Research Article Mathematical Model of MDR-TB and XDR-TB...
Transcript of Research Article Mathematical Model of MDR-TB and XDR-TB...
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Research ArticleMathematical Model of MDR-TB and XDR-TB with Isolationand Lost to Follow-Up
F. B. Agusto, J. Cook, P. D. Shelton, and M. G. Wickers
Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA
Correspondence should be addressed to F. B. Agusto; [email protected]
Received 24 February 2015; Accepted 31 May 2015
Academic Editor: Juan-Carlos Corteฬs
Copyright ยฉ 2015 F. B. Agusto et al. This 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.
We present a deterministic model with isolation and lost to follow-up for the transmission dynamics of three strains ofMycobacterium tuberculosis (TB), namely, the drug sensitive, multi-drug-resistant (MDR), and extensively-drug-resistant (XDR)TB strains. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoreticaland epidemiological findings indicate that the model has locally asymptotically stable (LAS) disease-free equilibrium when theassociated reproduction number is less than unity. Furthermore, the model undergoes in the presence of disease reinfection thephenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemicequilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection. The result of the global sensitivityanalysis indicates that the dominant parameters are the disease progression rate, the recovery rate, the infectivity parameter, theisolation rate, the rate of lost to follow-up, and fraction of fast progression rates. Our results also show that increase in isolation rateleads to a decrease in the total number of individuals who are lost to follow-up.
1. Introduction
Mycobacterium tuberculosis (TB) is caused by bacteria thatare transmitted from person to person through the air by aninfected personโs coughing, sneezing, speaking, or singing [1].TB usually affects the lungs, but it can also affect other parts ofthe body, such as the brain, the kidneys, or the spine [1]. TheTB bacteria can stay in the air for several hours, depending onthe environment. In 2013, 9 million people were ill with TB,and 1.5millionmortalities occurred from the disease [2]; over95% of deaths occurred in low- andmiddle-income countries[2]. About one-third of the worldโs population has latent TB[2]. TB is also among the top three causes of death in womenaged 15 to 44 [2]. Tuberculosis is second only to HIV/AIDSas the greatest killer worldwide due to a single infectiousagent [2]. Those who have a compromised immune system,like those who are living with HIV, malnutrition, or diabetes,or people who use tobacco products, have a much higherrisk of falling ill. Individuals who develop TB are providedwith a six-month course of four antimicrobial drugs alongwith supervision and support by a health worker. Improper
treatment compliance or use of poor quality medicines canall lead to the development of drug-resistant tuberculosis [2].
Multi-drug-resistant (MDR) TB is a form of TB causedby bacteria that do not respond to, at least, isoniazid andrifampicin, which are the two most powerful, standard anti-tuberculosis drugs.MDR-TB is treatable and curable by usingsecond-line drugs [3]. However, these treatment options arelimited and recommendedmedicines are not always available[3]. In some cases, more severe drug resistance can develop.Extensively-drug-resistant (XDR)TB is a formofmulti-drug-resistant TB that responds to even fewer available medicines,including the second-line drugs [3]. In 2013, there were about480,000 cases of MDR-TB present in the world [4]; it wasestimated that about 9%-10% of these cases were XDR-TB[2, 4โ6].
The increase in drug-resistant TB strains has called for anincreased urgency for isolating individuals infected with suchstrains of TB [7]. High priority is being placed on identifyingand curing these individuals. With proper identification andtreatment, about 40% of XDR-TB cases could potentially becured [6, 8]. However, only 10% of MDR-TB cases are ever
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 828461, 21 pageshttp://dx.doi.org/10.1155/2015/828461
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2 Abstract and Applied Analysis
identified, leaving the potential development of XDR-TB tobecome more prevalent worldwide [2]. The need for a mod-ern and effective approach to curtail the rise of drug-resistantTB strains is being sought after, one of which is constructedisolation, whether it is an at-home isolation or isolation ata medical facility [6]. An event that portrays the need forcarefully constructed isolation was the identification of anindividual infected with MDR-TB in Atlanta InternationalAirport in 2007 [9]. The individual had flown to Atlantaafter visiting Paris, Greece, Italy, CzechRepublic, andCanada.The individual unknowingly was infected withMDR-TB and,after twelve days of travel, the individual was involuntarilyisolated by the CDC in Atlanta under the Public HealthService Act [9]. The CDC held the individual in isolationfor one week and then moved him to a hospital in Denver,Colorado [9]. In this case, there are no reported infectionsresulting from the travel of the individual, perhaps due tothe slightly lower infection rate of MDR-TB as comparedto drug sensitive TB strains [9]. It should be noted herethat the isolation of this individual, albeit brief, potentiallyhelped prevent a rise in drug-resistant TB in the UnitedStates, which is currently at admittedly low levels [10]. Anoccurrence such as this one clearly demonstrates the need forcarefully executed isolation procedures for individuals withdrug-resistant TB strains.
Several reports have shown the effectiveness of isola-tion in reducing the number of people with TB [6, 7, 26,27]. Weis et al. [27] showed the effectiveness of isolation,reporting lower occurrences of primary and acquired drugresistance among individuals with TB. Historically, to treatthe infection, individuals were isolated in a sanatoriumwherethey would receive proper nutrition and a constant supplyof fresh air [28]. However, while this method is successfulin certain situations, this treatment methodology is difficultto implement without proper infrastructure in place [29].Locations without proper facilities such as South Africa havepoor treatment and success rates [29]; effective isolation isessential in areas such as these which have high treatmentfailure rates. Sutton et al. [26] studied three hospitals inCalifornia; they found that implementing the CDC isolationguidelines for hospitals was feasible; however, since not everyhospital could afford the necessary equipment, the isolationwas not the same for each hospital. Even though the resultsvaried, it is noticeably more efficient to isolate patients whoare infected with TB.
According to the National Committee of Fight againstTuberculosis of Cameroon [30], about 10% of infectious indi-viduals who start the recommended WHO DOTS treatmenttherapy in the hospital do not return to the hospital for therest of sputumexaminations and check-up and are thus lost tofollow-up.This can be attributed to the long duration of treat-ment regimen, negligence, or lack of information about TB[30], a brief relief from the long term treatment [22], poverty,and so forth. As such, health-care personnel do not knowtheir epidemiological status, that is, if they died, recovered,or are still infectious and this lack of epidemiological status ofthese individuals can affect the spread of TB in a population[30]. A number of mathematical models for tuberculosisdeveloped account for this population by the inclusion of
either the lost sight class [23, 30โ32] or lost to follow-up class[22].
The aim of this study is to develop a new deterministictransmission model for TB to gain qualitative insight intothe effects of isolation in the presence of individuals who arelost to follow-up on TB transmission dynamics. A notablefeature of themodel is the incorporation of isolated and lost tofollow-up classes for the three TB strains. The paper is orga-nized as follows.Themodel formulation and analysis is givenin Section 2. Sensitivity analysis of the model is considered inSection 3. Analysis of the reproduction number is carried outin Section 4.The effect of isolation is numerically investigatedin Section 5. The key theoretical and epidemiological resultsfrom this study are summarized in Section 6.
2. Model Formulation
Themodel is formulated as follows: the population is dividedinto susceptible (๐), latently infected (๐ธ
๐), symptomatically
infectious with drug sensitive strain (๐), MDR strain (๐),XDR strain (๐), symptomatically infectious individuals whoare lost to follow-up (๐ฟ
๐), isolated (๐ฝ), and recovered (๐ ),
where ๐ = ๐,๐,๐. Thus, the total population is ๐(๐ก) =๐(๐ก) + ๐ธ
๐(๐ก) + ๐(๐ก) + ๐ธ
๐(๐ก) + ๐(๐ก) + ๐ธ
๐(๐ก) + ๐(๐ก) +
๐ฟ๐(๐ก) + ๐ฟ
๐(๐ก) + ๐ฟ
๐(๐ก) + ๐ฝ(๐ก) + ๐ (๐ก).
As the disease evolves individuals move from one class tothe other with respect to their disease status. The populationof susceptible (๐) is generated by new recruits (either via birthor immigration) who enter the population at a rate ๐. Theparameter ๐ denotes the recruitment rate. It is assumed thatthere is no vertical transmission or immigration of infectious;thus, these new inflow does not enter the infectious classes.All individuals, whatever their status, are subject to naturaldeath, which occurs at a rate ๐. The susceptible population isreduced by infection following effective contact with infectedindividuals with drug sensitive, MDR-, and XDR-TB strainsat the rates ๐
๐, ๐๐, and ๐
๐, where
๐๐=๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐,
๐๐=๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐,
๐๐=๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐.
(1)
The parameters ๐ฝ๐, ๐ฝ๐, and ๐ฝ
๐are the effective transmission
probability per contact; we assume that ๐ฝ๐
< ๐ฝ๐
< ๐ฝ๐
[9] and the parameters ๐๐
> 1, ๐๐
> 1, and ๐๐
> 1are the modification parameter that indicates the increasedinfectivity of individuals who are lost to follow-up.
A fraction ๐๐1 of the newly infected individuals with drug
sensitive strainmove into the latently infected class (๐ธ๐), with
๐๐2 fractionmoving into the symptomatic-infectious class (๐)and the other fraction [1 โ (๐
๐1 + ๐๐2)] moving into the lostto follow-up class (๐ฟ
๐) with ๐
๐1 + ๐๐2 < 1. The latentlyinfected individuals become actively infectious as a result ofendogenous reactivation of the latent bacilli at the rate ๐
๐.
Similarly a fraction ๐๐1 of the newly infected individuals
with MDR stain move into the latently infected class with
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Abstract and Applied Analysis 3
MDR (๐ธ๐), with ๐
๐2 fractions moving into the symptomat-ically infectious class (๐) and the other fraction [1 โ (๐
๐1 +๐๐2)] moving into the symptomatically infectious lost tofollow-up class (๐ฟ
๐) with ๐
๐1 + ๐๐2 < 1.The latently infectedindividuals with MDR strain become actively infectious as aresult of endogenous reactivation of the latent bacilli at therate ๐
๐.
Lastly, a fraction ๐๐1 of the newly infected individualswith
XDR stain moves into the latently infected class with XDR(๐ธ๐), with ๐
๐2 fractions moving into the symptomaticallyinfectious class (๐) and the other fraction [1 โ (๐
๐1 + ๐๐2)]moving into the symptomatically infectious lost to follow-upclass (๐ฟ
๐) with ๐
๐1 + ๐๐2 < 1.The latently infected individualswith XDR strain become actively infectious as a result ofendogenous reactivation at the rate ๐
๐.
Members of the symptomatically infectious class withthe drug sensitive strain (๐) are lost to follow-up at therate ๐
๐and move to the class of lost to follow-up with
drug sensitive strain (๐ฟ๐). They are isolated into the isolated
class (๐ฝ) at the rate ๐ผ๐. They undergo the WHO recom-
mended DOTS treatment (Directly Observed Treatment,Short Course (DOTS)); however due to treatment failure(from treatment noncompliance) they move into the latentlyinfected class at the rate ๐
๐1๐พ๐ or the latently infected classwith MDR at the rate ๐
๐2๐พ๐. The remaining fraction moveinto the recovered class (๐ ) following effective treatment atthe rate (1 โ ๐
๐1 โ ๐๐2)๐พ๐ (where ๐๐1 + ๐๐2 < 1). Or they candie from the infection at the rate ๐ฟ
๐.
Similarlymembers of the symptomatically infectiouswithMDR strain (๐) are lost to follow-up at the rate ๐
๐and
move to the class of lost to follow-up with MDR strain (๐ฟ๐).
They are isolated at the rate ๐ผ๐. And a fraction of them
move into the population of the latently infected with XDRstrain as a result of treatment failure of the symptomaticallyinfectious individuals with MDR strain at the rate ๐
๐1๐พ๐.The remaining fraction move into the recovered class at therate (1 โ ๐
๐1)๐พ๐ (where ๐๐1 < 1). Or they can die from theinfection at the rate ๐ฟ
๐.
Lastly, members of the symptomatically infectious withXDR strain (๐) are lost to follow-up at the rate ๐
๐and move
to the class of lost to follow-up with XDR strain (๐ฟ๐). Or they
move into recovered class at the rate ๐พ๐. Or they can die from
the infection at the rate ๐ฟ๐. We assume that ๐พ
๐< ๐พ๐< ๐พ๐.
The individuals who are lost to follow-up (๐ฟ๐) with drug
sensitive strain return at the rate๐๐andmove into the class of
individuals with drug sensitive strain. Or they die at the rate๐ฟ๐ฟ๐. Similarly, the individuals who are lost to follow-up (๐ฟ
๐)
withMDR strain return at the rate๐๐andmove into the class
of individuals with MDR strain. Or they die at the rate ๐ฟ๐ฟ๐
.Lastly, the individuals who are lost to follow-up (๐ฟ
๐) with
XDR strain return at the rate ๐๐and move into the class of
individuals with XDR strain. Or they die at the rate ๐ฟ๐ฟ๐.
Recovered individuals (๐ ) are reinfected with drug sen-sitive strain at the rate ๐๐
๐, with ๐
๐1๐๐๐ fraction movinginto the latently infected class with drug sensitive strain,๐๐2๐๐๐ fraction moving into the symptomatically infectiousclass with drug sensitive strain, and the other [1 โ (๐
๐1 +๐๐2)]๐๐๐ moving into the symptomatically infectious lostto follow-up class with drug sensitive strain. Also, these
individuals experience reinfection with MDR and XDRstrains and fractions of these move into the latently infectedand symptomatically infectious classes, respectively.
It follows, from the above descriptions and assump-tions, that the model for the transmission dynamics of thetuberculosis with isolation and lost to follow-up is given bythe following deterministic system of nonlinear differentialequations (the variables and parameters of the model aredescribed in Table 1; a schematic diagram of the model isdepicted in Figure 1):
๐๐
๐๐ก= ๐โ[
๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐+๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐
+๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐] ๐โ๐๐,
๐๐ธ๐
๐๐ก=๐๐1๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐1๐พ๐๐โ (๐๐
+๐) ๐ธ๐,
๐๐
๐๐ก=๐๐2๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐๐ธ๐+๐๐๐ฟ๐
โ (๐๐+๐ผ๐+ ๐พ๐+๐+ ๐ฟ
๐) ๐,
๐๐ธ๐
๐๐ก=๐๐1๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐2๐พ๐๐โ (๐๐
+๐) ๐ธ๐,
๐๐
๐๐ก=๐๐2๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐๐ธ๐
+๐๐๐ฟ๐โ (๐๐+๐ผ๐+ ๐พ๐+๐+ ๐ฟ
๐)๐,
๐๐ธ๐
๐๐ก=๐๐1๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐1๐พ๐๐โ(๐๐
+๐) ๐ธ๐,
๐๐
๐๐ก=๐๐2๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐๐ธ๐+๐๐๐ฟ๐
โ (๐๐+๐ผ๐+ ๐พ๐+๐+ ๐ฟ
๐)๐,
๐๐ฟ๐
๐๐ก=(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐๐
โ (๐๐+๐+ ๐ฟ
๐ฟ๐) ๐ฟ๐,
๐๐ฟ๐
๐๐ก=(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐
+๐๐๐โ(๐
๐+๐+ ๐ฟ
๐ฟ๐) ๐ฟ๐,
๐๐ฟ๐
๐๐ก=(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ + ๐๐๐ฟ๐) (๐ + ๐๐ )
๐+๐๐๐
โ (๐๐+๐+ ๐ฟ
๐ฟ๐) ๐ฟ๐,
๐๐ฝ
๐๐ก= ๐ผ๐๐+๐ผ๐๐+๐ผ
๐๐โ (๐พ
๐ฝ+๐+ ๐ฟ
๐ฝ) ๐ฝ,
๐๐
๐๐ก= (1โ๐
๐1 โ๐๐2) ๐พ๐๐+ (1โ๐๐1) ๐พ๐๐+๐พ๐๐
+๐พ๐ฝ๐ฝ โ ๐๐ โ ๐ [
๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐+๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐
+๐ฝ๐(๐ + ๐
๐๐ฟ๐)
๐]๐ .
(2)
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4 Abstract and Applied Analysis
Table 1: Variables and parameters description of model (2).
Variable Description๐(๐ก) Susceptible individuals๐ธ๐(๐ก), ๐ธ๐(๐ก), ๐ธ๐(๐ก) Latently infected individuals with drug sensitive, MDR, and XDR strains
๐(๐ก),๐(๐ก), ๐(๐ก) Symptomatically infectious individuals with drug sensitive, MDR, and XDR strains๐ฟ๐(๐ก), ๐ฟ๐(๐ก), ๐ฟ๐(๐ก) Symptomatically infectious individuals who are lost to sight with drug sensitive, MDR, and XDR strains
๐ฝ(๐ก) Isolated individuals๐ (๐ก) Recovered individualsParameter Description๐ Recruitment rate๐ Natural death rate๐ TB reinfection rate๐ฝ๐, ๐ฝ๐, ๐ฝ๐
Transmission probability๐๐, ๐๐, ๐๐
Infectivity modification parameter๐๐1, ๐๐2, ๐๐1, ๐๐2, ๐๐1, ๐๐2 Fraction of fast disease progression๐๐1, ๐๐2, ๐๐1 Fraction that failed treatment๐๐, ๐๐, ๐๐
Disease progression rate๐พ๐, ๐พ๐, ๐พ๐, ๐พ๐ฝ
Recovery rate๐ผ๐, ๐ผ๐, ๐ผ๐
Isolation rate๐ฟ๐, ๐ฟ๐, ๐ฟ๐, ๐ฟ๐ฝ
Disease-induced death rate๐๐, ๐๐, ๐๐
Lost to follow-up rate๐๐, ๐๐, ๐๐
Return rate from lost to follow-up๐ฟ๐ฟ๐, ๐ฟ๐ฟ๐, ๐ฟ๐ฟ๐
Disease-induced death rate in individuals who are lost to follow-up
lT1๐T
lM1๐T lM2๐T
lX1๐X lX2๐X
๐lM1๐M ๐lM2๐M
pT1๐พT
pT2๐พT
๐๐
๐๐ ๐
๐
๐ ๐
๐
๐๐
๐
ET
EM
EX
S
T
M
X
LT
LM
LX
J R
๐T๐M
๐X
๐
๐X
๐T
๐M
๐ฟM
๐ฟX
๐ฟJ
๐พJ
๐T
๐M
๐X
๐ฟLM
lT2๐T(1 โ lT1 โ lT2)๐T ๐(1 โ lT1 โ lT2)๐T
(1 โ pT1 โ pT2)๐พT
(1 โ pM1)๐พM
(1 โ lM1 โ lM2)๐T
๐(1 โ lM1 โ lM2)๐M
๐(1 โ lX1 โ lX2)๐X
(1 โ lX1 โ lX2)๐X
pM1๐พM
๐lX1๐X๐lX2๐X
๐ฟLT
๐ฟLX
๐lT2๐T
๐ผM
๐ผT
๐ผX
๐lT1๐T
๐ฟT
๐X
๐T
๐M
Figure 1: Systematic flow diagram of the tuberculosis model (2).
2.1. Basic Properties
2.1.1. Positivity and Boundedness of Solutions. For TB model(2) to be epidemiologically meaningful, it can be shown(using the method in Appendix A of [33]) that all itsstate variables are nonnegative for all time. In other words,
solutions of the model system (2) with nonnegative initialdata will remain nonnegative for all time ๐ก > 0.
Lemma 1. Let the initial data ๐(0) โฅ 0, ๐ธ๐(0) โฅ 0, ๐(0) โฅ
0, ๐ธ๐(0) โฅ 0, ๐(0) โฅ 0, ๐ธ
๐(0) โฅ 0, ๐(0) โฅ 0, ๐ฟ
๐(0) โฅ
0, ๐ฟ๐(0) โฅ 0, ๐ฟ
๐(0) โฅ 0, ๐ฝ(0) โฅ 0, ๐ (0) โฅ 0.
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Abstract and Applied Analysis 5
Then the solutions (๐, ๐ธ๐, ๐, ๐ธ๐,๐, ๐ธ
๐, ๐, ๐ฟ๐, ๐ฟ๐, ๐ฟ๐, ๐ฝ, ๐ )
of the tuberculosis model (2) are nonnegative for all ๐ก > 0.Furthermore,
lim sup๐กโโ
๐(๐ก) โค๐
๐, (3)
with
๐ = ๐+๐ธ๐+๐+๐ธ
๐+๐+๐ธ
๐+๐+๐ฟ
๐+๐ฟ๐
+๐ฟ๐+ ๐ฝ +๐ .
(4)
2.1.2. Invariant Regions. Since model (2) monitors humanpopulations, all variables and parameters of the model arenonnegative. Model (2) will be analyzed in a biologicallyfeasible region as follows. Consider the feasible region
ฮฆ โ R12+
(5)
with
ฮฆ = {(๐, ๐ธ๐, ๐, ๐ธ๐,๐, ๐ธ
๐, ๐, ๐ฟ๐, ๐ฟ๐, ๐ฟ๐, ๐ฝ, ๐ )
โR12+: ๐ (๐ก) โค
๐
๐} .
(6)
The following steps are followed to establish the positiveinvariance ofฮฆ (i.e., solutions inฮฆ remain inฮฆ for all ๐ก > 0).The rate of change of the population is obtained by adding theequations of model (2) and this gives
๐๐ (๐ก)
๐๐ก= ๐โ๐๐ (๐ก) โ ๐ฟ๐๐ (๐ก) โ ๐ฟ๐๐(๐ก) โ ๐ฟ๐๐(๐ก)
โ ๐ฟ๐ฟ๐๐ฟ๐ (๐ก) โ ๐ฟ๐ฟ๐๐ฟ๐ (๐ก) โ ๐ฟ๐ฟ๐๐ฟ๐ (๐ก) .
(7)
And it follows that
๐๐ (๐ก)
๐๐กโค ๐โ๐๐ (๐ก) . (8)
A standard comparison theorem [34] can then be used toshow that
๐(๐ก) โค ๐ (0) ๐โ๐๐ก + ๐๐(1โ ๐โ๐๐ก) . (9)
In particular, ๐(๐ก) โค ๐/๐, if ๐(0) โค ๐/๐. Thus, region ฮฆis positively invariant. Hence, it is sufficient to consider thedynamics of the flow generated by (2) in ฮฆ. In this region,the model can be considered as being epidemiologicallyand mathematically well-posed [35]. Thus, every solution ofmodel (2) with initial conditions in ฮฆ remains in ฮฆ for all๐ก > 0. Therefore, the ๐-limit sets of system (2) are containedin ฮฆ. This result is summarized below.
Lemma 2. The region ฮฆ โ R12+ร is positively invariant for
model (2) with nonnegative initial conditions in R12+.
2.2. Stability of the Disease-Free Equilibrium (DFE). Tubercu-losis model (2) has a DFE, obtained by setting the right-handsides of the equations in the model to zero, given by
E0
= (๐โ, ๐ธโ
๐, ๐โ, ๐ธโ
๐,๐โ, ๐ธโ
๐, ๐โ, ๐ฟโ
๐, ๐ฟโ
๐, ๐ฟโ
๐, ๐ฝโ, ๐ โ)
= (๐
๐, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) .
(10)
The linear stability of E0 can be established using thenext generation operator method on system (2). Using thenotations in [36], the matrices ๐น and๐, for the new infectionterms and the remaining transfer terms, are, respectively,given by
๐น = [๐น1 | ๐น2] , (11)
where
๐น1 =
((((((((((((((((
(
0 ๐๐1๐ฝ๐ 0 0 0
0 ๐๐2๐ฝ๐ 0 0 0
0 0 0 ๐๐1๐ฝ๐ 0
0 0 0 ๐๐2๐ฝ๐ 0
0 0 0 0 00 0 0 0 00 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ 0 0 00 0 0 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ 00 0 0 0 00 0 0 0 0
))))))))))))))))
)
,
-
6 Abstract and Applied Analysis
๐น2 =
((((((((((((((((
(
0 ๐๐1๐ฝ๐๐๐ 0 0 0
0 ๐๐2๐ฝ๐๐๐ 0 0 0
0 0 ๐๐1๐ฝ๐๐๐ 0 0
0 0 ๐๐2๐ฝ๐๐๐ 0 0
๐๐1๐ฝ๐ 0 0 ๐๐1๐ฝ๐๐๐ 0๐๐2๐ฝ๐ 0 0 ๐๐2๐ฝ๐๐๐ 00 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐๐๐ 0 0 00 0 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐๐๐ 0 0(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ 0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐ 0
0 0 0 0 0
))))))))))))))))
)
,
๐ =
((((((((((((((((
(
๐1 โ๐๐1๐พ๐ 0 0 0 0 0 0 0 0โ๐๐
๐2 0 0 0 0 โ๐๐ 0 0 00 โ๐
๐2๐พ๐ ๐3 0 0 0 0 0 0 00 0 โ๐
๐๐4 0 0 0 โ๐๐ 0 0
0 0 0 โ๐๐1๐พ๐ ๐5 0 0 0 0 0
0 0 0 0 โ๐๐
๐6 0 0 โ๐๐ 00 โ๐
๐0 0 0 0 ๐7 0 0 0
0 0 0 โ๐๐
0 0 0 ๐8 0 00 0 0 0 0 โ๐
๐0 0 ๐9 0
0 โ๐ผ๐
0 โ๐ผ๐
0 โ๐ผ๐
0 0 0 ๐10
))))))))))))))))
)
,
(12)
where๐1 = ๐๐+๐, ๐2 = ๐๐+๐พ๐+๐ผ๐+๐๐+๐, ๐3 = ๐๐+๐, ๐4 =๐๐+ ๐พ๐+ ๐ผ๐+ ๐๐+ ๐, ๐5 = ๐๐ + ๐, ๐6 = ๐๐ + ๐พ๐ +
๐ผ๐+ ๐๐+ ๐, ๐7 = ๐๐ + ๐ + ๐ฟ๐ฟ๐, ๐8 = ๐๐ + ๐ + ๐ฟ๐ฟ๐, ๐9 =
๐๐+ ๐ + ๐ฟ
๐ฟ๐, ๐10 = ๐พ๐ฝ + ๐ + ๐ฟ๐ฝ.
It follows that the basic reproduction number of tubercu-losis model (2), denoted byR0, is given by
R0 = ๐ (๐น๐โ1) = max (R
๐,R๐,R๐) , (13)
where
R๐=๐ฝ๐{๐๐(๐7 + ๐๐๐๐) ๐๐1 + (๐7 + ๐๐๐๐) ๐1๐๐2 + [(๐2๐๐ + ๐๐) ๐1 โ ๐๐๐๐๐๐1๐พ๐] (1 โ ๐๐1 โ ๐๐2)}
[๐1 (๐7๐2 โ ๐๐๐๐) โ ๐๐๐๐๐๐1๐พ๐],
R๐=๐ฝ๐[๐๐(๐8 + ๐๐๐๐) ๐๐1 + (๐8 + ๐๐๐๐) ๐3๐๐2 + (๐4๐๐ + ๐๐) ๐3 (1 โ ๐๐1 โ ๐๐2)]
๐3 (๐8๐4 โ ๐๐๐๐),
R๐=๐ฝ๐[๐๐(๐9 + ๐๐๐๐) ๐๐1 + (๐9 + ๐๐๐๐) ๐5๐๐2 + (๐6๐๐ + ๐๐) ๐5 (1 โ ๐๐1 โ ๐๐2)]
๐5 (๐9๐6 โ ๐๐๐๐).
(14)
Quantity R๐represents the reproduction number of TB
drug sensitive-only population. Similarly, quantity R๐
isthe reproduction number for MDR-only population and thequantity R
๐represents the reproduction number for XDR-
only TB population.Further, using Theorem 2 in [36], the following result is
established.
Lemma 3. TheDFE of the tuberculosis model (2), given byE0,is locally asymptotically stable (LAS) ifR0 < 1, and unstable ifR0 > 1.
The threshold quantity (R0, i.e., the basic reproductionnumber) measures the average number of new infectionsgenerated by a single infected individual in a completely
-
Abstract and Applied Analysis 7
๐X๐M๐TlX2lX1lM2lM1lT2lT1
๐ฟJ๐ฟX๐ฟM๐ฟT๐X๐M๐T๐X๐M๐T๐
pM1pT2pT1๐ผT๐ผX๐ผM๐X๐M๐T๐ฝX๐ฝM๐ฝT๐พJ๐พX๐พM๐พT๐
0 0.5 1โ0.5
๐
๐ฟLX๐ฟLM๐ฟLT
(a)
๐X๐M๐TlX2lX1lM2lM1lT2lT1
๐ฟJ๐ฟX๐ฟM๐ฟT๐X๐M๐T๐X๐M๐T๐
pM1pT2pT1๐ผT๐ผX๐ผM๐X๐M๐T๐ฝX๐ฝM๐ฝT๐พJ๐พX๐พM๐พT๐
0 0.2 0.4 0.6 0.8 1โ0.8 โ0.6 โ0.4 โ0.2
๐
๐ฟLX๐ฟLM๐ฟLT
(b)๐X๐M๐TlX2lX1lM2lM1lT2lT1
๐ฟJ๐ฟX๐ฟM๐ฟT๐X๐M๐T๐X๐M๐T๐
pM1pT2pT1๐ผT๐ผX๐ผM๐X๐M๐T๐ฝX๐ฝM๐ฝT๐พJ๐พX๐พM๐พT๐
0 0.2 0.4 0.6 0.8 1โ0.8 โ0.6 โ0.4 โ0.2
๐
๐ฟLX๐ฟLM๐ฟLT
(c)
Figure 2: PRCC values for model (2), using as the response function (a) the basic reproduction number (R๐), (b) the basic reproduction
number (R๐), and (c) the basic reproduction number (R
๐). Parameter values (baseline) and ranges used are as given in Table 2.
susceptible population [35โ38]. Thus, Lemma 3 implies thattuberculosis can be eliminated from the population (whenR0 < 1) if the initial sizes of the subpopulations of model(2) are in the basin of attraction of the DFE, E0.
3. Sensitivity Analysis
A global sensitivity analysis [39โ42] is carried out, on theparameters of model (2), to determine which of the param-eters have the most significant impact on the outcome ofthe numerical simulations of the model. Figure 2(a) depictsthe partial rank correlation coefficient (PRCC) values foreach parameter of the models, using the ranges and baselinevalues tabulated in Table 2 (with the basic reproductionnumbers, R
๐, as the response function), from which it
follows that the parameters that have the most influence ondrug sensitive TB transmission dynamics are the fraction offast progression rate (๐
๐2) into the drug sensitive TB class,the infectivity modification parameter (๐
๐), the recovery rate
(๐พ๐) from drug sensitive TB, disease progression rate (๐
๐),
rate of lost to follow-up (๐๐) of those with drug sensitive
TB, fraction of latently infected with drug sensitive TB thatfailed treatment (๐
๐1), the fraction of fast progression rate(๐๐1) into the latently infected with drug sensitive TB class,
and the isolation rate (๐ผ๐) from drug sensitive TB class.
The identification of these key parameters is vital to theformulation of effective control strategies for combating thespread of the disease, as this study identifies the most impor-tant parameters that drive the transmissionmechanism of thedisease. In other words, the results of this sensitivity analysissuggest that, to effectively control the spread of drug sensitiveTB in the community, the effective strategy will be to reducethe disease progression rate (reduce๐
๐), increase the recovery
rate (increase ๐พ๐) from drug sensitive TB, reduce the disease
modification parameter (reduce ๐๐), increase the isolation
rate (increase ๐ผ๐) from drug sensitive TB class, and reduce
the fraction of fast progression rates (reduce ๐๐1 and ๐๐2) into
the drug sensitive TB class and lost to follow-up class andreduce the rate of lost to follow-up (reduce ๐
๐) of those with
drug sensitive TB.The result of this analysis suggests that thefraction of latently infected with drug sensitive TB that failedtreatment (๐
๐1) be increased, since this has a negative impactof the basic reproduction number, R
๐. This of course is
counter intuitive; however increasing this rate lowers the rateof development of drug-resistant TB due to treatment failure.
The sensitivity analysis was also carried out using model(2) with the basic reproduction number, R
๐, for drug-
resistant TB, as the response function (see Figure 2(b)). Thedominant parameters in this case are ๐ผ
๐, ๐๐, ๐๐, ๐พ๐, ๐๐,
๐๐, ๐, ๐๐1, and ๐๐2. Similarly, when using as response func-
tion the basic reproduction number, R๐, for extended drug
-
8 Abstract and Applied Analysis
resistance TB (see Figure 2(c)), the dominant parameters inthis case are ๐ผ
๐, ๐๐, ๐๐, ๐พ๐, ๐๐, ๐๐, ๐, ๐๐1, and ๐๐2.
The results from these analyses using as response func-tions the basic reproduction numbers,R
๐andR
๐, suggest
that the natural death rate (๐) be increased, since it has anegative impact on the basic reproduction numbers,R
๐and
R๐. However increasing this rate is not epidemiologically
relevant, as it implies reducing the population size thatwe wish to preserve by other means aside from death bytuberculosis and should therefore be ignored in any controlmeasures.
4. Analysis of the Reproduction Number
Following the result obtained in Section 3, we investigate inthis section whether or not treatment-only, isolation-onlyof individuals with tuberculosis or a combination of bothcan lead to tuberculosis elimination in the population. Theanalysis will be carried out using the reproduction numberfor drug sensitive TB (R
๐) since R0 is the maximum of the
reproduction number of drug sensitive TB (R๐), MDR-TB
(R๐), and XDR-TB (R
๐); similar result can be obtained for
the MDR- and XDR-TB.In the absence of isolation (๐ผ
๐= 0), the reproduction
number (R๐) reduces to
R๐๐ผ
=๐ฝ๐[(๐๐๐7 + ๐๐๐๐๐๐) ๐๐1 + (๐๐๐1๐๐ + ๐1๐7) ๐๐2 + (๐1๐๐ + ๐๐๐1 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) โ ๐๐๐๐๐๐ก1๐พ๐) (1 โ ๐๐1 โ ๐๐2)]
๐1 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) ๐7 โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7.
(15)
The reproduction number (R๐) can be written as
R๐= ๐ด๐ผR๐๐ผ
, (16)
where
๐ด๐ผ
=(๐1๐7 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7) [(๐๐๐7 + ๐๐๐๐๐๐) ๐๐1 + (๐๐๐1๐๐ + ๐1๐7) ๐๐2 + (๐1๐๐ + ๐๐๐1๐2 โ ๐๐๐๐๐๐1๐พ๐) (1 โ ๐๐1 โ ๐๐2)]{(๐1๐2๐7 โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7) [๐๐ (๐7 + ๐๐๐๐) ๐๐1 + (๐๐๐๐ + ๐7) ๐1๐๐2 + (๐๐๐1 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) + ๐1๐๐ โ ๐๐๐๐๐๐1๐พ๐) (1 โ ๐๐1 โ ๐๐2)]}
.
(17)
The difference between R๐and R
๐๐ผ
is in the isolation rate(๐ผ๐); as such the factor ๐ด
๐ผcompares a population with
and without isolation; however, this is in the presence oftreatment and individuals who are lost to follow-up andindividuals returning from lost to follow-up. If R
๐๐ผ
< 1,then drug sensitive TB cannot develop into an epidemicin the community. However, if R
๐๐ผ
> 1, it is imperative
to investigate the effect of isolation on the transmission ofdrug sensitive TB among the populace and determine thenecessary condition for slowing down its development in thecommunity. Following [43] we have
ฮ๐ผ= R๐๐ผ
โR๐= (1โ๐ด
๐ผ)R๐๐ผ
, (18)
where
ฮ๐ผ=[๐ฝ๐๐1 (๐7 + ๐๐๐๐) (๐๐ โ ๐2 + ๐พ๐ + ๐ + ๐ฟ๐) (๐๐๐7๐๐1 + ๐1๐7๐๐2 + ๐1๐๐ (1 โ ๐๐1 โ ๐๐2))]
{(๐1๐2๐7 โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7) [๐1๐7 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7]}. (19)
To slow down the spread of drug sensitive tuberculosis inthe population via effective isolation, proper treatment, andidentification of individuals who return from lost to follow-up and in the presence of individuals who are lost to follow-up, we expect that ฮ
๐ผ> 0, and this is satisfied if ๐ด
๐ผ< 1
in (18). Now, setting R๐= 1 and solving for ๐ด
๐ผgives the
threshold effectiveness of isolation and taking into account
treatment and identification of individuals who return fromlost to follow-up and who are also lost to follow-up:
๐ดโ
๐ผ=
1R๐๐ผ
. (20)
Hence, drug sensitive tuberculosis can be eradicated inthe community in the presence of isolation taking into
-
Abstract and Applied Analysis 9
consideration proper treatment and identification of individ-uals who return from lost to follow-up and are lost to follow-up if ๐ด
๐ผ< ๐ดโ
๐ผis attained. Note that ๐ดโ
๐ผis a decreasing
function ofR๐๐ผ
, thus indicating that higher values for๐ดโ๐ผwill
result in smaller values forR๐๐ผ
, a desired outcome. But a large
value for R๐๐ผ
results in a small value for ๐ดโ๐ผ, an indication
that eradication may not be attainable.The following limits of ๐ด
๐ผprovide a further insight into
possible ways of reducing the burden of drug sensitive TB inthe community:
lim๐ด๐ผ๐ผ๐โโ
=๐๐2๐๐ [(๐๐ + ๐ + ๐ฟ๐ + ๐พ๐) ๐1๐7 โ ๐๐๐๐๐1 โ ๐๐๐๐1๐พ๐๐7]
๐7 {๐๐ (๐7 + ๐๐๐๐) ๐๐1 + (๐๐๐๐ + ๐7) ๐1๐๐2 + [๐๐๐1 + ๐๐ (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) ๐1 โ ๐๐๐๐๐๐1๐พ๐] (1 โ ๐๐1 โ ๐๐2)},
(21)
lim๐ด๐ผ๐๐โโ
= 1,
lim๐ด๐ผ๐๐โโ
= 1,
lim๐ด๐ผ๐พ๐โโ
= 1.
(22)
The limits in (21) will be less than unity. In practice, ๐ผ๐โ โ
implies high rate of isolating individuals with drug sensitiveTB, ๐
๐โ โ implies high rate of individuals who are lost
to follow-up, ๐๐
โ โ implies a high rate of individualswho return from lost to follow-up, and ๐พ
๐โ โ implies
high treatment rate. Ideally, the results obtained from theselimits can be pursued for the reduction of the burden of
drug sensitive TB in the community, provided of course thatit is feasible and practicable economically. Therefore, with alook at factor ๐ด
๐ผ, one observes that an effective isolation will
lead to a reduction in the burden of drug sensitive TB in thepopulation.
Next, we consider the case when the treatment rate is setto zero (๐พ
๐= 0). The reproduction number is given as
R๐๐พ
=๐ฝ๐[(๐๐๐7 + ๐๐๐๐๐๐) ๐๐1 + (๐๐๐1๐๐ + ๐1๐7) ๐๐2 + [๐1๐๐๐๐๐1 (๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐)] (1 โ ๐๐1 โ ๐๐2)]
๐1 (๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐) ๐7 โ ๐1๐๐๐๐. (23)
The reproduction numberR๐can be expressed as
R๐= ๐ด๐พR๐๐พ
, (24)
where
๐ด๐พ
={[(๐๐๐7 + ๐๐๐๐๐๐) ๐๐1 + (๐๐๐1๐๐ + ๐1๐7) ๐๐2 + (๐1๐๐ + ๐๐๐1๐2 โ ๐๐๐๐๐๐1๐พ๐) (1 โ ๐๐1 โ ๐๐2)] ๐1 [(๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐) ๐7 โ ๐๐๐๐]}
{(๐1๐2๐7 โ ๐1๐๐๐๐ โ ๐๐๐๐ก1๐พ๐๐7) [(๐๐๐7 + ๐๐๐๐๐๐) ๐๐1 + (๐๐๐1๐๐ + ๐1๐7) ๐๐2 + (๐1๐๐ + (๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐) ๐1๐๐) (1 โ ๐๐1 โ ๐๐2)]}.
(25)
The difference between R๐๐พ
and R๐is in the treatment
rate (๐พ๐); thus ๐ด
๐พcompares a population with and without
treatment in the presence of isolation of infected individualswith drug sensitive TB, individuals who are lost to follow-up and who return from of lost to follow-up. If R
๐๐พ
< 1,then drug sensitive TB cannot develop into an epidemic in
the community and no control strategy will be required forits control. Take the difference betweenR
๐andR
๐๐พ
; that is,
ฮ๐พ= R๐โR๐๐พ
= (1โ๐ด๐พ)R๐๐พ
, (26)
where
ฮ๐พ={๐ฝ๐(๐7 + ๐๐๐๐) [๐1 (๐๐ + ๐ + ๐ฟ๐ + ๐ผ๐ โ ๐2) + ๐พ๐๐๐1๐๐] [๐1๐7๐๐2 + ๐๐๐7๐๐1 + ๐1๐๐ (1 โ ๐๐1 โ ๐๐2)]}
{๐1 (๐1๐2๐7 โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7) [๐7 (๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐) โ ๐๐๐๐]}. (27)
-
10 Abstract and Applied Analysis
To slow down the spread of drug sensitive tuberculosis inthe population using proper treatment, effective isolationand identification of individuals who return from lost tofollow-up and in the presence of those who are of lost to
follow-up, we expect that ฮ๐พ> 0, and this is satisfied if
๐ด๐พ< 1 in (18).Take the following limits of ๐ด
๐พ:
lim๐ด๐พ๐พ๐โโ
=[(๐๐+ ๐ผ๐+ ๐ + ๐ฟ
๐) ๐7 โ ๐๐๐๐] ๐๐๐1 (1 โ ๐๐1 โ ๐๐2)
๐7 {๐๐ (๐7 + ๐๐๐๐) ๐๐1 + (๐๐๐๐ + ๐7) ๐1๐๐2 + [๐๐๐1 + ๐๐ (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) ๐1 โ ๐๐๐๐๐๐1๐พ๐] (1 โ ๐๐1 โ ๐๐2)},
lim๐ด๐พ๐๐โโ
= 1,
lim๐ด๐พ๐๐โโ
= 1,
lim๐ด๐พ๐ผ๐โโ
= 1.
(28)
From the limit of๐ด๐พ, one observes that an effective treatment
will lead to a reduction in the burden of drug sensitive TB inthe population.
Comparing the quantities ๐ด๐ผand ๐ด
๐พshows that ๐ด
๐ผ<
๐ด๐พ; that is, size
๐ด๐ผโ๐ด๐พ= โ {[(๐7 + ๐๐๐๐) + (๐7 + ๐๐๐๐) ๐๐2๐1 + (๐๐๐1 + ๐๐๐1๐2 โ ๐๐๐๐๐๐1๐พ๐) (1โ ๐๐1 โ ๐๐2)] [๐1 (๐ผ๐ โ ๐พ๐)
+ ๐๐๐๐1๐พ๐] (๐7 + ๐๐๐๐) [๐๐๐7๐๐1 +๐1๐7๐๐2 +๐1๐๐ (1โ ๐๐1 โ ๐๐2)]} ({(๐1๐2๐7 โ๐1๐๐๐๐ โ๐๐๐๐1๐พ๐๐7)
โ [๐๐(๐7๐๐๐๐) ๐๐1 + (๐7 + ๐๐๐๐) ๐1๐๐2 + (๐๐๐1 + ๐๐๐1 (๐๐ + ๐ผ๐ + ๐ + ๐ฟ๐)) (1 โ ๐๐1 โ ๐๐2)]
โ [๐๐(๐7 + ๐๐๐๐) ๐๐1 + (๐7 + ๐๐๐๐) ๐1๐๐2 + (๐๐๐1 + ๐๐๐1 (๐๐ + ๐พ๐ + ๐ + ๐ฟ๐) โ ๐๐๐๐๐๐1๐พ๐) (1 โ ๐๐1 โ ๐๐2)]})
โ1< 0.
(29)
This implies that ๐ด๐พwill provide better results in slowing
down drug sensitive tuberculosis spread, using reduction inthe prevalence, than using๐ด
๐ผ. On the other hand if๐ด
๐ผโ๐ด๐พ>
0, this means that ๐ด๐ผwill give better outcome in slowing
down drug sensitive tuberculosis spread compared to using๐ด๐พ.Thus, from the above discussions, to slow down the
spread of the disease and reduce the number of secondaryinfections in the population, we require control strategieswith parameter values that would make ๐ด
๐ผ< 1 or ๐ด
๐พ<
1. Hence, the necessary condition for slowing down thedevelopment of drug sensitive TB at the population level isthat ฮ
๐ผ> 0 or ฮ
๐พ> 0. However, ฮ
๐ผgives a better result
in terms of reduction in the prevalence of the disease overฮ๐พprovided ฮ
๐พ> ฮ๐ผ; otherwise if ฮ
๐ผ> ฮ๐พ, then ฮ
๐พ
gives a better result over ฮ๐ผ. Using parameters in Table 1, we
have that ๐ด๐ผ= 0.8524, ๐ด
๐พ= 0.8802 and ฮ
๐ผ= 0.1061,
ฮ๐พ= 0.0833; thus ๐ด
๐ผโ ๐ด๐พ= โ0.02789. It follows that, the
isolation-only strategy provides more effective control mea-sures in curtailing the disease transmission in the community.
Using the threshold quantity, R๐, we determine how
isolation and treatment rates could lead to tuberculosiselimination in the population. Thus
limR๐๐ผ๐โโ
=๐ฝ๐(1 โ ๐๐1 โ ๐๐2) ๐๐
(๐๐+ ๐ + ๐ฟ
๐)
> 0,
limR๐๐พ๐โโ
=๐ฝ๐(1 โ ๐๐1 โ ๐๐2) ๐๐
(๐๐+ ๐ + ๐ฟ
๐)
> 0.
(30)
Thus a sufficient effective TB control program that isolates (ortreats) the identified cases at a high rate ๐ผ
๐โ โ (or ๐พ
๐โ
โ) can lead to effective disease control if it results in makingthe respective right-hand side of (30) less than unity.
Differentiating partially the reproduction number of ๐R๐
with respect to the key parameters (๐ผ๐and ๐พ๐), this gives
๐R๐
๐๐ผ๐
=โ๐ฝ๐๐1 (๐7 + ๐๐๐๐) [๐๐1๐๐๐7 + ๐๐2๐1๐7 + (1 โ ๐๐1 โ ๐๐2) ๐1๐๐]
[๐1๐7 (๐๐ + ๐ผ๐ + ๐พ๐ + ๐ + ๐ฟ๐) โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7]2 , (31)
๐R๐
๐๐พ๐
=โ๐ฝ๐(๐1 โ ๐๐๐๐ก1) (๐7 + ๐๐๐๐) [๐๐1๐๐๐7 + ๐๐2๐1๐7 + (1 โ ๐๐1 โ ๐๐2) ๐1๐๐][๐1๐7 (๐๐ + ๐ผ๐ + ๐พ๐ + ๐1๐7๐ + ๐ฟ๐) โ ๐1๐๐๐๐ โ ๐๐๐๐1๐พ๐๐7]
2 . (32)
-
Abstract and Applied Analysis 11
Table 2: Values of the parameters of the model (2).
Parameter Baselinevalues Range Reference
๐ ๐ ร 105 (0.0143 ร105, 0.04 ร 105) [11]
๐ 0.0159 (0.0143, 0.04) [12]๐ 0.06 (0.01, 0.5) [13]๐ฝ๐ 9.75 (4.5, 15.0) [14]๐ฝ๐ 1.5 (1.5, 3.5) [13, 15]๐ฝ๐ 0.0000085 (0.01, 0.1) [13]๐๐, ๐๐, ๐๐ 0.5 (0, 1) Assumed
๐๐1, ๐๐2, ๐๐1, ๐๐2, ๐๐1, ๐๐2 0.14 (0.02, 0.3) [12, 16, 17]๐๐1 0.3 (0.1, 0.5) [14, 18]๐๐2 0.03 (0.01, 0.05) [14, 18]๐๐1 0.03 (0.01, 0.1) [14, 18]
๐๐, ๐๐, ๐๐
0.05, 0.0018,0.013 (0.005, 0.05) [12, 19, 20]
๐พ๐, ๐พ๐ฝ 1.5 (1.5, 2.5) [11, 16, 21]
๐พ๐, ๐พ๐ 0.75 (0.5, 1.0) [13]
๐ผ๐, ๐ผ๐, ๐ผ๐ 0.6 (0.2, 1.0) Assumed
๐๐, ๐๐, ๐๐ 0.2511 (0.0022, 0.5) [22]
๐๐, ๐๐, ๐๐ 0.1 (0.5, 1.0) [23]
๐ฟ๐, ๐ฟ๐ฝ 0.365 (0.22, 0.39) [20, 24, 25]
๐ฟ๐, ๐ฟ๐ 0.028 (0.01, 0.03) [13]
๐ฟ๐ฟ๐, ๐ฟ๐ฟ๐
, ๐ฟ๐ฟ๐ 0.02 (0.01, 0.039) [22, 23]
Thus, it follows from (31) that ๐R๐/๐๐ผ๐
< 0, henceshowing further the effectiveness of the control measures.Thus, isolation (๐ผ
๐) of drug sensitive tuberculosis will have
a positive impact in reducing the drug sensitive TB burdenin the community, regardless of the values of the otherparameters. This result is stated in the following lemma.
Lemma 4. The use of isolation (๐ผ๐) will have a positive
impact on the reduction of the drug sensitive TB burden in acommunity regardless of the values of other parameters in thebasic reproduction number under isolation.
Similarly, from (32), we have that ๐R๐/๐๐พ๐
< 0.Thus, effective treatment (๐พ
๐) of drug sensitive tuberculosis
will have a positive impact in reducing the drug sensitivetuberculosis burden in the community, irrespective of thevalues of the other parameters. This result is summarizedbelow.
Lemma 5. The use of effective treatment (๐พ๐) will have a
positive impact on the reduction of the drug sensitive TB burdenin a community irrespective of the values of other parameters inthe basic reproduction number under treatment.
A contour plot of the reproduction number R๐, as a
function of the effective treatment rate (๐พ๐) and isolation
rate (๐ผ๐), is depicted in Figure 3(a). As expected, the plot
shows a decrease in R๐values with increasing values of
the treatment and isolation rates. For instance, if the useof effective treatment result in ๐พ
๐= 0.8 and ๐ผ
๐= 0.8,
drug sensitive TB burden will be reduced considerably in thepopulation. Similarly in Figure 3(b) the plot shows a decreasein R๐values with decreasing values of the return rate from
lost to follow-up (๐๐) and lost to follow-up rate (๐
๐).
A contour plot of the reproduction number R๐, as a
function of return rate from lost to follow-up (๐๐) and the
effective treatment rate (๐พ๐), is depicted in Figure 4(a). The
plot shows a decrease inR๐values with increasing values of
the treatment rate. Similarly the plot in Figure 4(b) shows adecrease inR
๐values with increasing values of the isolation
rate (๐ผ๐).
4.1. Backward Bifurcation Analysis. Model (2) is now investi-gated for the possibility of the existence of the phenomenonof backward bifurcation (where a stable DFE coexists with astable endemic equilibrium when the reproduction number,R0, is less than unity) [44โ53]. The epidemiological impli-cation of backward bifurcation is that the elimination (oreffective control) of the TB (and various strains) in the systemis no longer guaranteed when the reproduction number isless than unity but is dependent on the initial sizes of thesubpopulations. The possibility of backward bifurcation inmodel (2) is explored using the centre manifold theory [47],as described in [54] (Theorem 4.1).
Theorem 6. Model (2) undergoes a backward bifurcation atR๐= 1whenever inequality (A.9), given inAppendix A, holds.
Theproof ofTheorem 6 is given in Appendix A (the proofcan be similarly given for the case whenR
๐= 1 orR
๐= 1).
The backward bifurcation property of model (2) is illustratedby simulating themodel using a set of parameter values givenin Table 2 (such that the bifurcation parameters, ๐ and ๐,given in Appendix A, take the values ๐ = 558.61 > 0 and ๐ =1.59 > 0, resp.). The backward bifurcation phenomenon ofmodel (2) makes the effective control of the TB strains in thepopulation difficult, since, in this case, disease control whenR0 < 1 is dependent on the initial sizes of the subpopulationsof model (2). This phenomenon is illustrated numerically inFigures 5 and 6 for individuals with drug sensitive, MDR-,and XDR-TB, as well as individuals who are lost to follow-upwith drug sensitive, MDR-, and XDR-TB, respectively.
It is worth mentioning that when the reinfection param-eter of model (2), for the recovered individuals, is set to zero(i.e., ๐ = 0), the bifurcation parameter, ๐, becomes negative(see Appendix A). This rules out backward bifurcation (inline with Item (iv) of Theorem 4.1 of [54]) in this case. Thus,this study shows that the reinfection of recovered individualscauses backward bifurcation in the transmission dynamicsof TB in the system. To further confirm the absence of thebackward bifurcation phenomenon in model (2) for thiscase, the global asymptotic stability of the DFE of the modelis established below for the case when no reinfection ofrecovered individuals occurs.
-
12 Abstract and Applied Analysis
4
4
6
6
6
6
8
8
8
10
10
1214
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1๐พT
๐ผT
(a)
2.5
33
33
3.53.5
3.5
4
4
4.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
๐T
๐T
(b)
Figure 3: Contour plot of the reproduction number (R๐) of model (2) as: (a) a function of isolation rate (๐ผ
๐) and treatment rate (๐พ
๐); (b) a
function of return rate from lost to follow-up (๐๐) and lost to follow-up rate (๐
๐). Parameter values used are as given in Table 2.
4.5 4.5 4.55
5 5 55.5 5.5 5.56
6 6 66.5 6.5 6.57
7 7 77.5 7.5 7.58 8 8 88.5 8.5 8.590 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
๐พT
๐T
(a)
2.6
2.8 2.82.8
33 3 3
3.2 3.23.2
3.4 3.43.4
3.6 3.63.6
3.8 3.83.8
4 40 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1๐ผT
๐T
(b)
Figure 4: Contour plot of the reproduction number (R๐) of model (2) as: (a) a function of return rate from lost to follow-up (๐
๐) and
treatment rate (๐พ๐); (b) a function of return rate from lost to follow-up (๐
๐) and treatment rate (๐ผ
๐). Parameter values used are as given in
Table 2.
4.2. Global Stability of the DFE: Special Case. Consider thespecial case of model (2) where the reinfection parametersare set to zero (i.e., ๐ = 0). It is convenient to define thereproduction threshold Rฬ0 = R0|๐=0.
Theorem 7. The DFE of model (2), with ๐ = 0, is GAS in ฮฆwhenever Rฬ0 < 1.
The proof of Theorem 7 is given in Appendix B.The epidemiological significance ofTheorem 7 is that, for
the special case of model (2) with ๐ = 0, TB will be eliminatedfrom the community if the reproduction number (Rฬ0) can bebrought to (and maintained at) a value less than unity.
5. The Effects of Isolation
Following the result obtained from the sensitivity analysis, weinvestigate the impact of the isolation parameters ๐ผ
๐, ๐ผ๐, and
๐ผ๐, which are one of the dominant parameters of model (2).
We start by individually varying these parameters for (say)๐ผ๐= 0.2, 0.4, 0.6, 0.8, 1.0, with the other parameters given
in Table 2 kept constant. We observed that (see Figure 7) asthe isolation rate, ๐ผ
๐, for drug sensitive TB increases, the
total number of individuals (with drug sensitive, MDR, andXDR) isolatedwith each strain of TB increases, while the totalnumber of individuals who are lost to follow-up decreases.We observed similar result for the isolation rate ๐ผ
๐(see
Figure 8). However, for isolation rate ๐ผ๐, negligible change
was observed and the plots are not shown.
6. Conclusion
In this paper, we have developed and analyzed a system ofordinary differential equations for the transmission dynam-ics of drug-resistant tuberculosis with isolation. From ouranalysis, we have the following results which are summarizedbelow:
-
Abstract and Applied Analysis 13
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
800
700
500
300
100
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M
0
20
40
60
80
100
120
140
160
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
0
5
10
15
20
25
(c)
Figure 5: Backward bifurcation plot of model (2) as a function of time. (a) Individuals with drug sensitive TB (๐). (b) Individuals with MDR(๐). (c) Individuals with XDR (๐). Parameter values used are as given in Table 2.
(i) Themodel is locally asymptotically stable (LAS)R0 <1 and unstable whenR0 > 1.
(ii) The model exhibits in the presence of disease reinfec-tion the phenomenon of backward bifurcation, wherethe stable disease-free coexists with a stable endemicequilibrium, when the associated reproduction num-ber is less than unity.
(iii) As the isolation rate for each strain of TB increases,the total number of individuals infected with theparticular strain of TB decreases.
(iv) Model (2) in the absence of disease reinfection isglobally asymptotically stable (GAS)R0 < 1.
(v) The sensitivity analysis of the model shows that thedominant parameters for the drug sensitive TB are
the disease progression rate (๐๐), the recovery rate
(๐พ๐) from drug sensitive TB, the infectivity parameter
(๐๐), the isolation rate (๐ผ
๐) from drug sensitive TB
class, fraction of fast progression rates and (๐๐1 and
๐๐2) into the drug sensitive TB class and lost to follow-up class, and the rate of lost to follow-up (๐
๐). Similar
parameters and return rates from lost to follow-up(๐๐
and ๐๐) are dominant for MDR- and XDR-
TB. The natural death rate (๐), although dominantin MDR- and XDR-TB, is however epidemiologicallyirrelevant.
(vi) Increase in isolation rate leads to increase in totalnumber of individuals isolated with each TB strainresulting in decreases in the total number of individ-uals who are lost to follow-up.
-
14 Abstract and Applied Analysis
Appendices
A. Proof of Theorem 6
Proof. Theproof is based on using the centremanifold theory[47], as described in [54]. It is convenient to make thefollowing simplification and change of variables.
Let ๐ = ๐ฅ1, ๐ธ๐ = ๐ฅ2, ๐ = ๐ฅ3, ๐ธ๐ = ๐ฅ4, ๐ = ๐ฅ5, ๐ธ๐ =๐ฅ6, ๐ = ๐ฅ7, ๐ฟ๐ = ๐ฅ8, ๐ฟ๐ = ๐ฅ9, ๐ฟ๐ = ๐ฅ10, ๐ฝ = ๐ฅ11and ๐ = ๐ฅ12, so that ๐ = ๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 +๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12. Using the vectornotation x = (๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4, ๐ฅ5, ๐ฅ6, ๐ฅ7, ๐ฅ8, ๐ฅ9, ๐ฅ10, ๐ฅ11, ๐ฅ12)
๐,model (2) can be written in the form ๐x/๐๐ก = m(x), wherem = (๐1, ๐2, ๐3, ๐4, ๐5, ๐6, ๐7, ๐8, ๐9, ๐10, ๐11, ๐12)
๐, asfollows:
๐๐ฅ1๐๐ก
= ๐1 = ๐โ[๐ฝ๐(๐ฅ3 + ๐๐๐ฅ8) + ๐ฝ๐ (๐ฅ5 + ๐๐๐ฅ9) + ๐ฝ๐๐ฅ7 (๐ฅ7 + ๐๐๐ฅ10)] ๐ฅ1
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)โ ๐๐ฅ1,
๐๐ฅ2๐๐ก
= ๐2 =(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ฅ3 + ๐๐๐ฅ8) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)โ ๐1๐ฅ2,
๐๐ฅ3๐๐ก
= ๐3 =๐๐1๐ฝ๐ (๐ฅ3 + ๐๐๐ฅ8) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ2 +๐๐๐ฅ8 โ๐2๐ฅ3,
๐๐ฅ4๐๐ก
= ๐4 =(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ฅ5 + ๐๐๐ฅ9) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)โ ๐3๐ฅ4,
๐๐ฅ5๐๐ก
= ๐5 =๐๐1๐ฝ๐ (๐ฅ5 + ๐๐๐ฅ9) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ4 +๐๐๐ฅ3 +๐๐๐ฅ9 โ๐4๐ฅ5,
๐๐ฅ6๐๐ก
= ๐6 =(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ (๐ฅ7 + ๐๐๐ฅ10) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)โ ๐5๐ฅ6,
๐๐ฅ7๐๐ก
= ๐7 =๐๐1๐ฝ๐ (๐ฅ7 + ๐๐๐ฅ10) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ6 +๐๐๐ฅ5 +๐๐๐ฅ10 โ๐6๐ฅ7,
๐๐ฅ8๐๐ก
= ๐8 =๐๐2๐ฝ๐ (๐ฅ3 + ๐๐๐ฅ8) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ3 โ๐7๐ฅ8,
๐๐ฅ9๐๐ก
= ๐9 =๐๐2๐ฝ๐ (๐ฅ5 + ๐๐๐ฅ6) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ5 โ๐8๐ฅ9,
๐๐ฅ10๐๐ก
= ๐10 =๐๐2๐ฝ๐ (๐ฅ7 + ๐๐๐ฅ10) (๐ฅ1 + ๐๐ฅ12)
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12)+ ๐๐๐ฅ7 โ๐9๐ฅ10,
๐๐ฅ11๐๐ก
= ๐11 = ๐ผ๐๐ฅ3 +๐ผ๐๐ฅ5 +๐ผ๐๐ฅ7 โ๐10๐ฅ11,
๐๐ฅ12๐๐ก
= ๐12 = ๐พ๐๐ฅ3 + ๐พ๐๐ฅ5 + ๐พ๐๐ฅ7 โ๐11๐ฅ12 โ๐ [๐ฝ๐(๐ฅ3 + ๐๐๐ฅ8) + ๐ฝ๐ (๐ฅ5 + ๐๐๐ฅ9) + ๐ฝ๐ (๐ฅ7 + ๐๐๐ฅ10)] ๐ฅ12
(๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 + ๐ฅ5 + ๐ฅ6 + ๐ฅ7 + ๐ฅ8 + ๐ฅ9 + ๐ฅ10 + ๐ฅ11 + ๐ฅ12).
(A.1)
The Jacobian of the transformed system (A.1), at the disease-free equilibriumE1, is given by
๐ฝ (E1) = (๐ฝ1 | ๐ฝ2) , (A.2)
-
Abstract and Applied Analysis 15
where
๐ฝ1 =
((((((((((((((((((((((
(
โ๐ 0 โ๐ฝ๐
0 โ๐ฝ๐
00 โ๐1 ๐๐1๐ฝ๐ + ๐๐1๐พ๐ 0 0 00 ๐๐
๐๐2๐ฝ๐ โ ๐2 0 0 0
0 0 0 โ๐3 ๐๐1๐ฝ๐ 00 0 ๐
๐2๐พ๐ ๐๐ ๐๐2๐ฝ๐ โ ๐4 00 0 0 0 ๐
๐1๐พ๐ โ๐5
0 0 0 0 0 ๐๐
0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ + ๐๐ 0 0 0
0 0 0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ + ๐๐ 0
0 0 0 0 0 00 0 ๐ผ
๐0 ๐ผ
๐0
0 0 (1 โ ๐๐1 โ ๐๐2) ๐พ๐ 0 (1 โ ๐๐1) ๐พ๐ 0
))))))))))))))))))))))
)
,
๐ฝ2
=
((((((((((((((((((((((
(
โ๐ฝ๐
โ๐ฝ๐๐๐
โ๐ฝ๐๐๐
โ๐ฝ๐๐๐
0 00 ๐
๐1๐ฝ๐๐๐ 0 0 0 00 ๐
๐2๐ฝ๐๐๐ + ๐๐ 0 0 0 00 0 ๐
๐1๐ฝ๐๐๐ 0 0 00 0 ๐
๐2๐ฝ๐๐๐ + ๐๐ 0 0 0๐๐1๐ฝ๐ 0 0 ๐๐1๐ฝ๐๐๐ 0 0
๐๐2๐ฝ๐ โ ๐6 0 0 ๐๐2๐ฝ๐๐๐ + ๐๐ 0 0
0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐ โ ๐7 0 0 0 0
0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐ โ ๐8 0 0 0
(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ + ๐๐ 0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐ โ ๐9 0 0
๐ผ๐
0 0 0 โ๐10 0๐พ๐
0 0 0 0 โ๐11
))))))))))))))))))))))
)
.
(A.3)
Consider the case when R0 = 1. Suppose, further, that ๐ฝ๐is chosen as a bifurcation parameter. Solving (2) for ๐ฝ
๐from
R0 = 1 gives ๐ฝ๐ = ๐ฝโ
๐. The transformed system (A.1) at
the DFE evaluated at ๐ฝ๐= ๐ฝโ
๐has a simple zero eigenvalue
(and all other eigenvalues having negative real parts). Hence,the centre manifold theory [47] can be used to analyze thedynamics of (A.1) near ๐ฝ
๐= ๐ฝโ
๐. In particular, the theorem
in [54] (see also [36, 47, 48]) is used (it is reproduced inthe Appendix for convenience). To apply the theorem, thefollowing computations are necessary (it should be noted thatwe are using ๐ฝ
๐instead of ๐ for the bifurcation parameter).
Eigenvectors of ๐ฝ(E1)|๐ฝ๐=๐ฝโ
๐
. The Jacobian of (A.1) at ๐ฝ๐= ๐ฝโ
๐,
denoted by ๐ฝ(E1)|๐ฝ๐=๐ฝโ
๐
, has a right eigenvector (associatedwith the zero eigenvalue) given by
w = (๐ค1, ๐ค2, ๐ค3, ๐ค4, ๐ค5, ๐ค6, ๐ค7, ๐ค8, ๐ค9, ๐ค10, ๐ค11,
๐ค12)๐,
(A.4)
where
๐ค1 = โ1๐[๐ฝ๐๐ค3 +๐ฝ๐๐ค5 +๐ฝ๐๐ค7 +๐ฝ๐๐๐๐ค8
+๐ฝ๐๐๐๐ค9 +๐ฝ๐๐๐๐ค10] ,
๐ค2 =1๐1
{[๐๐1๐พ๐ + ๐๐1๐ฝ๐] ๐ค3 + ๐๐1๐ฝ๐๐๐๐ค8} ,
๐ค3 > 0,
๐ค5 > 0,
๐ค4 =1๐3
[๐๐2๐พ๐๐ค3 + ๐๐1๐ฝ๐๐ค5 + ๐๐1๐ฝ๐๐๐๐ค9] ,
๐ค6 =1๐5
[๐๐1๐พ๐๐ค5 + ๐๐1๐ฝ๐๐ค7 + ๐๐1๐ฝ๐๐๐๐ค10] ,
๐ค7 > 0,
๐ค8 =[(1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ + ๐๐] ๐ค3
[๐7 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐],
-
16 Abstract and Applied Analysis
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LT
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1000
800
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LM
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
80
70
60
50
40
30
20
10
0
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LX
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
15
10
5
0
(c)
Figure 6: Backward bifurcation plot of model (2) as a function of time. (a) Individuals who are lost to follow-up with drug sensitive TB (๐).(b) Individuals who are lost to follow-up with MDR (๐). (c) Individuals who are lost to follow-up with XDR (๐). Parameter values used areas given in Table 2.
๐ค9 =[(1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ + ๐๐] ๐ค5
[๐8 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐],
๐ค10 =[(1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ + ๐๐] ๐ค7
(๐9 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐),
๐ค11 =(๐ผ๐๐ค5 + ๐ผ๐๐ค7)
๐10,
๐ค12 =1๐11
[(1โ๐๐1 โ๐๐2) ๐พ๐๐ค3 + (1โ๐๐1) ๐พ๐๐ค5
+ ๐พ๐๐ค7] .
(A.5)
Also, ๐ฝ(E1)|๐ฝ๐=๐ฝโ
๐
has a left eigenvector k = (V1, V2, V3,V4, V5, V6, V7, V8, V9, V10, V11, V12) (associated with the zeroeigenvalue), where
V1 = 0,
V2 =๐๐V3๐1
,
V4 =๐๐V4
๐3,
V6 =๐๐V7
๐5,
V3 > 0,
-
Abstract and Applied Analysis 17
๐ผT = 0.20
๐ผT = 0.40
๐ผT = 0.60
๐ผT = 0.80
๐ผT = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
1200
1400
1600To
tal n
umbe
r of i
sola
ted
indi
vidu
als
(a)
๐ผT = 0.20
๐ผT = 0.40
๐ผT = 0.60
๐ผT = 0.80
๐ผT = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
2500
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 7: Simulation of model (2) as a function of time varying ๐ผ๐for (a) total number of isolated individuals and (b) total number of
individuals who are lost to follow-up. Parameter values used are as given in Table 2.
๐ผM = 0.20
๐ผM = 0.40
๐ผM = 0.60
๐ผM = 0.80
๐ผM = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
Tota
l num
ber o
f iso
late
d in
divi
dual
s
(a)
๐ผM = 0.20
๐ผM = 0.40
๐ผM = 0.60
๐ผM = 0.80
๐ผM = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 8: Simulation of model (2) as a function of time varying ๐ผ๐
for (a) total number of isolated individuals and (b) total number ofindividuals who are lost to follow-up. Parameter values used are as given in Table 2.
V7 > 0,
V5 =1
(๐4 โ ๐๐2๐ฝ๐)[๐๐1๐ฝ๐V4 +๐๐1๐พ๐V6
+ [(1โ ๐๐1 โ ๐๐2) ๐ฝ๐ +๐๐] V9] ,
V8 =1
[๐7 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐][๐๐1๐ฝ๐๐๐V2
+ (๐๐2๐ฝ๐๐๐ +๐๐) V3] ,
V9 =1
(๐8 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐)[๐๐1๐ฝ๐๐๐V4
+ (๐๐2๐ฝ๐๐๐ +๐๐) V5] ,
V10 =1
(๐9 โ (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐)[๐๐1๐ฝ๐๐๐V6
+ (๐๐2๐ฝ๐๐๐ +๐๐) V7] ,
V11 = 0,
V12 = 0.(A.6)
Computations of Bifurcation Coefficients ๐ and ๐. The appli-cation of the theorem (given in the Appendix) entails the
-
18 Abstract and Applied Analysis
computation of two bifurcation coefficients ๐ and ๐. It can beshown, after some algebraic manipulations, that
๐ = V212โ
๐,๐=1๐ค๐๐ค๐
๐2๐2
๐๐ฅ๐๐๐ฅ๐
+ V312โ
๐,๐=1๐ค๐๐ค๐
๐2๐3
๐๐ฅ๐๐๐ฅ๐
+ V412โ
๐,๐=1๐ค๐๐ค๐
๐2๐4
๐๐ฅ๐๐๐ฅ๐
+ V512โ
๐,๐=1๐ค๐๐ค๐
๐2๐5
๐๐ฅ๐๐๐ฅ๐
+ V612โ
๐,๐=1๐ค๐๐ค๐
๐2๐6
๐๐ฅ๐๐๐ฅ๐
+ V712โ
๐,๐=1๐ค๐๐ค๐
๐2๐7
๐๐ฅ๐๐๐ฅ๐
=2๐ฅ1
(โ๐ค2 โ๐ค3 โ๐ค4 โ๐ค5 โ๐ค6 โ๐ค7 โ๐ค8 โ๐ค9
โ๐ค10 โ๐ค11 โ๐ค12 +๐ค12๐) {๐ฝ๐ (๐ค3 +๐ค8๐๐)
โ [๐๐1V2 + ๐๐2V3 + (1โ ๐๐1 โ ๐๐2) V8]
+ ๐ฝ๐(๐ค5 +๐ค9๐๐)
โ [๐๐1V4 + ๐๐2V5 + (1โ ๐๐1 โ ๐๐2) V9]
+ ๐ฝ๐(๐ค7 +๐ค10๐๐)
โ [๐๐1V6 + ๐๐2V7 + (1โ ๐๐1 โ ๐๐2) V10]} .
(A.7)
Furthermore,
๐ = V212โ
๐=1๐ค๐
๐2๐2
๐๐ฅ๐๐๐ฝโ๐
+ V312โ
๐=1๐ค๐
๐2๐3
๐๐ฅ๐๐๐ฝโ๐
= (๐ค3 +๐ค8๐๐) [๐๐1V2 + ๐๐2V3 + (1โ ๐๐1 โ ๐๐2) V8]
> 0.
(A.8)
Hence, it follows from Theorem 4.1 of [54] that the trans-formed model (A.1) (or, equivalently, (2)) undergoes back-ward bifurcation atR0 = 1whenever the following inequalityholds:
๐ > 0. (A.9)
It is worth noting that if ๐ = 0 (i.e., reinfectionof recovered individuals does not occur), the bifurcationcoefficient, ๐, given in (A.7), reduces to
๐ = โ2๐ฅ1
(๐ค2 +๐ค3 +๐ค4 +๐ค5 +๐ค6 +๐ค7 +๐ค8 +๐ค9
+๐ค10 +๐ค11) {๐ฝ๐ (๐ค3 +๐ค8๐๐)
โ [๐๐1V2 + ๐๐2V3 + (1โ ๐๐1 โ ๐๐2) V8]
+ ๐ฝ๐(๐ค5 +๐ค9๐๐)
โ [๐๐1V4 + ๐๐2V5 + (1โ ๐๐1 โ ๐๐2) V9]
+ ๐ฝ๐(๐ค7 +๐ค10๐๐)
โ [๐๐1V6 + ๐๐2V7 + (1โ ๐๐1 โ ๐๐2) V10]} .
(A.10)
It follows from (A.10) that the bifurcation coefficient ๐ < 0(ruling out backward bifurcation in this case, in line withTheorem 4.1 in [54]). Thus, this study shows that the back-ward bifurcation phenomenon ofmodel (A.1) is caused by thereinfection of the recovered individuals in the population.
B. Proof of Theorem 7
Proof. The proof is based on using a comparison theorem.The equations for the infected components of model (2), with๐ = 0, can be rewritten as
((((((((((((((((((((((((((((((((((((
(
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฝ (๐ก)
๐๐ก
))))))))))))))))))))))))))))))))))))
)
= (๐นโ๐)
(((((((((((((((((((
(
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฝ (๐ก)
)))))))))))))))))))
)
โ๐๐
(((((((((((((((((((
(
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฝ (๐ก)
)))))))))))))))))))
)
,
(B.1)
-
Abstract and Applied Analysis 19
where ๐ = 1 โ ๐, the matrices ๐น and ๐ are as given inSection 2.2, and๐ is nonnegativematrices given, respectively,by
๐ = [๐1 | ๐2] , (B.2)
where
๐1 =
(((((((((((((
(
0 ๐๐1๐ฝ๐ 0 0 0
0 ๐๐2๐ฝ๐ 0 0 0
0 0 0 ๐๐1๐ฝ๐ 0
0 0 0 ๐๐2๐ฝ๐ 0
0 0 0 0 00 0 0 0 00 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ 0 0 00 0 0 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐ 00 0 0 0 00 0 0 0 0
)))))))))))))
)
,
๐2 =
(((((((((((((
(
0 ๐๐1๐ฝ๐๐๐ 0 0 0
0 ๐๐2๐ฝ๐๐๐ 0 0 0
0 0 ๐๐1๐ฝ๐๐๐ 0 0
0 0 ๐๐2๐ฝ๐๐๐ 0 0
๐๐1๐ฝ๐ 0 0 ๐๐1๐ฝ๐๐๐ 0๐๐2๐ฝ๐ 0 0 ๐๐2๐ฝ๐๐๐ 00 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐๐๐ 0 0 00 0 (1 โ ๐
๐1 โ ๐๐2) ๐ฝ๐๐๐ 0 0(1 โ ๐๐1 โ ๐๐2) ๐ฝ๐ 0 0 (1 โ ๐๐1 โ ๐๐2) ๐ฝ๐๐๐ 0
0 0 0 0 0
)))))))))))))
)
.
(B.3)
Thus, since ๐(๐ก) โค ๐(๐ก) in ฮฆ for all ๐ก โฅ 0, it follows from(B.1) that
((((((((((((((((((((((((((((((((
(
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ธ๐ (๐ก)
๐๐ก
๐๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฟ๐ (๐ก)
๐๐ก
๐๐ฝ (๐ก)
๐๐ก
))))))))))))))))))))))))))))))))
)
โค (๐นโ๐)
(((((((((((((((((((
(
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ธ๐ (๐ก)
๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฟ๐ (๐ก)
๐ฝ (๐ก)
)))))))))))))))))))
)
. (B.4)
Using the fact that the eigenvalues of the matrix ๐น โ ๐ allhave negative real parts (see the local stability result givenin Lemma 3, where ๐(๐น๐โ1) < 1 if R0 < 1, which isequivalent to ๐น โ ๐ having eigenvalues with negative realparts when R0 < 1 [36]), it follows that the linearizeddifferential inequality system (B.4) is stable whenever R0 <1. Consequently, by comparison of theorem [34] (Theorem1.5.2, p. 31),
(๐ธ๐ (๐ก) , ๐ (๐ก) , ๐ฟ๐ (๐ก) , ๐ธ๐ (๐ก) ,๐ (๐ก) , ๐ฟ๐ (๐ก) , ๐ธ๐ (๐ก) ,
๐ (๐ก) , ๐ฟ๐ (๐ก) , ๐ฝ (๐ก)) โ (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) ,
as ๐ก โ โ.
(B.5)
Substituting ๐ธ๐= ๐ = ๐ฟ
๐= ๐ธ๐= ๐ = ๐ฟ
๐= ๐ธ๐= ๐ =
๐ฟ๐= ๐ฝ = 0 into the equations of ๐ and ๐ in model (2), and
noting that ๐ = 0, gives ๐(๐ก) โ ๐โ, ๐ (๐ก) โ 0 as ๐ก โ โ.Thus, in summary,
(๐ (๐ก) , ๐ธ๐ (๐ก) , ๐ (๐ก) , ๐ฟ๐ (๐ก) , ๐ธ๐ (๐ก) ,๐ (๐ก) , ๐ฟ๐ (๐ก) ,
๐ธ๐ (๐ก) , ๐ (๐ก) , ๐ฟ๐ (๐ก) , ๐ฝ (๐ก) , ๐ (๐ก)) โ (๐
โ, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0) ,
(B.6)
-
20 Abstract and Applied Analysis
as ๐ก โ โ. Hence, the DFE (E0) of model (2), with ๐ = 0, isGAS in ฮฆ if Rฬ0 < 1.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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