597Γ = 11987211986810

cup11987211986820

cup119872119904

120597Γ = 119872119868120

(72)

Assuming the boundary equilibria of (1) are globally asymp-totically stable we have

119860120597119872119868120

= cup(11987801198900(sdot)11986811198682119869)isin120597Γ120596 ((119878

0 1198900(sdot) 119868

1 1198682 119869))

= 1198640 119864

1 119864

2

(73)

By the above conclusions it follows that 119860120597119872119868120

is isolatedand has an acyclic covering119872 = 119864

0 119864

1 119864

2 Since the orbit

of any bounded set is bounded and from Theorem 42 in[17] we only need to show that 119864

0 119864

1 119864

2are ejective in119872

119868120

if globally stable conditions of 1198640 119864

1 119864

2are not satisfied

Therefore we have the following lemmas

Lemma 11 Let Assumption 1 be satisfied and let 1198640be globally

asymptotically stable If 1198771gt 1 then 119864

0is ejective in119872

119868120

for119880

11986812

(119905)119905ge0

Proof Let 120575 gt 0 1198791gt 0 and 120576 isin (0 1) satisfy

1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 gt 1 (74)

Let 119909 = (1198780 0

119877 11986810 11986820 1198690) isin 119872

119868120

with 119909 minus 119909119878 lt 120576 Assume

that

1003817100381710038171003817119880 (119905) 119909 minus 1199091199041003817100381710038171003817 le 120576 119905 ge 0 (75)

Defining 119878(119905) = 119875119878119880(119905)119909 and 119868

1(119905) = 119875

1198681

119880(119905)119909 for all 119905 ge 0from (75) it follows that

119878 (119905) geΛ

120583minus 120576 forall119905 ge 0 (76)

From the second and third equations of (1) and integratingthem from the characteristic line 119905 minus 119886 = 119888 we obtain

119890 (119886 119905) ge

1205731(Λ

120583minus 120576) 119868

1(119905 minus 119886) 120587

1(119886) 119890

minus120583119886 119905 ge 119886

1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905 119905 lt 119886

(77)

where from the fourth equation of (1) we get

1198681015840

1(119905) ge 120573

1

Λ

120583int

119905

0

1198681(119905 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886

+ 1205731int

infin

119905

120574 (119886) 1198900(119886 minus 119905)

1205871(119886)

1205871(119886 minus 119905)

119890minus120583119905119889119886

minus (120583 + 1205721) 119868

1

(78)

Solving it we have

1198681(119905) ge 120573

1(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

119905minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(79)

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

10 Journal of Applied Mathematics

There exist a 1198791gt 0 such that

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

119905+1198791

0

119890minus(120583+120572

1)119904

times int

119905+1198791minus119904

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

119905

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(80)

Thus for 119905 ge 120575 we have

1198681(119905 + 119879

1)

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904 forall119905 ge 0

(81)

By Assumption 1 there exists 1199051ge 0 such that 119868

1(119905+119879

1) ge

0 for all 119905 + 1198791ge 119905

1 Hence there exists 120585 gt 0 such that

1198681(119905 + 119879

1) ge 120585 for all 119905 + 119879

1isin [2119905

1 2119905

1+ 120575] Set

1199052= sup 119905 + 119879

1ge 2119905

1+ 120575 119868 (119897) ge 120585 forall119897 isin [2119905

1+ 120575 119905]

(82)

Assume that 1199052lt infin Then

1198681(1199052) ge 120573

1(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1198681(119905 minus 119904 minus 119886) 120587

1(119886) 119890

minus120583119886119889119886 119889119904

ge 1205731(Λ

120583minus 120576)int

120575

0

119890minus(120583+120572

1)119904

times int

1198791

0

120574 (119886) 1205871(119886) 119890

minus120583119886119889119886 119889119904 120585

(83)

Thus 1198681(1199052) gt 120585 By the continuity for 119905 rarr 119868

1(119905) it follows

that there exists an 1205762gt 0 such that 119868

1(119905) ge 120585 for all 119905 isin

[1199052 1199052+ 120576

2] which contradicts the definition of 119905

2 Therefore

1198681(119905) ge 120585 for all 119905 gt 2119905

1 Denote 119868

1infin= lim inf

119905rarr+infin1198681(119905) ge

120585 gt 0 Using (81) it follows that 1198681infinge 119868

1infin1205731(Λ120583 minus

120576) int120575

0119890minus(120583+120572

1)119904int1198791

0120574(119886)119868

1(119905 minus 119904 minus 119886)120587

1(119886)119890

minus120583119886119889119886 119889119904 which is

impossible

Lemma 12 Let Assumption 1 be satisfied and let 1198641and 119864

2be

globally asymptotically stable Then one has the following

(i) If 1198771gt 1 then 119909

1198681

= 1198641is ejective in119872

119868120

(ii) If 119877

2gt 1 then 119909

1198682

= 1198642is ejective in119872

119868120

Theorem 13 Let Assumption 1 be satisfied and let 1198640 119864

1 and

1198642be globally asymptotically stable Assume 119877

1gt 1 119877

2gt 1

Then there exists 120576 gt 0 such that for all 119909 isin 119872119868120

lim inf 1003817100381710038171003817100381711987511986812119880 (119905) 11990910038171003817100381710038171003817ge 120576 (84)

Proof It is a consequence ofTheorem 42 in [18] applied withΩ(120597119872

119868120

) = 1198640 cup 119864

1 cup 119864

2 Using Lemma 12 the result

follows

6 Simulation

In this section we use (1) to examine how the prevalenceof HIV impacts on TB dynamics We also present somenumerical results on the stability of 119864

0(the disease-free

equilibrium) 1198641(the TB dominated equilibrium) and 119864

2

(the HIV dominated equilibrium) We perform a numericalanalysis to exhibit the TB impact on HIV under differenttreatments with (1) We now give three examples to illustratethe main results mentioned in the above section

Example 14 In (1) we set Λ = 2 120583 = 06 120575 = 7313 1205731=

049 1205732= 015 120584 = 09

120574 (119886) = 002 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(85)

1205721= 001 120572

2= 003 We have 119877

1= 007438 lt 1 and

1198772= 079365 lt 1 which satisfy the conditions of Theorems

2 and 3 1198640should be globally asymptotically stable (see

Figure 3(a)) In this case both TB andHIVwill be eliminated

Example 15 In (1) Λ = 2 120583 = 1 120575 = 4 1205731= 5049 120573

2=

015 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(86)

1205721= 001 120572

2= 003 We have 119877

1= 3448770 gt 1 gt

1198772= 079365 which satisfy the conditions of Theorem 4 119864

1

should be globally asymptotically stable (see Figure 3(b)) Inthis case TB is dominated in the coinfected dynamics

Example 16 In (1) Λ = 2 120583 = 06 120575 = 00073 1205731= 049

1205732= 415 120584 = 09

120574 (119886) = 01 119886 ge 149

0 0 le 119886 le 149

1205720(119886) =

01 119886 ge 149

0 0 le 119886 le 149

(87)

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 11

35

3

25

2

15

1

05

0

minus05

Case

s

0 5 10 15 20 25 30 35 40 45 50

119878

1198681

1198682

119864

119869

Time 119905

(a)

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

119864

119869

07

06

05

04

03

02

01

0

Case

s

Time 119905

(b)

Figure 3 (a) show that 1198640is globally asymptotically stable (b) shows that 119864

1is globally asymptotically stable

35

3

25

2

15

1

05

0

Case

s

0 5 10 15 20 25 30 35 40 45 50

1198781198681

1198682

1198682

119869

Time 119905

Figure 4 shows that 1198642is globally asymptotically stable

1205721= 001 120572

2= 003 We have 119877

2= 1137566 gt 1 gt

1198771= 033470 which satisfies the conditions ofTheorem 7119864

2

should be globally asymptotically stable (see left of Figure 4)In this case HIV is dominated in the coinfected dynamics

61 Effect of Treatment Parameter 120572119894 119894=0 12 We examined

hypothetical treatment to gain insight into the underlying co-epidemic dynamicsWe considered nine treatment scenariosno treatment 120572

0= 0 (latent TB treatment) (IPT) 120572

0= 05

(latent TB treatment) (IPT) 1205720= 1 (latent TB treatment)

(IPT) no treatment 1205722= 0 120572

2= 05 (HAART) and 120572

2= 1

(HAART) where 120572119894 119894 = 0 1 denotes the fraction of the

eligible population receiving the corresponding treatmentFirst we considered the effect of each type of treatment

in isolation (eg IPT for individuals with treatment for latentor active TB but not AIDS) Exclusively treating peoplewith latent TB reduced the number of new active TB caseswhich subsequently decreased the number of new latent TBinfections (Figure 5(a)) However latent TB treatment hadan adverse effect on the HIV epidemic the number of new

HIV cases increased because individuals coinfectedwithHIVand latent TB lived longer (due to latent TB treatment) andthus could infect more people with HIV(see Figure 5(c))Similarly active TB treatment reduced the number of peoplewith infectious TB which subsequently reduced the numberof new latent and active TB cases (Figure 5(b)) Once againnewHIV cases increased due to longer life expectancy amongthose who were coinfected with HIV Hence the coinfectedpeople have same effect of people infected by latent TB andactive TB

Second we considered the effect of each type of treatmentin isolation (eg HAART for individuals with AIDS butno treatment for latent or active TB) The provision ofHAART significantly slowed the HIV epidemic (Figure 6(c))However exclusively treating HIV-infected individuals withHAART adversely affects the TB epidemic (Figures 6(a)and 6(b)) Because HAART reduces AIDS-related mortalitytreated individuals have a longer time to potentially infectothers with TB especially in the absence of any TB treatmentThus the numbers of new latent and active TB cases increasedas more people were given HAART However the coinfectedpeople have the same effect of people infected by HIV (seeFigure 6(d))

7 Discussion

We have developed a mathematical model for modelling thecoepidemics calculated the basic reproduction number thedisease-free equilibrium the dominated equilibria (definedas a state where one disease is eradicated while the otherdisease remains endemic) and got the conditions for localand global stability We presented the sufficient conditionsfor the local stability of the TB dominated equilibrium inTheorem 4 We also obtained the persistence In the case ofthe HIV dominated equilibrium we presented the sufficientcondition for the local stability in Theorem 7 We alsoobtained the persistence of (1) We simulated and illustratedour analyzed results in Section 6

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

12 Journal of Applied Mathematics

Different latent TB treatment

1205720 = 11205720 = 051205720 = 0

0 1 2 3 4 5 6 7 8 9 10

0807060504030201

0

119864

Time 119905

(a)

0275025

022502

0175015

012501

0075005

00250

1198681

Different latent treatment

1205720 = 11205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

09

030201

0807

0506

04

0

1198682

Different latent treatment

1205720 = 1

1205720 = 05

1205720 = 0

0 1 2 3 4 5 6 7 8 9 10Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205720 = 1 1205720 = 0

1205720 = 05

Different latent treatment

Time 119905

(d)

Figure 5 We take different latent TB treatments while we take no treatment of HAART and active TB

1090807060504030201

0

119864

0 1 2 3 4 5 6 7 8 9 10

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

Time 119905

(a)

Different HAART

1205722 = 0

1205722 = 1

1205722 = 05

04035

03025

02015

01005

0

1198681

0 1 2 3 4 5 6 7 8 9 10Time 119905

(b)

Different HAART1205722 = 0

1205722 = 1

1205722 = 05

09

030201

0807

0506

04

00 1 2 3 4 5 6 7 8 9 10

1198682

Time 119905

(c)

119869

0275025

022502

0175015

012501

0075005

00250

0 1 2 3 4 5 6 7 8 9 10

1205722 = 0

1205722 = 11205722 = 05

Different HAART

Time 119905

(d)

Figure 6 We take different HAART while we take no treatment of IPT

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Journal of Applied Mathematics 13

Our models have several limitations We simplified thecomplicated infection dynamics of TB and HIV to developa tractable framework which helped us gain insights aboutthe basic reproduction number and equilibria We assumeduniform patterns within compartments that is the mixingbetween compartments is homogeneous To appropriatelyguide policy recommendations our coepidemicmodelwouldneed to be significantly expandedWe also apply a coepidemicmodel to describe the HIV coinfected with other diseasessuch as HIV and hepatitis C (HCV) Some 50ndash90 of HIV-infected injection drug users are coinfected with HCV [19]The successful HIV treatment may be adversely impactedby the presence of HCV and HCV may cause liver damageto occur more quickly in HIV-infected individuals [19]Modelling this kind of coinfection may play particularlyimportant role on evaluating interventions time targeted toinjection drug

Unfortunately we are unable yet to study the models thatintroduce a discrete delay and an impulsive perturbation inour model We will explore them in the future

Acknowledgments

This paper is supported by the NSF of China (6120322811071283) Mathematical Tianyuan Foundation (11226258)the Young Sciences Foundation of Shanxi (2011021001-1) andthe Foundation of University (YQ-2011045 JY-2011036)

References

[1] National Institute of Allergy and Infectious Diseases (NIAID)HIV Infection and AIDS An Overview United States NationalInstitutes of Health Bethesda Md USA 2006

[2] G D Sanders A M Bayoumi V Sundaram et al ldquoCost-effectiveness of screening for HIV in the era of highly activeantiretroviral therapyrdquo New England Journal of Medicine vol352 no 6 pp 570ndash585 2005

[3] World Health Organization (WHO) Antiretroviral Therapy(ART) WHO Geneva The Switzerland 2006

[4] D Kirschner ldquoDynamics of co-infection with M tuberculosisandHIV-1rdquoTheoretical Population Biology vol 55 no 1 pp 94ndash109 1999

[5] S M Moghadas and A B Gumel ldquoAn epidemic model for thetransmission dynamics of hiv and another pathogenrdquoANZIAMJournal vol 45 no 2 pp 181ndash193 2003

[6] B G Williams and C Dye ldquoAntiretroviral drugs for tuberculo-sis control in the era of HIVAIDSrdquo Science vol 301 no 5639pp 1535ndash1537 2003

[7] T C Porco P M Small and S M Blower ldquoAmplificationdynamics predicting the effect of HIV on tuberculosis out-breaksrdquo Journal of Acquired Immune Deficiency Syndromes vol28 no 5 pp 437ndash444 2001

[8] R W West and J R Thompson ldquoModeling the impact of HIVon the spread of tuberculosis in theUnited StatesrdquoMathematicalBiosciences vol 143 no 1 pp 35ndash60 1997

[9] R Naresh and A Tripathi ldquoModelling and analysis of HIV-TBco-infection in a variable size Prdquo Mathematical Modelling andAnalysis vol 10 no 3 pp 275ndash286 2005

[10] R Naresh D Sharma and A Tripathi ldquoModelling the effectof tuberculosis on the spread of HIV infection in a population

with density-dependent birth anddeath raterdquoMathematical andComputer Modelling vol 50 no 7-8 pp 1154ndash1166 2009

[11] E F Long N K Vaidya and M L Brandeau ldquoControlling Co-epidemics analysis of HIV and tuberculosis infection dynam-icsrdquo Operations Research vol 56 no 6 pp 1366ndash1381 2008

[12] M Martcheva and S S Pilyugin ldquoThe role of coinfection inmultidisease dynamicsrdquo SIAM Journal on Applied Mathematicsvol 66 no 3 pp 843ndash872 2006

[13] G J Butler and P Waltman ldquoBifurcation from a limit cycle ina two predator-one prey ecosystem modeled on a chemostatrdquoJournal of Mathematical Biology vol 12 no 3 pp 295ndash310 1981

[14] P Magal C C McCluskey and G F Webb ldquoLyapunovfunctional and global asymptotic stability for an infection-agemodelrdquo Applicable Analysis vol 89 no 7 pp 1109ndash1140 2010

[15] EMCDrsquoAgata PMagal and SG Ruan ldquoAsymptotic behaviorin nosocomial epidemic models with antibiotic resistancerdquoDifferential and Integral Equations vol 19 no 5 pp 573ndash6002006

[16] M Iannelli ldquoMathematical theory of age-structured populationdynamicsrdquo in Applied Mathematics Monographs 7 comitatoNazionale per le Scienze Matematiche Consiglio Nazionaledelle Ricerche (CNR) Pisa Italy 1995

[17] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo SIAM Journal onMathematical Analysis vol 24 no 2 pp 407ndash435 1993

[18] J K Hale and P Waltman ldquoPersistence in finite dimensionalsystemsrdquo SIAM Journal onMathematical Analysis vol 20 no 2pp 388ndash395 1989

[19] Centers for Disease Control and Prevention (CDC)CoinfectionWith HIV and Hepatitis C Virus CDC Atlanta Ga USA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Dynamic of a TB-HIV Coinfection …downloads.hindawi.com/journals/jam/2013/429567.pdfDynamic of a TB-HIV Coinfection Epidemic Model with Latent Age XiaoyanWang,JunyuanYang,andFengqinZhang

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


HIV / Hepatitis C Coinfection : From guesswork to Guidelines

HIV / Hepatitis C Coinfection : From guesswork to Guidelines

HIV and HCV coinfection - Barriers in Central and Eastern ...

HIV and HCV coinfection - Barriers in Central and Eastern ...

MILANO post CROI - simit.org Carosi, Massimo Galli ... Discussants: Raffaele Bruno, Massimo Puoti • HIV/TB Coinfection: reducing the burden of TB in people living with HIV

MILANO post CROI - simit.org Carosi, Massimo Galli ... Discussants: Raffaele Bruno, Massimo Puoti • HIV/TB Coinfection: reducing the burden of TB in people living with HIV

HCV-HIV Coinfection: Pathogenesis of Accelerated Liver ...regist2.virology-education.com › 2015 › 11COINFECTION › 15_Chung.… · HCV-HIV Coinfection: Pathogenesis of Accelerated

HCV-HIV Coinfection: Pathogenesis of Accelerated Liver ...regist2.virology-education.com › 2015 › 11COINFECTION › 15_Chung.… · HCV-HIV Coinfection: Pathogenesis of Accelerated

HIV and HCV Coinfection

HIV and HCV Coinfection

Ledipasvir-Sofosbuvir in GT-1 and HIV Coinfection ERADICATE Trial

Ledipasvir-Sofosbuvir in GT-1 and HIV Coinfection ERADICATE Trial

HCV and HIV Coinfection - Dr. Cave

HCV and HIV Coinfection - Dr. Cave

Recent Advances in Multidrug-Resistant TB of HIV/TB coinfection.2013

Recent Advances in Multidrug-Resistant TB of HIV/TB coinfection.2013

Solution of HIV and Malaria Coinfection Model Using Msgdtm

Solution of HIV and Malaria Coinfection Model Using Msgdtm

TB/HIV - brocher.ch · TB/HIV Distinct Histories, Entangled Futures Towards an Epistemology of Coinfection 27 / 28 February 2014, Fondation Brocher, Geneva Symposium Organized by

TB/HIV - brocher.ch · TB/HIV Distinct Histories, Entangled Futures Towards an Epistemology of Coinfection 27 / 28 February 2014, Fondation Brocher, Geneva Symposium Organized by

Ooo1tuberculosis-Hiv Coinfection Auto Saved]

Ooo1tuberculosis-Hiv Coinfection Auto Saved]

Chastain HCV-HIV Coinfection Webinar 10-26-16€¦ · § HIV/HCV coinfection is a significant cause of morbidity and mortality in people infected with HIV. § HCV should be treated

Chastain HCV-HIV Coinfection Webinar 10-26-16€¦ · § HIV/HCV coinfection is a significant cause of morbidity and mortality in people infected with HIV. § HCV should be treated

TUBERCULOSIS WITH HIV CO-INFECTION Chest, …. TB and HIV Coinfection. March...Miliary TB. Chest X-ray in a child demonstrating the classic miliary infiltrates (nodules < 1mm), one

TUBERCULOSIS WITH HIV CO-INFECTION Chest, …. TB and HIV Coinfection. March...Miliary TB. Chest X-ray in a child demonstrating the classic miliary infiltrates (nodules < 1mm), one

Management of Tuberculosis and HIV Coinfection

Management of Tuberculosis and HIV Coinfection

Challenges of HIV-TB Coinfection Sinata Koulla Shiro (MD) ANRS Satellite Meeting, IAC 2012 Washington DC, 23 July 2012.

Challenges of HIV-TB Coinfection Sinata Koulla Shiro (MD) ANRS Satellite Meeting, IAC 2012 Washington DC, 23 July 2012.

HiV and Hepatitis coinfection in Greece

HiV and Hepatitis coinfection in Greece

The effect of HIV coinfection, HAART and TB treatment on ... · IGRA have low sensitivity and specificity especially in malnutri-tioned and HIV infected individuals [8]. The fact

The effect of HIV coinfection, HAART and TB treatment on ... · IGRA have low sensitivity and specificity especially in malnutri-tioned and HIV infected individuals [8]. The fact

Care of the Treatment-naive Patient With HCV/HIV Coinfection

Care of the Treatment-naive Patient With HCV/HIV Coinfection

TB and HIV - Curry International Tuberculosis Center Coinfection Page 3 of 20 This image cannot currently be displayed. TB Diagnosis Symptoms • Prolonged cough, hemoptysis, fevers,

TB and HIV - Curry International Tuberculosis Center Coinfection Page 3 of 20 This image cannot currently be displayed. TB Diagnosis Symptoms • Prolonged cough, hemoptysis, fevers,

14 niaid-eradicate-hiv-coinfection

14 niaid-eradicate-hiv-coinfection