Research Article Successive Vaccination and Difference in...
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Research Article Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate Yongzhen Pei, 1 Li Changguo, 2 Qianyong Wu, 1 and Yunfei Lv 3 1 School of Computer Science and Soſtware Engineering, Tianjin Polytechnic University, Tianjin 300387, China 2 Department of Basic Science, Military Transportation University, Tianjin 300161, China 3 School of Science, Tianjin Polytechnic University, Tianjin 300387, China Correspondence should be addressed to Yongzhen Pei; [email protected] Received 28 December 2013; Revised 7 May 2014; Accepted 8 May 2014; Published 16 June 2014 Academic Editor: Kaifa Wang Copyright © 2014 Yongzhen Pei 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 delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on the disease spread. From theorems, it is obtained that the basic reproduction number governs the dynamic behavior of the system. e existence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproduction number. By use of new computational techniques for delay differential equations, we prove that the system is permanent. Our results indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion, and to illustrate possible behavioral scenarios of the model. 1. Introduction e current threat of some new type diseases has raised our awareness that curbing the spread of some emerging and reemerging human diseases is of public health importance such as H1N1. is emerging disease, which was first reported in Mexico, spread very quickly, due to the travel of infected persons by airplanes, trains, and buses to some other regions. It continued to spread around the world and caused about 5000 deaths. In recent years, many mathematical models have been developed for the transmission dynamics of infectious diseases such as SARS, HIV/AIDS, measles, and smallpox ([1–6], to name a few, and the references therein). ese models have provided understanding of the underlying mechanisms which influence the spread of diseases and suggested some control strategies. Moreover, to our knowl- edge, the first effective control strategy for the elimination of infectious diseases is obtaining immunity. It has been reported that “People’s immunity to A/H1N1 flu virus is greater than previously thought aſter access vaccines. e WHO is working to give more nations access vaccines to fight the H1N1 flu pandemic.” Besides these above studies, many authors formulated and analyzed SIR epidemic models for the control of diseases [7–20]. In particular, some authors have studied the effects of vaccination on the spread of diseases [7–11]; others have studied the effects of treatment on the spread of diseases [12–15]. Gao et al. have proposed an epidemic model with density-dependent birth pulses and seasonal prevention [16]. Recently, some works have investigated permanent and tem- porary immunity [17–20]. However, in these SIR models, an unrealistic assumption is that all the rest of infected indi- viduals acquire immunity besides death. Measles encephalitis in adults in [21, 22] shows that there is difference in immunity of infected individuals. at is, some infected individuals can acquire immunity aſter recovery, but some do not acquire immunity and can be infected once more. At the same time, vaccination is an important strategy for the elimination of infectious diseases [7–11]. Vaccinations have many types; impulsive vaccination and successive vaccination are two main policies. Successive vaccination is that people have been vaccinated at birth to protect themselves from disease; the studies can be found in Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 678723, 10 pages http://dx.doi.org/10.1155/2014/678723
Transcript of Research Article Successive Vaccination and Difference in...
Research ArticleSuccessive Vaccination and Difference in Immunity of a DelaySIR Model with a General Incidence Rate
Yongzhen Pei1 Li Changguo2 Qianyong Wu1 and Yunfei Lv3
1 School of Computer Science and Software Engineering Tianjin Polytechnic University Tianjin 300387 China2Department of Basic Science Military Transportation University Tianjin 300161 China3 School of Science Tianjin Polytechnic University Tianjin 300387 China
Correspondence should be addressed to Yongzhen Pei yongzhenpei163com
Received 28 December 2013 Revised 7 May 2014 Accepted 8 May 2014 Published 16 June 2014
Academic Editor Kaifa Wang
Copyright copy 2014 Yongzhen Pei 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 delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on thedisease spread From theorems it is obtained that the basic reproduction number governs the dynamic behavior of the systemTheexistence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproductionnumber By use of new computational techniques for delay differential equations we prove that the system is permanent Ourresults indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system Numericalsimulations are carried out to investigate the influence of the key parameters on the spread of the disease to support the analyticalconclusion and to illustrate possible behavioral scenarios of the model
1 Introduction
The current threat of some new type diseases has raised ourawareness that curbing the spread of some emerging andreemerging human diseases is of public health importancesuch asH1N1This emerging disease whichwas first reportedin Mexico spread very quickly due to the travel of infectedpersons by airplanes trains and buses to some other regionsIt continued to spread around the world and caused about5000 deaths In recent years many mathematical modelshave been developed for the transmission dynamics ofinfectious diseases such as SARS HIVAIDS measles andsmallpox ([1ndash6] to name a few and the references therein)Thesemodels have provided understanding of the underlyingmechanisms which influence the spread of diseases andsuggested some control strategies Moreover to our knowl-edge the first effective control strategy for the eliminationof infectious diseases is obtaining immunity It has beenreported that ldquoPeoplersquos immunity to AH1N1 flu virus isgreater than previously thought after access vaccines TheWHO is working to givemore nations access vaccines to fightthe H1N1 flu pandemicrdquo
Besides these above studies many authors formulatedand analyzed SIR epidemic models for the control of diseases[7ndash20] In particular some authors have studied the effectsof vaccination on the spread of diseases [7ndash11] others havestudied the effects of treatment on the spread of diseases[12ndash15] Gao et al have proposed an epidemic model withdensity-dependent birth pulses and seasonal prevention [16]Recently some works have investigated permanent and tem-porary immunity [17ndash20] However in these SIR models anunrealistic assumption is that all the rest of infected indi-viduals acquire immunity besides deathMeasles encephalitisin adults in [21 22] shows that there is difference in immunityof infected individualsThat is some infected individuals canacquire immunity after recovery but some do not acquireimmunity and can be infected once more At the same timevaccination is an important strategy for the elimination ofinfectious diseases [7ndash11]
Vaccinations have many types impulsive vaccinationand successive vaccination are two main policies Successivevaccination is that people have been vaccinated at birth toprotect themselves from disease the studies can be found in
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 678723 10 pageshttpdxdoiorg1011552014678723
2 Abstract and Applied Analysis
[23 24] Makinde in [23] studied a SIR model for the trans-mission dynamics of a childhood disease in the presence of apreventive vaccine and analyzed the vaccination reproductivenumber for disease control and eradication qualitativelyImpulsive vaccination (only at fixed time sequencewe executeeffectively the vaccination for the disease) is an important andeffective strategy for the elimination of infectious diseasesand has been studied in the literature For example see[10 17 25ndash27] In above-mentioned papers authors almostconsidered the vaccination of susceptible population Butin fact under a certain situation the vaccine treatment alsoshould be considered for the newborns of the susceptiblethe exposed and the removed We find that there are fewstudies on the aspect of the vaccination of newborns Inthis case successive vaccination seems more reasonable thanimpulsive vaccination Therefore in this paper a SIR modelwith difference in immunity and successive vaccination isconsidered
As far as disease transmission is concerned the incidencerate defined as the rate of new infection plays a veryimportant role in modelling infectious diseases Bilinearincidence rate 120573119878119868 in [28 29] and standard incidence rates120582119878119868119873 in [30 31] have often been used in epidemic modelsHowever it is unreasonable to consider the bilinear incidencerate (based on the law of mass action) as the number ofsusceptibles is large owing to the number of susceptibles withwhich every infective contact within a certain time is limitedStandard incidence rate may be a good approximation if thenumber of available partners is large enough but it is notpossible to make more contacts when the population 119873 issmall Combine the two previous approaches by assumingthat if the number of available partners119873 is low the numberof actual per capita partners is proportional to119873 whereas ifthe number of available partners is large there is a saturationeffect which makes the number of actual partners constantConsidering this case a saturation incidence rate of type119891(119868)119878 with 119891(119868) = 119896119868(1 + 120572119868) is being proposed in [32]More general incidence rate used in the literature is the onefor which 119891(119868) = 119896119868119897(1 + 120572119868ℎ) [33 34] where 119868119897 measuresthe infection force of the disease and 119891(119868) = 1(1 + 120572119868ℎ)measures the inhibitory effect caused by behavioral changesNote that if 119891(119868) is decreasing when 119868 is large this may beinterpreted as the fact that susceptibles tend to reduce theirsocial contacts if the perceived number of infectives increasesover a psychologically significant valueThe above saturationincidence rate depends also on the size of the infectives119868 termed as infectives-dependent Particular examples ofsusceptibles-dependent incidence rate are 119891(119878) = 119896119878(1+120572119878)[35] Very general incidence rates which are not linear in 119878 arealso used in Derrick and van denDriessche [36] (119891(119878 119868119873) =119868Φ(119878 119868119873) where119873 = 119878 + 119868) Korobeinikov and Maini [37](119891(119878 119868) = ℎ1(119868)ℎ2(119878)) and Moghadas and Alexander [38](119891(119878 119868) = 120573(1 + 119892(119868 V))119868119878)
Nie et al [19] and Ji et al [39] respectively considereda delay SIR epidemic model with nonlinear incidence rateand density-dependent birth and death rates Motivatedby the main idea described in [6 39] in this paper weconsider a delay SIR model with difference in immunity and
successive vaccination and an abstract incidence rate Themain difference between our study and those described in [639] is the difference in immunity and successive vaccinationand an abstract incidence rate An abstract incidence rate oftype 119891(119868)119878 is employed to model the spread of the diseasewhich is propagated through the infective individuals undera few biologically feasible assumptions upon 119891(119868)
In view of above facts we will formulate a mathematicalmodel in Section 2 We provide the region of biologicallyfeasible solutions in Section 3 Then we study the existenceand stability of the steady states in the next section analyzethe permanence result in Section 5 and give some numericalsimulations in Section 6 Lastly we end the paper with a briefdiscussion of our results in Section 6
2 Model Formulation and Invariant Region
In this section we will present a delay SIR epidemic modelwith a general nonlinear incidence rate The total population119873(119905) is divided into three subclasses namely the susceptibles119878(119905) the infectives 119868(119905) and the recovered individuals 119877(119905)Based on the SIR model in [12 39] we considered followingsystem
119889119878
119889119905= [119887 minus
119886119903119873
119870]119873 minus 120573119890minus1198891120591119878119891 (119868 (119905 minus 120591))
minus [119889 +(1 minus 119886) 119903119873
119870] 119878 minus 120579119878 + 1205831119868
119889119868
119889119905= 120573119890minus1198891120591119891 (119868 (119905 minus 120591)) 119878
minus [119889 +(1 minus 119886) 119903119873
119870] 119868 minus 1205831119868 minus 119890
minus1198891120596120575119868 (119905 minus 120596)
119889119877
119889119905= 119890minus1198891120596120575119868 (119905 minus 120596) minus [119889 +
(1 minus 119886) 119903119873
119870]119877 + 120579119878
119873 (0) = 1198780 gt 0 119868 (0) = 120593 (120579) ge 0
forall120579 isin [minus120591 0] 119877 (0) = 1198770 ge 0
(1)
where 120591 = max120591 120596 and 120593 isin 119862([minus120591 0] 119877) We give thefollowing useful assumptions
(1) There are no disease induced deaths and all thenewborns are susceptible
(2) 119891(119868) is the nonlinear incidence rate satisfying thefollowing assumptions
119891 (0) = 0 1198911015840 (119868) gt 0 11989110158401015840 (119868) lt 0
lim119905rarrinfin
119891 (119868) = 119888 lt +infin(2)
(3) The force of infection at any time 119905 is dominated by120573119890minus1198891120591119878(119905)119891(119868(119905minus120591)) where 120591 is incubation period and0 lt 119890minus1198891120591 le 1 represents the survival probability ofindividuals in the population after time 120591 [20] It isalso assumed that 1198891 le 119889 in [minus120591 0] where 119889 is thedeath rate and 1198891 is the death rate in the time interval[minus120591 0]
Abstract and Applied Analysis 3
(4) The parameters 119886 are a convex combination constant119903 = 119887 minus 119889 gt 0 is the intrinsic growth rate (119887 is thebirth rate) and 119870 gt 0 is the carrying capacity of thepopulationThe term (119887 minus (119886119903119873(119905)119870)) has a density-dependent per capita birth rate and the term (119889+((1minus119886)119903119873(119905)119870))has a density-dependent per capita deathrate [39]
(5) For 0 lt 119886 lt 1 the birth and death rates are consistentwith the limited resources associated with densitydependence The birth rate is density independentwhen 119886 = 0 and the death rate is density independentwhen 119886 = 1 Thus the spread of the disease (animalssuch as rodents etc) is assumed to be governed by thefollowing system of logistic equations with time delay
(6) The total population is assumed to be large enough tobe adequately described by a deterministic model andis divided into compartments based on the diseasestatus [40]
(7) The successive vaccination rate 120579 is positive The pos-itive constant 1205831 is the recovery rate of the infectiousindividuals from compartment 119868 to 119878The parameters120573 are the effective per capita contact rate constantof infected individuals The parameters 120575 are therecovery rate of infected individuals
(8) Models are formulated as functional differentialandor integral equations when time delay is included[40] Ours follows the former with the assumptionthat the 119868-equation satisfies a certain integral condi-tion [41]
Models with multiple delays are not common but fewauthors have in the past considered these-Beretta et al [42]-to name but a few Since 119873(119905) = 119878(119905) + 119868(119905) + 119877(119905) thus thegoverning equation (1) can be rewritten as
119889119873
119889119905= 119903 [1 minus
119873
119870]119873
119889119868
119889119905= 120573119890minus1198891120591 (119873 minus 119868 minus 119877) 119891 (119868 (119905 minus 120591))
minus [119889 +(1 minus 119886) 119903119873
119870] 119868 minus 1205831119868 minus 119890
minus1198891120596120575119868 (119905 minus 120596)
119889119877
119889119905= 119890minus1198891120596120575119868 (119905 minus 120596)
minus [119889 +(1 minus 119886) 119903119873
119870]119877 + 120579 (119873 minus 119868 minus 119877)
(3)
Let 120591 = max(120596 120591) Then (3) satisfies the following initialconditions
119873(0) = 1198780 gt 0 119868 (0) = 120593 (120579) ge 0
forall120579 isin [minus120591 0] 119877 (0) = 1198770 ge 0(4)
In this paper we will consider two different delays 120591 120596which are important parameters on the dynamic behavior Sothe present study is continuation of the previous work 120591 = 120596by Naresh et al [43]
Lemma 1 All solutions of the model system (3) starting in 1198773+
are bounded and eventually enter the compact attracting set
Φ = (119878 119868 119877) isin 1198773
+ 119878 (119905) + 119868 (119905) + 119877 (119905) = 119873 (119905) le 119870 (5)
Lemma 2 Let the initial data be 119873(0) = 1198780 gt 0 119868(0) =1198680(119906) ge 0 for all 119906 isin [minus120591 0] with 1198680(0) gt 0 119877(0) = 1198770 ge0 Then the solution (119878(119905) 119868(119905) 119877(119905)) of the model remainspositive for all time 119905 gt 0
Lemma 3 (see [44]) For the characteristic equation in theform 119901(120582) + 119902(120582)119890minus119903120582 = 0 where 119901 and 119902 are polynomials withreal coefficients and 119903 gt 0 is the delay suppose
(a) 119901(120582) = 0 119877(120582) gt 0
(b) |119902(119894119910)| lt |119901(119894119910)| 0 le 119910 lt infin
(c) lim|120582|rarrinfin119877(120582)ge0|119902(120582)119901(120582)| = 0
Then 119877(120582) lt 0 for every root 120582 and all 119903 gt 0
3 Equilibrium and Stability Analysis
In this section we focus on the existence and local stability ofequilibria Let the right-hand side of equalities in model (3)be zero Then there are two equilibria namely
(i) 1198640 = (119870 0 119901) 119901 = 119870120579[119889+(1minus119886)119903+120579]) disease-freeequilibrium
(ii) 119864lowast = (119873lowast 119868lowast 119877lowast) endemic equilibrium
where the values of119873lowast 119868lowast and 119877lowast are given in Section 32
31 Community Matrix Firstly after computing the Jacobianor community matrix of model (3) at point (119873 119868 119877) thecharacteristic equation is given by
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119903 minus2119903
119870119873 minus 120582 0 0
120573119890minus1198891120591119891 (119868) minus(1 minus 119886) 119903
119870119868 119898119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868)
120579 minus(1 minus 119886) 119903
119870119877 120575119890minus(1198891+120582)120596 minus 120579 minus [119889 +
(1 minus 119886) 119903
119870119873 + 120579] minus 120582
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (6)
4 Abstract and Applied Analysis
where 119898 = 120573(119873 minus 119868 minus 119877)1198911015840(119868) 119899 = 120573119890minus1198891120591119891(119868) + 119889 + ((1 minus119886)119903119870)119873 + 1205831
Now we analyze the equilibria stability of system (3)Computing the Jacobian of system (3) evaluated at 1198640 onegets the following matrix
119869 (1198640) = (
minus119903 minus 120582 0 00 11988622 minus 120582 0
(119889 + 120579) 120579
119889 + (1 minus 119886) 119903 + 120579120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
) (7)
where
11988622 = 120573 (119870 minus 119901)1198911015840(0) 119890minus(1198891+120582)120591
minus 120575119890minus(1198891+120582)120596 minus [119889 + (1 minus 119886) 119903 + 1205831] (8)
Denote
119860 = 120573 (119870 minus 119901)1198911015840 (0) 119862 = 119889 + (1 minus 119886) 119903 + 1205831 (9)
then
11988622 = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 (10)
Denote
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 minus 120582 (11)
The eigenvalues of the system (3) about the steady state1198640 are 1205821 = minus119903 ℎ(120582) = 0 and 1205823 = minus[119889 + (1 minus 119886)119903 + 120579] Allthe parameters of the model are assumed to be nonnegativeTherefore 1205821 and 1205823 are negative Next we discuss the rootsof ℎ(120582) = 0 in five cases
Case 1 For 120591 = 120596 = 0 from the second equation of the system(3) we can get the following
Proposition 4 For 120591 = 120596 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
(119860 minus 120575) 119890minus1198891120591 lt 119862 (12)
Proof From the above analysis 1205822 satisfies the followingcharacteristic equation
119892 (120582) = (119860 minus 120575) 119890minus(1198891+120582)120591 minus 119862 minus 120582 = 0 (13)
(1) Clearly 120582 = 0 is not a root of 119892(120582) = 0(2) From the fact that 119892(0) lt 0 1198921015840(120582) lt 0 for 120582 gt 0 it is
obtained that 119892(120582) = 0 has no positive real root(3) It is sufficient to show that 119892(120582) = 0 does not admit
a purely imaginary root In fact if 120582 = 119894V (V gt 0) isa root of (119892(120582) = 0) then by separating the real partone gets
(119860 minus 120575) 119890minus1198891120591 cos (V120591) = 119862 (14)
Together with the condition of Proposition 4 we have
cos (V120591) gt 1 (15)
This is impossible
(4) It is easy to show that 119892(120582) = 0 has no imaginaryroot whose real part is positive Otherwise there isan imaginary root 120582 = 119906 + 119894V with 119906 gt 0 Without anyloss of generality we consider V gt 0Then we take thereal and imaginary parts of 119892(120582) = 0 namely
(119860 minus 120575) 119890minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (16)
Combined with (9) we have
119862 gt (119860 minus 120575) 119890minus1198891120591 gt (119860 minus 120575) 119890
minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (17)
This is a contradictionwhich implies that all eigenvalues rootsof 119892(120582) have negative real parts Therefore the disease-freeequilibrium of the system (3) is locally asymptotically stablewhen (9) holds The proof is completed
Case 2 For 120591 = 0 120596 = 0 by the same way as in Case 1 onegets the following
Proposition 5 For all 120591 = 0 120596 = 0 119877(120582) lt 0 for every root 120582of ℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575 lt 119862 (18)
Case 3 For 120596 = 0 120591 = 0 one gets the following
Proposition 6 For 120596 = 0 120591 = 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
120575119890minus1198891120596 lt |119860 minus 119862| (19)
Proof By the fact that ℎ(120582) = 0 is equivalent to 119901(120582) +119902(120582)119890minus120582120596 = 0 with 119901(120582) = 119860 minus 119862 minus 120582 119902(120582) = minus120575119890minus1198891120596
(i) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then 119901(120582) = 119860minus119862minus 119906 minus119894V = 0
(ii) By |119902(119894V)| = 120575119890minus1198891120596 |119901(119894V)| = |119860 minus 119862 minus
119894V| = radic(119860 minus 119888)2 + V2 together with the condition ofProposition 6 we know that |119902(119894V)| lt |119901(119894V)|
Abstract and Applied Analysis 5
(iii) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then
lim1199062+V2rarr+infin
10038161003816100381610038161003816100381610038161003816
119902 (120582)
119901 (120582)
10038161003816100381610038161003816100381610038161003816= 120575119890minus1198891120596
lim1199062+V2rarr+infin
1
radic(119860 minus 119862 minus 119906)2 + V2= 0
(20)
Then using Lemma 3 we have 119877(120582) lt 0 for all 120596
Case 4 For 120596 gt 120591 gt 0 let 120576 = 120596 minus 120591 Then 120596 = 120591 + 120576 Forfixed 120591
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)(120591+120576) minus 119862 minus 120582 (21)
Let 120582 = 119906 + 119894V (119906 gt 0) Then we take the real and imaginaryparts of ℎ(120582) = 0 namely
119860119890minus(1198891+119906)120591 cos (V120591) minus 120575119890minus(1198891+119906)(120591+120576) cos (V (120591 + 120576))
= 119862 + 119906
minus119860119890minus(1198891+119906)120591 sin (V120591) minus 120575119890minus(1198891+119906)(120591+120576) sin (V (120591 + 120576)) = V
(22)
Sum of squares of the above equalities is
1198602119890minus2(1198891+119906)120591 + 1205752119890minus2(1198891+119906)(120591+120576)
minus 2119860120575119890minus(1198891+119906)(2120591+120576) cos (V120576) minus (119862 + 119906)2 minus V2 = 0(23)
Then we have
120597119906
120597120576
10038161003816100381610038161003816100381610038161003816120576=0=
120575 (119906 + 1198891) (119860 minus 120575)
(119860 minus 120575)2120591 + (119862 + 119906) 1198902120591(119906+120575) (24)
Obviously 120597119906120597120576|120576=0 lt 0 when 119860 minus 120575 lt 0 Combined with(119860 minus 120575)119890minus1198891120591 lt 119860119890minus1198891120591 minus 120575119890minus1198891120596 we have the following
Proposition 7 For 120596 gt 120591 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1 (25)
Case 5 For 0 lt 120591 lt 120596 in the same way as in Case 4 we havethe following
Proposition 8 For 0 lt 120591 lt 120596 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1 (26)
From what has been discussed above we get the follow-ing
Theorem 9 The disease-free equilibrium of the system (3) islocally asymptotically stable if one of the following conditionsholds
(a) 120591 = 120596 = 0 (119860 minus 120575)119890minus1198891120591 lt 119862(b) 120591 = 0 120596 = 0 119860119890minus1198891120591 minus 120575 lt 119862(c) 120591 = 0 120596 = 0 120575119890minus1198891120596 lt |119860 minus 119862|(d) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1(e) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1
32 Existence of Endemic Equilibrium Thus by Theorem 9we may define the basic reproduction number as
1198770 =119860119890minus1198891120591 minus 120575119890minus1198891120596
119862 (27)
This threshold 1198770 defines the average number of sec-ondary infections generated by a typical infectious individualin a completely susceptible population in a steady demo-graphic state
In Theorem 9 we have already shown that the system(3) has an infection-free steady state which is locally asymp-totically stable under condition 1198770 lt 1 The disease-freeequilibrium is unstable when 1198770 gt 1 and the system (3) hasa nontrivial endemic equilibrium 119864lowast = (119873lowast 119868lowast 119877lowast) when1198770 gt 1 From (3)
119873lowast = 119870 gt 0
119877lowast =120575119890minus1198891120596 minus 120579
119889 + (1 minus 119886) 119903 + 120579119868lowast
+119870120579
119889 + (1 minus 119886) 119903 + 120579≐ 119902119868lowast + 119901
(28)
where 119902 = (119890minus1198891120596120575 minus 120579)(119889 + (1 minus 119886)119903 + 120579) Substituting thesevalues of119873lowast and 119877lowast in the second equation of (3) we get thefollowing equation for 119868
119866 (119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901) 119891 (119868) minus [119862 + 120575119890
minus1198891120596] 119868
(29)
Obviously 119868 = 0 is one of the roots of (29) as 119891(0) = 0Therefore to exclude that root choose
119867(119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901)
119891 (119868)
119868minus [119862 + 120575119890minus1198891120596]
(30)
It can easily be seen that the function119867(119868) is negative forlarge positive 119868 that is
119867(119870) = minus120573119890minus1198891120591 (119870119902 + 119901)
119891 (119896)
119870minus [119862 + 120575119890minus1198891120596] lt 0 (31)
Next we determine the sign of its derivative
1198671015840 (119868) = 120573119890minus1198891120591 (119870 minus 119901)
1198911015840 (119868) 119868 minus 119891 (119868)
1198682
minus 120573119890minus1198891120591 (1 + 119902) 1198911015840 (119868)
(32)
It can easily be seen that 119870 gt 119901 In addition from theproperties of the function 119891(119868) in particular from 119891(0) = 0and 11989110158401015840(0) lt 0 it follows that 119891(119868) minus 1198911015840(119868)119868 gt 0 andconsequently1198671015840(119868) lt 0 for all 119868 gt 0 Therefore for a positiveroot of119867(119868) = 0 to exist119867(119868) has to satisfy119867(0) gt 0 that is
119867(0) = 119860119890minus1198891120591 minus 120575119890minus1198891120596 minus 119862
= (119860119890minus1198891120591 minus 119890minus1198891120596120575
119862minus 1)119862
= (1198770 minus 1)119862
(33)
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
[23 24] Makinde in [23] studied a SIR model for the trans-mission dynamics of a childhood disease in the presence of apreventive vaccine and analyzed the vaccination reproductivenumber for disease control and eradication qualitativelyImpulsive vaccination (only at fixed time sequencewe executeeffectively the vaccination for the disease) is an important andeffective strategy for the elimination of infectious diseasesand has been studied in the literature For example see[10 17 25ndash27] In above-mentioned papers authors almostconsidered the vaccination of susceptible population Butin fact under a certain situation the vaccine treatment alsoshould be considered for the newborns of the susceptiblethe exposed and the removed We find that there are fewstudies on the aspect of the vaccination of newborns Inthis case successive vaccination seems more reasonable thanimpulsive vaccination Therefore in this paper a SIR modelwith difference in immunity and successive vaccination isconsidered
As far as disease transmission is concerned the incidencerate defined as the rate of new infection plays a veryimportant role in modelling infectious diseases Bilinearincidence rate 120573119878119868 in [28 29] and standard incidence rates120582119878119868119873 in [30 31] have often been used in epidemic modelsHowever it is unreasonable to consider the bilinear incidencerate (based on the law of mass action) as the number ofsusceptibles is large owing to the number of susceptibles withwhich every infective contact within a certain time is limitedStandard incidence rate may be a good approximation if thenumber of available partners is large enough but it is notpossible to make more contacts when the population 119873 issmall Combine the two previous approaches by assumingthat if the number of available partners119873 is low the numberof actual per capita partners is proportional to119873 whereas ifthe number of available partners is large there is a saturationeffect which makes the number of actual partners constantConsidering this case a saturation incidence rate of type119891(119868)119878 with 119891(119868) = 119896119868(1 + 120572119868) is being proposed in [32]More general incidence rate used in the literature is the onefor which 119891(119868) = 119896119868119897(1 + 120572119868ℎ) [33 34] where 119868119897 measuresthe infection force of the disease and 119891(119868) = 1(1 + 120572119868ℎ)measures the inhibitory effect caused by behavioral changesNote that if 119891(119868) is decreasing when 119868 is large this may beinterpreted as the fact that susceptibles tend to reduce theirsocial contacts if the perceived number of infectives increasesover a psychologically significant valueThe above saturationincidence rate depends also on the size of the infectives119868 termed as infectives-dependent Particular examples ofsusceptibles-dependent incidence rate are 119891(119878) = 119896119878(1+120572119878)[35] Very general incidence rates which are not linear in 119878 arealso used in Derrick and van denDriessche [36] (119891(119878 119868119873) =119868Φ(119878 119868119873) where119873 = 119878 + 119868) Korobeinikov and Maini [37](119891(119878 119868) = ℎ1(119868)ℎ2(119878)) and Moghadas and Alexander [38](119891(119878 119868) = 120573(1 + 119892(119868 V))119868119878)
Nie et al [19] and Ji et al [39] respectively considereda delay SIR epidemic model with nonlinear incidence rateand density-dependent birth and death rates Motivatedby the main idea described in [6 39] in this paper weconsider a delay SIR model with difference in immunity and
successive vaccination and an abstract incidence rate Themain difference between our study and those described in [639] is the difference in immunity and successive vaccinationand an abstract incidence rate An abstract incidence rate oftype 119891(119868)119878 is employed to model the spread of the diseasewhich is propagated through the infective individuals undera few biologically feasible assumptions upon 119891(119868)
In view of above facts we will formulate a mathematicalmodel in Section 2 We provide the region of biologicallyfeasible solutions in Section 3 Then we study the existenceand stability of the steady states in the next section analyzethe permanence result in Section 5 and give some numericalsimulations in Section 6 Lastly we end the paper with a briefdiscussion of our results in Section 6
2 Model Formulation and Invariant Region
In this section we will present a delay SIR epidemic modelwith a general nonlinear incidence rate The total population119873(119905) is divided into three subclasses namely the susceptibles119878(119905) the infectives 119868(119905) and the recovered individuals 119877(119905)Based on the SIR model in [12 39] we considered followingsystem
119889119878
119889119905= [119887 minus
119886119903119873
119870]119873 minus 120573119890minus1198891120591119878119891 (119868 (119905 minus 120591))
minus [119889 +(1 minus 119886) 119903119873
119870] 119878 minus 120579119878 + 1205831119868
119889119868
119889119905= 120573119890minus1198891120591119891 (119868 (119905 minus 120591)) 119878
minus [119889 +(1 minus 119886) 119903119873
119870] 119868 minus 1205831119868 minus 119890
minus1198891120596120575119868 (119905 minus 120596)
119889119877
119889119905= 119890minus1198891120596120575119868 (119905 minus 120596) minus [119889 +
(1 minus 119886) 119903119873
119870]119877 + 120579119878
119873 (0) = 1198780 gt 0 119868 (0) = 120593 (120579) ge 0
forall120579 isin [minus120591 0] 119877 (0) = 1198770 ge 0
(1)
where 120591 = max120591 120596 and 120593 isin 119862([minus120591 0] 119877) We give thefollowing useful assumptions
(1) There are no disease induced deaths and all thenewborns are susceptible
(2) 119891(119868) is the nonlinear incidence rate satisfying thefollowing assumptions
119891 (0) = 0 1198911015840 (119868) gt 0 11989110158401015840 (119868) lt 0
lim119905rarrinfin
119891 (119868) = 119888 lt +infin(2)
(3) The force of infection at any time 119905 is dominated by120573119890minus1198891120591119878(119905)119891(119868(119905minus120591)) where 120591 is incubation period and0 lt 119890minus1198891120591 le 1 represents the survival probability ofindividuals in the population after time 120591 [20] It isalso assumed that 1198891 le 119889 in [minus120591 0] where 119889 is thedeath rate and 1198891 is the death rate in the time interval[minus120591 0]
Abstract and Applied Analysis 3
(4) The parameters 119886 are a convex combination constant119903 = 119887 minus 119889 gt 0 is the intrinsic growth rate (119887 is thebirth rate) and 119870 gt 0 is the carrying capacity of thepopulationThe term (119887 minus (119886119903119873(119905)119870)) has a density-dependent per capita birth rate and the term (119889+((1minus119886)119903119873(119905)119870))has a density-dependent per capita deathrate [39]
(5) For 0 lt 119886 lt 1 the birth and death rates are consistentwith the limited resources associated with densitydependence The birth rate is density independentwhen 119886 = 0 and the death rate is density independentwhen 119886 = 1 Thus the spread of the disease (animalssuch as rodents etc) is assumed to be governed by thefollowing system of logistic equations with time delay
(6) The total population is assumed to be large enough tobe adequately described by a deterministic model andis divided into compartments based on the diseasestatus [40]
(7) The successive vaccination rate 120579 is positive The pos-itive constant 1205831 is the recovery rate of the infectiousindividuals from compartment 119868 to 119878The parameters120573 are the effective per capita contact rate constantof infected individuals The parameters 120575 are therecovery rate of infected individuals
(8) Models are formulated as functional differentialandor integral equations when time delay is included[40] Ours follows the former with the assumptionthat the 119868-equation satisfies a certain integral condi-tion [41]
Models with multiple delays are not common but fewauthors have in the past considered these-Beretta et al [42]-to name but a few Since 119873(119905) = 119878(119905) + 119868(119905) + 119877(119905) thus thegoverning equation (1) can be rewritten as
119889119873
119889119905= 119903 [1 minus
119873
119870]119873
119889119868
119889119905= 120573119890minus1198891120591 (119873 minus 119868 minus 119877) 119891 (119868 (119905 minus 120591))
minus [119889 +(1 minus 119886) 119903119873
119870] 119868 minus 1205831119868 minus 119890
minus1198891120596120575119868 (119905 minus 120596)
119889119877
119889119905= 119890minus1198891120596120575119868 (119905 minus 120596)
minus [119889 +(1 minus 119886) 119903119873
119870]119877 + 120579 (119873 minus 119868 minus 119877)
(3)
Let 120591 = max(120596 120591) Then (3) satisfies the following initialconditions
119873(0) = 1198780 gt 0 119868 (0) = 120593 (120579) ge 0
forall120579 isin [minus120591 0] 119877 (0) = 1198770 ge 0(4)
In this paper we will consider two different delays 120591 120596which are important parameters on the dynamic behavior Sothe present study is continuation of the previous work 120591 = 120596by Naresh et al [43]
Lemma 1 All solutions of the model system (3) starting in 1198773+
are bounded and eventually enter the compact attracting set
Φ = (119878 119868 119877) isin 1198773
+ 119878 (119905) + 119868 (119905) + 119877 (119905) = 119873 (119905) le 119870 (5)
Lemma 2 Let the initial data be 119873(0) = 1198780 gt 0 119868(0) =1198680(119906) ge 0 for all 119906 isin [minus120591 0] with 1198680(0) gt 0 119877(0) = 1198770 ge0 Then the solution (119878(119905) 119868(119905) 119877(119905)) of the model remainspositive for all time 119905 gt 0
Lemma 3 (see [44]) For the characteristic equation in theform 119901(120582) + 119902(120582)119890minus119903120582 = 0 where 119901 and 119902 are polynomials withreal coefficients and 119903 gt 0 is the delay suppose
(a) 119901(120582) = 0 119877(120582) gt 0
(b) |119902(119894119910)| lt |119901(119894119910)| 0 le 119910 lt infin
(c) lim|120582|rarrinfin119877(120582)ge0|119902(120582)119901(120582)| = 0
Then 119877(120582) lt 0 for every root 120582 and all 119903 gt 0
3 Equilibrium and Stability Analysis
In this section we focus on the existence and local stability ofequilibria Let the right-hand side of equalities in model (3)be zero Then there are two equilibria namely
(i) 1198640 = (119870 0 119901) 119901 = 119870120579[119889+(1minus119886)119903+120579]) disease-freeequilibrium
(ii) 119864lowast = (119873lowast 119868lowast 119877lowast) endemic equilibrium
where the values of119873lowast 119868lowast and 119877lowast are given in Section 32
31 Community Matrix Firstly after computing the Jacobianor community matrix of model (3) at point (119873 119868 119877) thecharacteristic equation is given by
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119903 minus2119903
119870119873 minus 120582 0 0
120573119890minus1198891120591119891 (119868) minus(1 minus 119886) 119903
119870119868 119898119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868)
120579 minus(1 minus 119886) 119903
119870119877 120575119890minus(1198891+120582)120596 minus 120579 minus [119889 +
(1 minus 119886) 119903
119870119873 + 120579] minus 120582
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (6)
4 Abstract and Applied Analysis
where 119898 = 120573(119873 minus 119868 minus 119877)1198911015840(119868) 119899 = 120573119890minus1198891120591119891(119868) + 119889 + ((1 minus119886)119903119870)119873 + 1205831
Now we analyze the equilibria stability of system (3)Computing the Jacobian of system (3) evaluated at 1198640 onegets the following matrix
119869 (1198640) = (
minus119903 minus 120582 0 00 11988622 minus 120582 0
(119889 + 120579) 120579
119889 + (1 minus 119886) 119903 + 120579120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
) (7)
where
11988622 = 120573 (119870 minus 119901)1198911015840(0) 119890minus(1198891+120582)120591
minus 120575119890minus(1198891+120582)120596 minus [119889 + (1 minus 119886) 119903 + 1205831] (8)
Denote
119860 = 120573 (119870 minus 119901)1198911015840 (0) 119862 = 119889 + (1 minus 119886) 119903 + 1205831 (9)
then
11988622 = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 (10)
Denote
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 minus 120582 (11)
The eigenvalues of the system (3) about the steady state1198640 are 1205821 = minus119903 ℎ(120582) = 0 and 1205823 = minus[119889 + (1 minus 119886)119903 + 120579] Allthe parameters of the model are assumed to be nonnegativeTherefore 1205821 and 1205823 are negative Next we discuss the rootsof ℎ(120582) = 0 in five cases
Case 1 For 120591 = 120596 = 0 from the second equation of the system(3) we can get the following
Proposition 4 For 120591 = 120596 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
(119860 minus 120575) 119890minus1198891120591 lt 119862 (12)
Proof From the above analysis 1205822 satisfies the followingcharacteristic equation
119892 (120582) = (119860 minus 120575) 119890minus(1198891+120582)120591 minus 119862 minus 120582 = 0 (13)
(1) Clearly 120582 = 0 is not a root of 119892(120582) = 0(2) From the fact that 119892(0) lt 0 1198921015840(120582) lt 0 for 120582 gt 0 it is
obtained that 119892(120582) = 0 has no positive real root(3) It is sufficient to show that 119892(120582) = 0 does not admit
a purely imaginary root In fact if 120582 = 119894V (V gt 0) isa root of (119892(120582) = 0) then by separating the real partone gets
(119860 minus 120575) 119890minus1198891120591 cos (V120591) = 119862 (14)
Together with the condition of Proposition 4 we have
cos (V120591) gt 1 (15)
This is impossible
(4) It is easy to show that 119892(120582) = 0 has no imaginaryroot whose real part is positive Otherwise there isan imaginary root 120582 = 119906 + 119894V with 119906 gt 0 Without anyloss of generality we consider V gt 0Then we take thereal and imaginary parts of 119892(120582) = 0 namely
(119860 minus 120575) 119890minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (16)
Combined with (9) we have
119862 gt (119860 minus 120575) 119890minus1198891120591 gt (119860 minus 120575) 119890
minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (17)
This is a contradictionwhich implies that all eigenvalues rootsof 119892(120582) have negative real parts Therefore the disease-freeequilibrium of the system (3) is locally asymptotically stablewhen (9) holds The proof is completed
Case 2 For 120591 = 0 120596 = 0 by the same way as in Case 1 onegets the following
Proposition 5 For all 120591 = 0 120596 = 0 119877(120582) lt 0 for every root 120582of ℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575 lt 119862 (18)
Case 3 For 120596 = 0 120591 = 0 one gets the following
Proposition 6 For 120596 = 0 120591 = 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
120575119890minus1198891120596 lt |119860 minus 119862| (19)
Proof By the fact that ℎ(120582) = 0 is equivalent to 119901(120582) +119902(120582)119890minus120582120596 = 0 with 119901(120582) = 119860 minus 119862 minus 120582 119902(120582) = minus120575119890minus1198891120596
(i) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then 119901(120582) = 119860minus119862minus 119906 minus119894V = 0
(ii) By |119902(119894V)| = 120575119890minus1198891120596 |119901(119894V)| = |119860 minus 119862 minus
119894V| = radic(119860 minus 119888)2 + V2 together with the condition ofProposition 6 we know that |119902(119894V)| lt |119901(119894V)|
Abstract and Applied Analysis 5
(iii) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then
lim1199062+V2rarr+infin
10038161003816100381610038161003816100381610038161003816
119902 (120582)
119901 (120582)
10038161003816100381610038161003816100381610038161003816= 120575119890minus1198891120596
lim1199062+V2rarr+infin
1
radic(119860 minus 119862 minus 119906)2 + V2= 0
(20)
Then using Lemma 3 we have 119877(120582) lt 0 for all 120596
Case 4 For 120596 gt 120591 gt 0 let 120576 = 120596 minus 120591 Then 120596 = 120591 + 120576 Forfixed 120591
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)(120591+120576) minus 119862 minus 120582 (21)
Let 120582 = 119906 + 119894V (119906 gt 0) Then we take the real and imaginaryparts of ℎ(120582) = 0 namely
119860119890minus(1198891+119906)120591 cos (V120591) minus 120575119890minus(1198891+119906)(120591+120576) cos (V (120591 + 120576))
= 119862 + 119906
minus119860119890minus(1198891+119906)120591 sin (V120591) minus 120575119890minus(1198891+119906)(120591+120576) sin (V (120591 + 120576)) = V
(22)
Sum of squares of the above equalities is
1198602119890minus2(1198891+119906)120591 + 1205752119890minus2(1198891+119906)(120591+120576)
minus 2119860120575119890minus(1198891+119906)(2120591+120576) cos (V120576) minus (119862 + 119906)2 minus V2 = 0(23)
Then we have
120597119906
120597120576
10038161003816100381610038161003816100381610038161003816120576=0=
120575 (119906 + 1198891) (119860 minus 120575)
(119860 minus 120575)2120591 + (119862 + 119906) 1198902120591(119906+120575) (24)
Obviously 120597119906120597120576|120576=0 lt 0 when 119860 minus 120575 lt 0 Combined with(119860 minus 120575)119890minus1198891120591 lt 119860119890minus1198891120591 minus 120575119890minus1198891120596 we have the following
Proposition 7 For 120596 gt 120591 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1 (25)
Case 5 For 0 lt 120591 lt 120596 in the same way as in Case 4 we havethe following
Proposition 8 For 0 lt 120591 lt 120596 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1 (26)
From what has been discussed above we get the follow-ing
Theorem 9 The disease-free equilibrium of the system (3) islocally asymptotically stable if one of the following conditionsholds
(a) 120591 = 120596 = 0 (119860 minus 120575)119890minus1198891120591 lt 119862(b) 120591 = 0 120596 = 0 119860119890minus1198891120591 minus 120575 lt 119862(c) 120591 = 0 120596 = 0 120575119890minus1198891120596 lt |119860 minus 119862|(d) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1(e) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1
32 Existence of Endemic Equilibrium Thus by Theorem 9we may define the basic reproduction number as
1198770 =119860119890minus1198891120591 minus 120575119890minus1198891120596
119862 (27)
This threshold 1198770 defines the average number of sec-ondary infections generated by a typical infectious individualin a completely susceptible population in a steady demo-graphic state
In Theorem 9 we have already shown that the system(3) has an infection-free steady state which is locally asymp-totically stable under condition 1198770 lt 1 The disease-freeequilibrium is unstable when 1198770 gt 1 and the system (3) hasa nontrivial endemic equilibrium 119864lowast = (119873lowast 119868lowast 119877lowast) when1198770 gt 1 From (3)
119873lowast = 119870 gt 0
119877lowast =120575119890minus1198891120596 minus 120579
119889 + (1 minus 119886) 119903 + 120579119868lowast
+119870120579
119889 + (1 minus 119886) 119903 + 120579≐ 119902119868lowast + 119901
(28)
where 119902 = (119890minus1198891120596120575 minus 120579)(119889 + (1 minus 119886)119903 + 120579) Substituting thesevalues of119873lowast and 119877lowast in the second equation of (3) we get thefollowing equation for 119868
119866 (119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901) 119891 (119868) minus [119862 + 120575119890
minus1198891120596] 119868
(29)
Obviously 119868 = 0 is one of the roots of (29) as 119891(0) = 0Therefore to exclude that root choose
119867(119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901)
119891 (119868)
119868minus [119862 + 120575119890minus1198891120596]
(30)
It can easily be seen that the function119867(119868) is negative forlarge positive 119868 that is
119867(119870) = minus120573119890minus1198891120591 (119870119902 + 119901)
119891 (119896)
119870minus [119862 + 120575119890minus1198891120596] lt 0 (31)
Next we determine the sign of its derivative
1198671015840 (119868) = 120573119890minus1198891120591 (119870 minus 119901)
1198911015840 (119868) 119868 minus 119891 (119868)
1198682
minus 120573119890minus1198891120591 (1 + 119902) 1198911015840 (119868)
(32)
It can easily be seen that 119870 gt 119901 In addition from theproperties of the function 119891(119868) in particular from 119891(0) = 0and 11989110158401015840(0) lt 0 it follows that 119891(119868) minus 1198911015840(119868)119868 gt 0 andconsequently1198671015840(119868) lt 0 for all 119868 gt 0 Therefore for a positiveroot of119867(119868) = 0 to exist119867(119868) has to satisfy119867(0) gt 0 that is
119867(0) = 119860119890minus1198891120591 minus 120575119890minus1198891120596 minus 119862
= (119860119890minus1198891120591 minus 119890minus1198891120596120575
119862minus 1)119862
= (1198770 minus 1)119862
(33)
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
(4) The parameters 119886 are a convex combination constant119903 = 119887 minus 119889 gt 0 is the intrinsic growth rate (119887 is thebirth rate) and 119870 gt 0 is the carrying capacity of thepopulationThe term (119887 minus (119886119903119873(119905)119870)) has a density-dependent per capita birth rate and the term (119889+((1minus119886)119903119873(119905)119870))has a density-dependent per capita deathrate [39]
(5) For 0 lt 119886 lt 1 the birth and death rates are consistentwith the limited resources associated with densitydependence The birth rate is density independentwhen 119886 = 0 and the death rate is density independentwhen 119886 = 1 Thus the spread of the disease (animalssuch as rodents etc) is assumed to be governed by thefollowing system of logistic equations with time delay
(6) The total population is assumed to be large enough tobe adequately described by a deterministic model andis divided into compartments based on the diseasestatus [40]
(7) The successive vaccination rate 120579 is positive The pos-itive constant 1205831 is the recovery rate of the infectiousindividuals from compartment 119868 to 119878The parameters120573 are the effective per capita contact rate constantof infected individuals The parameters 120575 are therecovery rate of infected individuals
(8) Models are formulated as functional differentialandor integral equations when time delay is included[40] Ours follows the former with the assumptionthat the 119868-equation satisfies a certain integral condi-tion [41]
Models with multiple delays are not common but fewauthors have in the past considered these-Beretta et al [42]-to name but a few Since 119873(119905) = 119878(119905) + 119868(119905) + 119877(119905) thus thegoverning equation (1) can be rewritten as
119889119873
119889119905= 119903 [1 minus
119873
119870]119873
119889119868
119889119905= 120573119890minus1198891120591 (119873 minus 119868 minus 119877) 119891 (119868 (119905 minus 120591))
minus [119889 +(1 minus 119886) 119903119873
119870] 119868 minus 1205831119868 minus 119890
minus1198891120596120575119868 (119905 minus 120596)
119889119877
119889119905= 119890minus1198891120596120575119868 (119905 minus 120596)
minus [119889 +(1 minus 119886) 119903119873
119870]119877 + 120579 (119873 minus 119868 minus 119877)
(3)
Let 120591 = max(120596 120591) Then (3) satisfies the following initialconditions
119873(0) = 1198780 gt 0 119868 (0) = 120593 (120579) ge 0
forall120579 isin [minus120591 0] 119877 (0) = 1198770 ge 0(4)
In this paper we will consider two different delays 120591 120596which are important parameters on the dynamic behavior Sothe present study is continuation of the previous work 120591 = 120596by Naresh et al [43]
Lemma 1 All solutions of the model system (3) starting in 1198773+
are bounded and eventually enter the compact attracting set
Φ = (119878 119868 119877) isin 1198773
+ 119878 (119905) + 119868 (119905) + 119877 (119905) = 119873 (119905) le 119870 (5)
Lemma 2 Let the initial data be 119873(0) = 1198780 gt 0 119868(0) =1198680(119906) ge 0 for all 119906 isin [minus120591 0] with 1198680(0) gt 0 119877(0) = 1198770 ge0 Then the solution (119878(119905) 119868(119905) 119877(119905)) of the model remainspositive for all time 119905 gt 0
Lemma 3 (see [44]) For the characteristic equation in theform 119901(120582) + 119902(120582)119890minus119903120582 = 0 where 119901 and 119902 are polynomials withreal coefficients and 119903 gt 0 is the delay suppose
(a) 119901(120582) = 0 119877(120582) gt 0
(b) |119902(119894119910)| lt |119901(119894119910)| 0 le 119910 lt infin
(c) lim|120582|rarrinfin119877(120582)ge0|119902(120582)119901(120582)| = 0
Then 119877(120582) lt 0 for every root 120582 and all 119903 gt 0
3 Equilibrium and Stability Analysis
In this section we focus on the existence and local stability ofequilibria Let the right-hand side of equalities in model (3)be zero Then there are two equilibria namely
(i) 1198640 = (119870 0 119901) 119901 = 119870120579[119889+(1minus119886)119903+120579]) disease-freeequilibrium
(ii) 119864lowast = (119873lowast 119868lowast 119877lowast) endemic equilibrium
where the values of119873lowast 119868lowast and 119877lowast are given in Section 32
31 Community Matrix Firstly after computing the Jacobianor community matrix of model (3) at point (119873 119868 119877) thecharacteristic equation is given by
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119903 minus2119903
119870119873 minus 120582 0 0
120573119890minus1198891120591119891 (119868) minus(1 minus 119886) 119903
119870119868 119898119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868)
120579 minus(1 minus 119886) 119903
119870119877 120575119890minus(1198891+120582)120596 minus 120579 minus [119889 +
(1 minus 119886) 119903
119870119873 + 120579] minus 120582
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (6)
4 Abstract and Applied Analysis
where 119898 = 120573(119873 minus 119868 minus 119877)1198911015840(119868) 119899 = 120573119890minus1198891120591119891(119868) + 119889 + ((1 minus119886)119903119870)119873 + 1205831
Now we analyze the equilibria stability of system (3)Computing the Jacobian of system (3) evaluated at 1198640 onegets the following matrix
119869 (1198640) = (
minus119903 minus 120582 0 00 11988622 minus 120582 0
(119889 + 120579) 120579
119889 + (1 minus 119886) 119903 + 120579120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
) (7)
where
11988622 = 120573 (119870 minus 119901)1198911015840(0) 119890minus(1198891+120582)120591
minus 120575119890minus(1198891+120582)120596 minus [119889 + (1 minus 119886) 119903 + 1205831] (8)
Denote
119860 = 120573 (119870 minus 119901)1198911015840 (0) 119862 = 119889 + (1 minus 119886) 119903 + 1205831 (9)
then
11988622 = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 (10)
Denote
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 minus 120582 (11)
The eigenvalues of the system (3) about the steady state1198640 are 1205821 = minus119903 ℎ(120582) = 0 and 1205823 = minus[119889 + (1 minus 119886)119903 + 120579] Allthe parameters of the model are assumed to be nonnegativeTherefore 1205821 and 1205823 are negative Next we discuss the rootsof ℎ(120582) = 0 in five cases
Case 1 For 120591 = 120596 = 0 from the second equation of the system(3) we can get the following
Proposition 4 For 120591 = 120596 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
(119860 minus 120575) 119890minus1198891120591 lt 119862 (12)
Proof From the above analysis 1205822 satisfies the followingcharacteristic equation
119892 (120582) = (119860 minus 120575) 119890minus(1198891+120582)120591 minus 119862 minus 120582 = 0 (13)
(1) Clearly 120582 = 0 is not a root of 119892(120582) = 0(2) From the fact that 119892(0) lt 0 1198921015840(120582) lt 0 for 120582 gt 0 it is
obtained that 119892(120582) = 0 has no positive real root(3) It is sufficient to show that 119892(120582) = 0 does not admit
a purely imaginary root In fact if 120582 = 119894V (V gt 0) isa root of (119892(120582) = 0) then by separating the real partone gets
(119860 minus 120575) 119890minus1198891120591 cos (V120591) = 119862 (14)
Together with the condition of Proposition 4 we have
cos (V120591) gt 1 (15)
This is impossible
(4) It is easy to show that 119892(120582) = 0 has no imaginaryroot whose real part is positive Otherwise there isan imaginary root 120582 = 119906 + 119894V with 119906 gt 0 Without anyloss of generality we consider V gt 0Then we take thereal and imaginary parts of 119892(120582) = 0 namely
(119860 minus 120575) 119890minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (16)
Combined with (9) we have
119862 gt (119860 minus 120575) 119890minus1198891120591 gt (119860 minus 120575) 119890
minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (17)
This is a contradictionwhich implies that all eigenvalues rootsof 119892(120582) have negative real parts Therefore the disease-freeequilibrium of the system (3) is locally asymptotically stablewhen (9) holds The proof is completed
Case 2 For 120591 = 0 120596 = 0 by the same way as in Case 1 onegets the following
Proposition 5 For all 120591 = 0 120596 = 0 119877(120582) lt 0 for every root 120582of ℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575 lt 119862 (18)
Case 3 For 120596 = 0 120591 = 0 one gets the following
Proposition 6 For 120596 = 0 120591 = 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
120575119890minus1198891120596 lt |119860 minus 119862| (19)
Proof By the fact that ℎ(120582) = 0 is equivalent to 119901(120582) +119902(120582)119890minus120582120596 = 0 with 119901(120582) = 119860 minus 119862 minus 120582 119902(120582) = minus120575119890minus1198891120596
(i) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then 119901(120582) = 119860minus119862minus 119906 minus119894V = 0
(ii) By |119902(119894V)| = 120575119890minus1198891120596 |119901(119894V)| = |119860 minus 119862 minus
119894V| = radic(119860 minus 119888)2 + V2 together with the condition ofProposition 6 we know that |119902(119894V)| lt |119901(119894V)|
Abstract and Applied Analysis 5
(iii) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then
lim1199062+V2rarr+infin
10038161003816100381610038161003816100381610038161003816
119902 (120582)
119901 (120582)
10038161003816100381610038161003816100381610038161003816= 120575119890minus1198891120596
lim1199062+V2rarr+infin
1
radic(119860 minus 119862 minus 119906)2 + V2= 0
(20)
Then using Lemma 3 we have 119877(120582) lt 0 for all 120596
Case 4 For 120596 gt 120591 gt 0 let 120576 = 120596 minus 120591 Then 120596 = 120591 + 120576 Forfixed 120591
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)(120591+120576) minus 119862 minus 120582 (21)
Let 120582 = 119906 + 119894V (119906 gt 0) Then we take the real and imaginaryparts of ℎ(120582) = 0 namely
119860119890minus(1198891+119906)120591 cos (V120591) minus 120575119890minus(1198891+119906)(120591+120576) cos (V (120591 + 120576))
= 119862 + 119906
minus119860119890minus(1198891+119906)120591 sin (V120591) minus 120575119890minus(1198891+119906)(120591+120576) sin (V (120591 + 120576)) = V
(22)
Sum of squares of the above equalities is
1198602119890minus2(1198891+119906)120591 + 1205752119890minus2(1198891+119906)(120591+120576)
minus 2119860120575119890minus(1198891+119906)(2120591+120576) cos (V120576) minus (119862 + 119906)2 minus V2 = 0(23)
Then we have
120597119906
120597120576
10038161003816100381610038161003816100381610038161003816120576=0=
120575 (119906 + 1198891) (119860 minus 120575)
(119860 minus 120575)2120591 + (119862 + 119906) 1198902120591(119906+120575) (24)
Obviously 120597119906120597120576|120576=0 lt 0 when 119860 minus 120575 lt 0 Combined with(119860 minus 120575)119890minus1198891120591 lt 119860119890minus1198891120591 minus 120575119890minus1198891120596 we have the following
Proposition 7 For 120596 gt 120591 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1 (25)
Case 5 For 0 lt 120591 lt 120596 in the same way as in Case 4 we havethe following
Proposition 8 For 0 lt 120591 lt 120596 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1 (26)
From what has been discussed above we get the follow-ing
Theorem 9 The disease-free equilibrium of the system (3) islocally asymptotically stable if one of the following conditionsholds
(a) 120591 = 120596 = 0 (119860 minus 120575)119890minus1198891120591 lt 119862(b) 120591 = 0 120596 = 0 119860119890minus1198891120591 minus 120575 lt 119862(c) 120591 = 0 120596 = 0 120575119890minus1198891120596 lt |119860 minus 119862|(d) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1(e) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1
32 Existence of Endemic Equilibrium Thus by Theorem 9we may define the basic reproduction number as
1198770 =119860119890minus1198891120591 minus 120575119890minus1198891120596
119862 (27)
This threshold 1198770 defines the average number of sec-ondary infections generated by a typical infectious individualin a completely susceptible population in a steady demo-graphic state
In Theorem 9 we have already shown that the system(3) has an infection-free steady state which is locally asymp-totically stable under condition 1198770 lt 1 The disease-freeequilibrium is unstable when 1198770 gt 1 and the system (3) hasa nontrivial endemic equilibrium 119864lowast = (119873lowast 119868lowast 119877lowast) when1198770 gt 1 From (3)
119873lowast = 119870 gt 0
119877lowast =120575119890minus1198891120596 minus 120579
119889 + (1 minus 119886) 119903 + 120579119868lowast
+119870120579
119889 + (1 minus 119886) 119903 + 120579≐ 119902119868lowast + 119901
(28)
where 119902 = (119890minus1198891120596120575 minus 120579)(119889 + (1 minus 119886)119903 + 120579) Substituting thesevalues of119873lowast and 119877lowast in the second equation of (3) we get thefollowing equation for 119868
119866 (119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901) 119891 (119868) minus [119862 + 120575119890
minus1198891120596] 119868
(29)
Obviously 119868 = 0 is one of the roots of (29) as 119891(0) = 0Therefore to exclude that root choose
119867(119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901)
119891 (119868)
119868minus [119862 + 120575119890minus1198891120596]
(30)
It can easily be seen that the function119867(119868) is negative forlarge positive 119868 that is
119867(119870) = minus120573119890minus1198891120591 (119870119902 + 119901)
119891 (119896)
119870minus [119862 + 120575119890minus1198891120596] lt 0 (31)
Next we determine the sign of its derivative
1198671015840 (119868) = 120573119890minus1198891120591 (119870 minus 119901)
1198911015840 (119868) 119868 minus 119891 (119868)
1198682
minus 120573119890minus1198891120591 (1 + 119902) 1198911015840 (119868)
(32)
It can easily be seen that 119870 gt 119901 In addition from theproperties of the function 119891(119868) in particular from 119891(0) = 0and 11989110158401015840(0) lt 0 it follows that 119891(119868) minus 1198911015840(119868)119868 gt 0 andconsequently1198671015840(119868) lt 0 for all 119868 gt 0 Therefore for a positiveroot of119867(119868) = 0 to exist119867(119868) has to satisfy119867(0) gt 0 that is
119867(0) = 119860119890minus1198891120591 minus 120575119890minus1198891120596 minus 119862
= (119860119890minus1198891120591 minus 119890minus1198891120596120575
119862minus 1)119862
= (1198770 minus 1)119862
(33)
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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4 Abstract and Applied Analysis
where 119898 = 120573(119873 minus 119868 minus 119877)1198911015840(119868) 119899 = 120573119890minus1198891120591119891(119868) + 119889 + ((1 minus119886)119903119870)119873 + 1205831
Now we analyze the equilibria stability of system (3)Computing the Jacobian of system (3) evaluated at 1198640 onegets the following matrix
119869 (1198640) = (
minus119903 minus 120582 0 00 11988622 minus 120582 0
(119889 + 120579) 120579
119889 + (1 minus 119886) 119903 + 120579120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
) (7)
where
11988622 = 120573 (119870 minus 119901)1198911015840(0) 119890minus(1198891+120582)120591
minus 120575119890minus(1198891+120582)120596 minus [119889 + (1 minus 119886) 119903 + 1205831] (8)
Denote
119860 = 120573 (119870 minus 119901)1198911015840 (0) 119862 = 119889 + (1 minus 119886) 119903 + 1205831 (9)
then
11988622 = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 (10)
Denote
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)120596 minus 119862 minus 120582 (11)
The eigenvalues of the system (3) about the steady state1198640 are 1205821 = minus119903 ℎ(120582) = 0 and 1205823 = minus[119889 + (1 minus 119886)119903 + 120579] Allthe parameters of the model are assumed to be nonnegativeTherefore 1205821 and 1205823 are negative Next we discuss the rootsof ℎ(120582) = 0 in five cases
Case 1 For 120591 = 120596 = 0 from the second equation of the system(3) we can get the following
Proposition 4 For 120591 = 120596 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
(119860 minus 120575) 119890minus1198891120591 lt 119862 (12)
Proof From the above analysis 1205822 satisfies the followingcharacteristic equation
119892 (120582) = (119860 minus 120575) 119890minus(1198891+120582)120591 minus 119862 minus 120582 = 0 (13)
(1) Clearly 120582 = 0 is not a root of 119892(120582) = 0(2) From the fact that 119892(0) lt 0 1198921015840(120582) lt 0 for 120582 gt 0 it is
obtained that 119892(120582) = 0 has no positive real root(3) It is sufficient to show that 119892(120582) = 0 does not admit
a purely imaginary root In fact if 120582 = 119894V (V gt 0) isa root of (119892(120582) = 0) then by separating the real partone gets
(119860 minus 120575) 119890minus1198891120591 cos (V120591) = 119862 (14)
Together with the condition of Proposition 4 we have
cos (V120591) gt 1 (15)
This is impossible
(4) It is easy to show that 119892(120582) = 0 has no imaginaryroot whose real part is positive Otherwise there isan imaginary root 120582 = 119906 + 119894V with 119906 gt 0 Without anyloss of generality we consider V gt 0Then we take thereal and imaginary parts of 119892(120582) = 0 namely
(119860 minus 120575) 119890minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (16)
Combined with (9) we have
119862 gt (119860 minus 120575) 119890minus1198891120591 gt (119860 minus 120575) 119890
minus(1198891+119906)120591 cos (V120591) = 119862 + 119906 (17)
This is a contradictionwhich implies that all eigenvalues rootsof 119892(120582) have negative real parts Therefore the disease-freeequilibrium of the system (3) is locally asymptotically stablewhen (9) holds The proof is completed
Case 2 For 120591 = 0 120596 = 0 by the same way as in Case 1 onegets the following
Proposition 5 For all 120591 = 0 120596 = 0 119877(120582) lt 0 for every root 120582of ℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575 lt 119862 (18)
Case 3 For 120596 = 0 120591 = 0 one gets the following
Proposition 6 For 120596 = 0 120591 = 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
120575119890minus1198891120596 lt |119860 minus 119862| (19)
Proof By the fact that ℎ(120582) = 0 is equivalent to 119901(120582) +119902(120582)119890minus120582120596 = 0 with 119901(120582) = 119860 minus 119862 minus 120582 119902(120582) = minus120575119890minus1198891120596
(i) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then 119901(120582) = 119860minus119862minus 119906 minus119894V = 0
(ii) By |119902(119894V)| = 120575119890minus1198891120596 |119901(119894V)| = |119860 minus 119862 minus
119894V| = radic(119860 minus 119888)2 + V2 together with the condition ofProposition 6 we know that |119902(119894V)| lt |119901(119894V)|
Abstract and Applied Analysis 5
(iii) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then
lim1199062+V2rarr+infin
10038161003816100381610038161003816100381610038161003816
119902 (120582)
119901 (120582)
10038161003816100381610038161003816100381610038161003816= 120575119890minus1198891120596
lim1199062+V2rarr+infin
1
radic(119860 minus 119862 minus 119906)2 + V2= 0
(20)
Then using Lemma 3 we have 119877(120582) lt 0 for all 120596
Case 4 For 120596 gt 120591 gt 0 let 120576 = 120596 minus 120591 Then 120596 = 120591 + 120576 Forfixed 120591
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)(120591+120576) minus 119862 minus 120582 (21)
Let 120582 = 119906 + 119894V (119906 gt 0) Then we take the real and imaginaryparts of ℎ(120582) = 0 namely
119860119890minus(1198891+119906)120591 cos (V120591) minus 120575119890minus(1198891+119906)(120591+120576) cos (V (120591 + 120576))
= 119862 + 119906
minus119860119890minus(1198891+119906)120591 sin (V120591) minus 120575119890minus(1198891+119906)(120591+120576) sin (V (120591 + 120576)) = V
(22)
Sum of squares of the above equalities is
1198602119890minus2(1198891+119906)120591 + 1205752119890minus2(1198891+119906)(120591+120576)
minus 2119860120575119890minus(1198891+119906)(2120591+120576) cos (V120576) minus (119862 + 119906)2 minus V2 = 0(23)
Then we have
120597119906
120597120576
10038161003816100381610038161003816100381610038161003816120576=0=
120575 (119906 + 1198891) (119860 minus 120575)
(119860 minus 120575)2120591 + (119862 + 119906) 1198902120591(119906+120575) (24)
Obviously 120597119906120597120576|120576=0 lt 0 when 119860 minus 120575 lt 0 Combined with(119860 minus 120575)119890minus1198891120591 lt 119860119890minus1198891120591 minus 120575119890minus1198891120596 we have the following
Proposition 7 For 120596 gt 120591 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1 (25)
Case 5 For 0 lt 120591 lt 120596 in the same way as in Case 4 we havethe following
Proposition 8 For 0 lt 120591 lt 120596 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1 (26)
From what has been discussed above we get the follow-ing
Theorem 9 The disease-free equilibrium of the system (3) islocally asymptotically stable if one of the following conditionsholds
(a) 120591 = 120596 = 0 (119860 minus 120575)119890minus1198891120591 lt 119862(b) 120591 = 0 120596 = 0 119860119890minus1198891120591 minus 120575 lt 119862(c) 120591 = 0 120596 = 0 120575119890minus1198891120596 lt |119860 minus 119862|(d) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1(e) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1
32 Existence of Endemic Equilibrium Thus by Theorem 9we may define the basic reproduction number as
1198770 =119860119890minus1198891120591 minus 120575119890minus1198891120596
119862 (27)
This threshold 1198770 defines the average number of sec-ondary infections generated by a typical infectious individualin a completely susceptible population in a steady demo-graphic state
In Theorem 9 we have already shown that the system(3) has an infection-free steady state which is locally asymp-totically stable under condition 1198770 lt 1 The disease-freeequilibrium is unstable when 1198770 gt 1 and the system (3) hasa nontrivial endemic equilibrium 119864lowast = (119873lowast 119868lowast 119877lowast) when1198770 gt 1 From (3)
119873lowast = 119870 gt 0
119877lowast =120575119890minus1198891120596 minus 120579
119889 + (1 minus 119886) 119903 + 120579119868lowast
+119870120579
119889 + (1 minus 119886) 119903 + 120579≐ 119902119868lowast + 119901
(28)
where 119902 = (119890minus1198891120596120575 minus 120579)(119889 + (1 minus 119886)119903 + 120579) Substituting thesevalues of119873lowast and 119877lowast in the second equation of (3) we get thefollowing equation for 119868
119866 (119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901) 119891 (119868) minus [119862 + 120575119890
minus1198891120596] 119868
(29)
Obviously 119868 = 0 is one of the roots of (29) as 119891(0) = 0Therefore to exclude that root choose
119867(119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901)
119891 (119868)
119868minus [119862 + 120575119890minus1198891120596]
(30)
It can easily be seen that the function119867(119868) is negative forlarge positive 119868 that is
119867(119870) = minus120573119890minus1198891120591 (119870119902 + 119901)
119891 (119896)
119870minus [119862 + 120575119890minus1198891120596] lt 0 (31)
Next we determine the sign of its derivative
1198671015840 (119868) = 120573119890minus1198891120591 (119870 minus 119901)
1198911015840 (119868) 119868 minus 119891 (119868)
1198682
minus 120573119890minus1198891120591 (1 + 119902) 1198911015840 (119868)
(32)
It can easily be seen that 119870 gt 119901 In addition from theproperties of the function 119891(119868) in particular from 119891(0) = 0and 11989110158401015840(0) lt 0 it follows that 119891(119868) minus 1198911015840(119868)119868 gt 0 andconsequently1198671015840(119868) lt 0 for all 119868 gt 0 Therefore for a positiveroot of119867(119868) = 0 to exist119867(119868) has to satisfy119867(0) gt 0 that is
119867(0) = 119860119890minus1198891120591 minus 120575119890minus1198891120596 minus 119862
= (119860119890minus1198891120591 minus 119890minus1198891120596120575
119862minus 1)119862
= (1198770 minus 1)119862
(33)
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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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
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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
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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
(iii) Suppose 120582 = 119906 + 119894V (119906 gt 0) Then
lim1199062+V2rarr+infin
10038161003816100381610038161003816100381610038161003816
119902 (120582)
119901 (120582)
10038161003816100381610038161003816100381610038161003816= 120575119890minus1198891120596
lim1199062+V2rarr+infin
1
radic(119860 minus 119862 minus 119906)2 + V2= 0
(20)
Then using Lemma 3 we have 119877(120582) lt 0 for all 120596
Case 4 For 120596 gt 120591 gt 0 let 120576 = 120596 minus 120591 Then 120596 = 120591 + 120576 Forfixed 120591
ℎ (120582) = 119860119890minus(1198891+120582)120591 minus 120575119890minus(1198891+120582)(120591+120576) minus 119862 minus 120582 (21)
Let 120582 = 119906 + 119894V (119906 gt 0) Then we take the real and imaginaryparts of ℎ(120582) = 0 namely
119860119890minus(1198891+119906)120591 cos (V120591) minus 120575119890minus(1198891+119906)(120591+120576) cos (V (120591 + 120576))
= 119862 + 119906
minus119860119890minus(1198891+119906)120591 sin (V120591) minus 120575119890minus(1198891+119906)(120591+120576) sin (V (120591 + 120576)) = V
(22)
Sum of squares of the above equalities is
1198602119890minus2(1198891+119906)120591 + 1205752119890minus2(1198891+119906)(120591+120576)
minus 2119860120575119890minus(1198891+119906)(2120591+120576) cos (V120576) minus (119862 + 119906)2 minus V2 = 0(23)
Then we have
120597119906
120597120576
10038161003816100381610038161003816100381610038161003816120576=0=
120575 (119906 + 1198891) (119860 minus 120575)
(119860 minus 120575)2120591 + (119862 + 119906) 1198902120591(119906+120575) (24)
Obviously 120597119906120597120576|120576=0 lt 0 when 119860 minus 120575 lt 0 Combined with(119860 minus 120575)119890minus1198891120591 lt 119860119890minus1198891120591 minus 120575119890minus1198891120596 we have the following
Proposition 7 For 120596 gt 120591 gt 0 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1 (25)
Case 5 For 0 lt 120591 lt 120596 in the same way as in Case 4 we havethe following
Proposition 8 For 0 lt 120591 lt 120596 119877(120582) lt 0 for every root 120582 ofℎ(120582) = 0 when
119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1 (26)
From what has been discussed above we get the follow-ing
Theorem 9 The disease-free equilibrium of the system (3) islocally asymptotically stable if one of the following conditionsholds
(a) 120591 = 120596 = 0 (119860 minus 120575)119890minus1198891120591 lt 119862(b) 120591 = 0 120596 = 0 119860119890minus1198891120591 minus 120575 lt 119862(c) 120591 = 0 120596 = 0 120575119890minus1198891120596 lt |119860 minus 119862|(d) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 lt 0 0 lt 120596 minus 120591 ≪ 1(e) 119860119890minus1198891120591 minus 120575119890minus1198891120596 lt 119862 119860 minus 120575 gt 0 0 lt 120591 minus 120596 ≪ 1
32 Existence of Endemic Equilibrium Thus by Theorem 9we may define the basic reproduction number as
1198770 =119860119890minus1198891120591 minus 120575119890minus1198891120596
119862 (27)
This threshold 1198770 defines the average number of sec-ondary infections generated by a typical infectious individualin a completely susceptible population in a steady demo-graphic state
In Theorem 9 we have already shown that the system(3) has an infection-free steady state which is locally asymp-totically stable under condition 1198770 lt 1 The disease-freeequilibrium is unstable when 1198770 gt 1 and the system (3) hasa nontrivial endemic equilibrium 119864lowast = (119873lowast 119868lowast 119877lowast) when1198770 gt 1 From (3)
119873lowast = 119870 gt 0
119877lowast =120575119890minus1198891120596 minus 120579
119889 + (1 minus 119886) 119903 + 120579119868lowast
+119870120579
119889 + (1 minus 119886) 119903 + 120579≐ 119902119868lowast + 119901
(28)
where 119902 = (119890minus1198891120596120575 minus 120579)(119889 + (1 minus 119886)119903 + 120579) Substituting thesevalues of119873lowast and 119877lowast in the second equation of (3) we get thefollowing equation for 119868
119866 (119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901) 119891 (119868) minus [119862 + 120575119890
minus1198891120596] 119868
(29)
Obviously 119868 = 0 is one of the roots of (29) as 119891(0) = 0Therefore to exclude that root choose
119867(119868) = 120573119890minus1198891120591 (119870 minus (1 + 119902) 119868 minus 119901)
119891 (119868)
119868minus [119862 + 120575119890minus1198891120596]
(30)
It can easily be seen that the function119867(119868) is negative forlarge positive 119868 that is
119867(119870) = minus120573119890minus1198891120591 (119870119902 + 119901)
119891 (119896)
119870minus [119862 + 120575119890minus1198891120596] lt 0 (31)
Next we determine the sign of its derivative
1198671015840 (119868) = 120573119890minus1198891120591 (119870 minus 119901)
1198911015840 (119868) 119868 minus 119891 (119868)
1198682
minus 120573119890minus1198891120591 (1 + 119902) 1198911015840 (119868)
(32)
It can easily be seen that 119870 gt 119901 In addition from theproperties of the function 119891(119868) in particular from 119891(0) = 0and 11989110158401015840(0) lt 0 it follows that 119891(119868) minus 1198911015840(119868)119868 gt 0 andconsequently1198671015840(119868) lt 0 for all 119868 gt 0 Therefore for a positiveroot of119867(119868) = 0 to exist119867(119868) has to satisfy119867(0) gt 0 that is
119867(0) = 119860119890minus1198891120591 minus 120575119890minus1198891120596 minus 119862
= (119860119890minus1198891120591 minus 119890minus1198891120596120575
119862minus 1)119862
= (1198770 minus 1)119862
(33)
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Hence one needs the requirement that 1198770 gt 1 to ensurethe existence of the endemic equilibrium From the aboveanalysis we have the following theorem
Theorem 10 The system (3) has a nontrivial endemic equilib-rium 119864lowast = (119873lowast 119868lowast 119877lowast) when 1198770 gt 1
33 Local Stability of the Endemic Equilibrium In this sec-tion we analyze the local stability of the endemic equilibrium119864lowast for 120591 = 120596 Its characteristic equation is given by
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus119903 minus 120582 0 0
120573119890minus1198891120591119891 (119868lowast) minus(1 minus 119886) 119903119868lowast
119870119898119890minus(1198891+120582)120591 minus 119899 minus 120582 120573119890minus1198891120591119891 (119868lowast)
120579 minus(1 minus 119886) 119903119877lowast
119870120575119890minus(1198891+120582)120591 minus 120579 minus [119889 + (1 minus 119886) 119903 + 120579] minus 120582
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0 (34)
where119898 = 120573(119870 minus 119868lowast minus 119877lowast)1198911015840(119868lowast) minus 120575 119899 = 120573119890minus1198891120591119891(119868lowast) + 119889 +(1 minus 119886)119903 + 1205831
The Jacobin matrix leads to the characteristic equation
(120582 + 119903) [1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890
minus120582120591] = 0 (35)
where
1198981 = 120573119890minus1198891120591119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 gt 0
1198980 = [120573119890minus1198891120591119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831]
times (119889 + (1 minus 119886) 119903 + 120579) + (119889 + (1 minus 119886) 119903 + 1205831) 120579 gt 0
1198991 = minus [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] 119890minus1198891120591
1198990 = [1205731198911015840 (119868lowast) (119870 minus 119868lowast minus 119877lowast) minus 120575] [119889 + (1 minus 119886) 119903 + 120579]
times 119890minus1198891120591 + 120575119891 (119868lowast) 119890minus21198891120591
(36)
Since all the model parameters are assumed to be nonnega-tive it follows that one eigenvalue is negative that is 1205821 = minus119903Thus the stability of 119864lowast depends on the roots of the quasi-polynomial
1205822 + 1198981120582 + 1198980 + (1198991120582 + 1198990) 119890minus120582120591 = 0 (37)
We note that 1198981 gt 0 and 1198980 gt 0 whereas 1198991 and 1198990 may bepositive or negative For 120591 = 0 we state the following resultsthat follow directly from (39) The endemic steady state islocally asymptotically stable if the following conditions hold
120573119891 (119868lowast) + 2119889 + 2 (1 minus 119886) 119903 + 120579 + 1205831 + 1205751205731198911015840 (119868lowast)
gt 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast) (1198671)
[120573119891 (119868lowast) + 119889 + (1 minus 119886) 119903 + 1205831 + 120575 minus 120573 (119870 minus 119868lowast minus 119877lowast) 1198911015840 (119868lowast)]
times [119889 + (1 minus 119886) 119903 + 120579] + 119891 (119868lowast) 120575 gt 0
(1198672)
The main purpose of this paper is to study the stabilitybehavior of 119864lowast in the case 120591 = 0 Obviously 119894120578 (120578 gt 0) is theroot of (29) if and only if 120578 satisfies
minus1205782 + 1198981119894120578 + 1198980 = minus (1198991119894120578 + 1198990) (cos 120578120591 minus 119894 sin 120578120591) (38)
Separating the real and imaginary parts we have
minus1205782 + 1198980 = minus1198990 cos 120578120591 minus 1198991120578 sin 120578120591 (39)
1198981120578 = minus1198991120578 cos 120578120591 + 1198990 sin 120578120591 (40)
Eliminating 120591 by squaring and adding (39) and (40) weobtain a polynomial in 120578 as
1205784 + (11989821minus 11989921minus 21198980) 120578
2 + 11989820minus 11989920= 0 (41)
Suppose that the conditions
11989821gt 11989921+ 21198980 1198982
0gt 11989920
(1198673)
hold for all 120591 ge 0Then the infected steady state of the system(3) is locally asymptotically stable
Theorem 11 For 120591 = 120596 if 1198770 gt 1 then the endemicequilibrium of the system (3) is locally asymptotically stablewhen conditions (1198671)ndash(1198673) hold
Corollary 12 For 120591 = 120596 if 1205831 lt 120583lowast1or 120579 lt 120579lowast then the
endemic equilibrium of the system (3) is locally asymptoticallystable when conditions (1198671)ndash(1198673) hold
4 Permanence
In this section we investigate a permanence result [5] Thefollowing is our main result of this paper We will give thefollowing result by using some techniques given in [8 11]The proof of the permanence with nonlinear incidence is adaunting task Consequently for simplicity andmathematicalconvenience let us choose a linear incidence rate 119891(119868) =119868 The result holds with the nonlinear incidence as shownnumerically but the algebraic proof is long and tedious andthe conditions to impose on some of the parameters may bevery restrictive Now let us firstly give the following theorem
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Theorem 13 If 1198770 gt 1 holds then the system (1) with 120591 = 120596is permanent that is there are positive constants 119888119894 (119894 = 1 2 3)such that
1198881 lt lim119905rarrinfin
inf 119878 (119905) le lim119905rarrinfin
sup 119878 (119905) le 119870
1198882 lt lim119905rarrinfin
inf 119868 (119905) le lim119905rarrinfin
sup 119868 (119905) le 119870
1198883 lt lim119905rarrinfin
inf 119877 (119905) le lim119905rarrinfin
sup119877 (119905) le 119870
(42)
hold for any solution of (1) with (1206011(120579) 1206012(120579) 1206013(120579)) in theinterior of Φ for all 120579 isin [minus120591 0] In fact 119888119894 (119894 = 1 2 3) canbe chosen explicitly as
1198881 =(119887 minus 119886119903)119870
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579
1198882 = 119868lowast119890minus(119889+(1minus119886)119903+1205831+119890
minus1198891120591120575)
1198883 =120575119890minus11988911205911198882 + 1205791198881119889 + (1 minus 119886) 119903
(43)
Proof Note that 0 lt 119873(119905) lt 119870 for all 119905 ge 0 and thatlim119905rarrinfin119873(119905) = 119870 It is easy to see that lim119905rarrinfin inf 119878(119905) ge 1198881In fact let 120598 lt 119870 be arbitrary Choose 1198791 gt 120591 so large that119873(119905) gt 119870 minus 120598 for 119905 gt 1198791 We have the following inequality
119878 (119905) gt minus [120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905)
+ (119887 minus 119886119903) (119870 minus 120598) (44)
for all 119905 ge 1198791 which implies that
lim inf119905rarrinfin
119878 (119905) ge(119887 minus 119886119903) (119870 minus 120598)
120573119870119890minus1198891120591 + 119889 + (1 minus 119886) 119903 + 120579 (45)
Note that 120598may be arbitrarily small so that lim119905rarrinfin inf 119878(119905) ge1198881
Next we will show lim119905rarrinfin inf 119868(119905) ge 1198882 For any 120585 0 lt 120585 lt 1 we see the inequality 119878lowast lt [(119887 minus 119886119903)119870 +
1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579) There exist sufficiently
large 120588 ge 1 and sufficiently small 120598 such that 119878lowast lt [(119887 minus
119886119903)(119870 minus 120598) + 1205831119868lowast](120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886)119903 + 120579)(1 minus
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)120588120591) equiv 119878Δ We show that 119868(1199050) gt 119902119868
lowast forsome 1199050 ge 120588120591 In fact if not it follows from the first equationof (1) that for all 119905 ge 120588120591 + 120591 ge 1198791 + 120591
119878 (119905) ge (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast
minus [120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579] 119878 (119905) (46)
Hence for 119905 ge 120588120591 + 120591
119878 (119905) ge 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)
times [119878 (120588120591 + 120591) + (119887 minus 119886119903) (119870 minus 120598) + 1205831119868lowast]
times int119905
120588120591+120591
119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591)119889120579
gt[(119887 minus 119886119903) (119870 minus 120598) + 1205831119868
lowast]
120573119890minus1198891120591120585119868lowast + 119889 + (1 minus 119886) 119903 + 120579
times (1 minus 119890minus(120573119890minus1198891120591120585119868
lowast+119889+(1minus119886)119903+120579)(119905minus120588120591minus120591))
(47)
which gives us for 119905 ge 2120588120591 + 120591
119878 (119905) gt 119878Δ gt 119878lowast (48)
For 119905 ge 0 we define a positive differentiable function 119881(119905) asfollows
119881 (119905) = 119868 (119905) +[120573 (119870 minus 119901) minus 120575] 119890minus1198891120591
1198770int119905
119905minus120591
119868 (119904) 119889119904 (49)
We obtain the inequality for 119905 ge 2120588120591 + 120591
(119905) = [120573119890minus1198891120591 (119878 (119905) minus 119878
lowast) 119868 (119905 minus 120591) + (1 minus 119886) 119903]
times (1 minus119873 (119905)
119870) 119868 (119905)
gt 120573119890minus1198891120591 (119878 (119905) minus 119878lowast) 119868 (119905 minus 120591)
gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119868 (119905 minus 120591)
(50)
Let 119894 = min120579isin[minus1205910]119868(2120588120591 + 2120591 + 120579) Now let us show that119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 In fact if there exists 1198792 ge 0 suchthat 119868(119905) ge 119894 for 2120588120591+120591 le 119905 le 2120588120591+2120591+1198792 119868(2120588120591+2120591+1198792) = 119894and 119868(2120588120591 + 2120591 +1198792) le 0 Direct calculation using the secondequation of (1) and (48) gives
119868 (2120588120591 + 2120591 + 1198792)
gt [120573119890minus1198891120591 (119878 (2120588120591 + 2120591 + 1198792)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 119894
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 119894 gt 0
(51)
This contradicts the definition of 1198792 Thus we have shownthat 119868(119905) ge 119894 for all 119905 ge 2120588120591 + 120591 Hence for all 119905 ge 2120588120591 + 2120591
(119905) gt 120573119890minus1198891120591 (119878Δ minus 119878lowast) 119894 (52)
which implies that 119881(119905) rarr +infin as 119905 rarr +infin Thiscontradicts the boundedness of 119881(119905) Consequently 119868(1199050) gt120585119868lowast for some 1199050 ge 120588120591
In the rest we now need to consider two cases
(i) 119868(119905) ge 120585119868lowast for all large 119905(ii) 119868(119905) oscillates about 120585119868lowast for all large 119905
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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
8 Abstract and Applied Analysis
Wenowneed to show that 119868(119905) ge 1205851198882 for large 119905 Obviouslyit suffices to show that it holds only for case (ii) We supposethat for any large 119879 there exists 1199051 1199052 gt 119879 such that 119868(1199051) =119868(1199052) = 120585119868
lowast and 119868(119905) lt 120585119868lowast for 1199051 lt 119905 lt 1199052 If 1199052 minus 1199051 le 120591 thesecond equation (1) gives us 119868(119905) gt minus(119889 + (1 minus 119886)119903 + 1205831)119868(119905)which implies that 119868(119905) gt 119868(1199051)119890
minus(119889+(1minus119886)119903+1205831)(119905minus1199051) on (1199051 1199052)
Thus 119868(119905) gt 1205851198882 On the other hand if 1199052 minus 1199051 gt 120591 applyingthe same manner gives 119868(119905) ge 1205851198882 on [1199051 1199051 + 120591] and hencethe remaining work is to show 119868(119905) ge 1205851198882 on [1199051 + 120591 1199052] Infact assuming that there exists 1198793 gt 0 such that 119868(119905) ge 1205851198882 on[1199051 1199051 + 120591 + 1198793] 119868(1199051 + 120591 + 1198793) = 1205851198882 and 119868(1199051 + 120591 + 1198793) le 0 itfollows from (1) that
119868 (1199051 + 120591 + 1198793)
ge [120573119890minus1198891120591119878 (1199051 + 120591 + 1198793)
minus (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575)] 1205851198882
gt (119889 + (1 minus 119886) 119903 + 1205831 + 119890minus1198891120591120575) [
119878Δ
119878lowastminus 1] 1205851198882 gt 0
(53)
This contradicts the definition of 1198793 Hence 119868(119905) ge 1205851198882 on[1199051 1199052] Consequently 119868(119905) ge 1205851198882 for large 119905 in the case (ii)Therefore lim119905rarrinfin inf 119868(119905) ge 1205851198882 Note that 119902may be so closeto 1 that lim119905rarrinfin inf 119868(119905) ge 1198882
Finally let us show that lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911198882 +1205791198881)(119889 + (1 minus 119886)119903) The third equation gives us
(119905) ge [120575119890minus1198891120591119868 + 120579119878 minus [119889 + (1 minus 119886) 119903] 119877
ge [120575119890minus11988911205911205851198882 + 1205791205851198881 minus [119889 + (1 minus 119886) 119903] 119877(54)
for large 119905 Hence lim119905rarrinfin inf 119877(119905) ge (120575119890minus11988911205911205851198882 +1205791205851198881)(119889 + (1 minus 119886)119903) In a similar manner we could showlim119905rarrinfin inf 119877(119905) ge 1198883 This proves the theorem
Corollary 14 If 1205831 lt 120583lowast1 and 120579 lt 120579lowast then the system (1) with
120591 = 120596 is permanent
5 Numerical Analysis
Since it is important to visualize the dynamical behavior ofthe model the model system (3) is integrated numericallywith the help of MATLAB 70 using the following set ofparameters
(1) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 08 120579 = 001 and 120591 = 5 It iseasy to compute that 1198640 = (8 0 02) and 1198770 = 094 lt 1 InFigure 1 the infective population and recovered populationrespectively are plotted against the total population We seefrom the figure that for any initial start the solution curvestend to the equilibrium 1198640 Hence we infer that the system(3) may be stable about the disease-free equilibrium point 1198640which satisfies Theorem 9
(2) Let 119903 = 05 119896 = 8 119889 = 004 1198891 = 004 120573 = 1119886 = 03 120575 = 02 1205831 = 02 120579 = 002 and 120591 = 5 Weget 119864lowast = (8 223 111) and this set of parameter valuessatisfies the local asymptotic stability conditions of 119864lowast It is
34 5 6 7
8
0
05
10
02
04
06
08
1
The disease-free equilibrium
R(t)
I(t) N(t)
Figure 1 The disease-free equilibrium 1198640 is locally asymptoticallystable Variation of infective population 119868(119905) and recovered popula-tion 119877(119905) with total population119873(119905)
4 5 6 78
9
01
230
05
1
15
2
25
The endemic equilibrium
R(t)
I(t)N(t)
Figure 2The endemic equilibrium 119864lowast is locally asymptotically sta-ble Variation of infective population 119868(119905) and recovered population119877(119905) with total population119873(119905)
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
1 R(t)2 R(t)
3 I(t)
4 I(t)
1205831 = 02
1205831 = 04
1205831 = 02
1205831 = 04
Figure 3 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 1205831
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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
Abstract and Applied Analysis 9
0 50 100 150 200 2500
05
1
15
2
25
Infe
ctiv
esI(t)
and
reco
vere
d in
divi
dual
sR(t)
Time (t)
120579 = 003
120579 = 002
120579 = 003
120579 = 002
1 R(t)2 R(t)
3 I(t)
4 I(t)
Figure 4 Variation of infective population 119868(119905) and recoveredpopulation 119877(119905) with time for different values of 120579
easy to verify that 1198770 = 183 gt 1 and all other conditions ofTheorem 11 are satisfied So we can obtain from Figure 2 thatthe system (3) is stable at the endemic equilibrium point 119864lowast
(3) The results of numerical simulation are displayedgraphically in Figures 3 and 4 In Figure 3 the variationof the infective population and recovered population isshown with time for different values of the removal rateconstant from groups 119868 to 119878 1205831 It is found that both theinfective population and the recovered population decreaseas 1205831 increases Figure 4 depicts the variation of infectivepopulation and recovered population respectively with timefor the different successive vaccination rate 120579 As 120579 increasesthe infective population decreases whereas the recoveredpopulation increases
6 Discussion
In this paper we will consider two different delays whichare important parameters on the dynamic behavior So thepresent study is continuation of the previous work by [43]Furthermore from biological epidemic point of view weinvestigate successive vaccination and difference in immunityin our system From mathematical point of view we studythe stability of disease-free equilibrium and the existenceof endemic equilibrium for different delay and consider thepermanence of the system in the new paper
In Theorems 9 10 11 and 13 corresponding to theircorollaries when the effect of the successive vaccinationrate and the transfer rate from the infectious group to thesusceptible group after treatment is strong that is 120579 gt 120579lowast
and 1205831 gt 120583lowast1 the basic reproduction number 1198770 being
unity is a strict threshold for the control of the diseasethe disease will be extinct or otherwise will tend to breakout and persist The other results are displayed graphicallyfrom our numerical simulation We show the variation of
the infective population and recovered population with timefor different values of 1205831 It is found that both the infectivepopulation and the recovered population decrease as 1205831increases The infective population decreases whereas therecovered population increases as the successive vaccinationrate increases 120579 respectively
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is supported by National Natural Science Founda-tion of China (11101305)
References
[1] S RuanWWang and S A Levin ldquoThe effect of global travel onthe spread of SARSrdquoMathematical Biosciences and Engineeringvol 3 no 1 pp 205ndash218 2006
[2] WWang and S Ruan ldquoSimulating the SARS outbreak in Beijingwith limited datardquo Journal of Theoretical Biology vol 227 no 3pp 369ndash379 2004
[3] S M Moghadas and A B Gumel ldquoGlobal stability of a two-stage epidemic model with generalized non-linear incidencerdquoMathematics and Computers in Simulation vol 60 no 1-2 pp107ndash118 2002
[4] X Zhou X Song and X Shi ldquoAnalysis of stability and Hopfbifurcation for anHIV infectionmodelwith time delayrdquoAppliedMathematics and Computation vol 199 no 1 pp 23ndash38 2008
[5] A Tripathi R Naresh and D Sharma ldquoModelling the effect ofscreening of unaware infectives on the spread of HIV infectionrdquoAppliedMathematics and Computation vol 184 no 2 pp 1053ndash1068 2007
[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007
[7] X Meng and L Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[8] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[9] F H Chen ldquoA susceptible-infected epidemicmodel with volun-tary vaccinationsrdquo Journal of Mathematical Biology vol 53 no2 pp 253ndash272 2006
[10] I A Moneim and D Greenhalgh ldquoThreshold and stabilityresults for an SIRS epidemic model with a general periodicvaccination strategyrdquo Journal of Biological Systems vol 13 no2 pp 131ndash150 2005
[11] E Shim Z Feng M Martcheva and C Castillo-Chavez ldquoAnage-structured epidemic model of rotavirus with vaccinationrdquoJournal of Mathematical Biology vol 53 no 4 pp 719ndash7462006
[12] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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
10 Abstract and Applied Analysis
[13] Y Pei S Liu C Li and L Chen ldquoThe dynamics of an impulsivedelay SImodel with variable coefficientsrdquoAppliedMathematicalModelling vol 33 no 6 pp 2766ndash2776 2009
[14] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[15] F Brauer ldquoEpidemic models with heterogeneous mixing andtreatmentrdquo Bulletin of Mathematical Biology vol 70 no 7 pp1869ndash1885 2008
[16] S Gao L Chen and L Sun ldquoDynamic complexities in aseasonal prevention epidemic model with birth pulsesrdquo ChaosSolitons amp Fractals vol 26 no 4 pp 1171ndash1181 2005
[17] Y Pei and S Liu ldquoPulse vaccination strategy in a delayedSIRS epidemic modelrdquo in Proceedings of the 6th Conference ofBiomathematics Advanced in Biomathematics vol 2 pp 775ndash778 2008
[18] T Zhang and Z Teng ldquoPermanence and extinction for anonautonomous SIRS epidemicmodel with time delayrdquoAppliedMathematical Modelling vol 33 no 2 pp 1058ndash1071 2009
[19] L Nie Z Teng L Hu and J Peng ldquoPermanence and stabilityin non-autonomous predator-prey Lotka-Volterra systems withfeedback controlsrdquo Computers amp Mathematics with Applica-tions vol 58 no 3 pp 436ndash448 2009
[20] Y N Kyrychko and K B Blyuss ldquoGlobal properties of a delayedSIR model with temporary immunity and nonlinear incidenceraterdquo Nonlinear Analysis Real World Applications vol 6 no 3pp 495ndash507 2005
[21] Y Baba Y Tsuboi H Inoue T Yamada Z K Wszolek and DF Broderick ldquoAcute measles encephalitis in adultsrdquo Journal ofNeurology vol 253 no 1 pp 121ndash124 2006
[22] A Schreurs E V Stalberg and A R Punga ldquoIndicationof peripheral nerve hyperexcitability in adult-onset subacutesclerosing panencephalitis (SSPE)rdquo Neurological Sciences vol29 no 2 pp 121ndash124 2008
[23] O D Makinde ldquoAdomian decomposition approach to a SIRepidemic model with constant vaccination strategyrdquo AppliedMathematics and Computation vol 184 no 2 pp 842ndash8482007
[24] D Greenhalgh Q J A Khan and F I Lewis ldquoHopf bifurcationin two SIRS density dependent epidemicmodelsrdquoMathematicaland Computer Modelling vol 39 no 11-12 pp 1261ndash1283 2004
[25] S Gao L Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[26] Z Lu X Chi and L Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[27] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[28] G Rost and J Wu ldquoSEIR epidemiological model with varyinginfectivity and infinite delayrdquo Mathematical Biosciences andEngineering vol 5 no 2 pp 389ndash402 2008
[29] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001
[30] S A Gourley Y Kuang and J D Nagy ldquoDynamics of adelay differential equation model of hepatitis B virus infectionrdquoJournal of Biological Dynamics vol 2 no 2 pp 140ndash153 2008
[31] J Arino J R Davis DHartley R Jordan JMMiller and P vanden Driessche ldquoA multi-species epidemic model with spatialdynamicsrdquo Mathematical Medicine and Biology vol 22 no 2pp 129ndash142 2005
[32] V Capasso and G Serio ldquoA generalization of the Kermack-McKendrick deterministic epidemic modelrdquoMathematical Bio-sciences vol 42 no 1-2 pp 43ndash61 1978
[33] S Ruan and W Wang ldquoDynamical behavior of an epidemicmodel with a nonlinear incidence raterdquo Journal of DifferentialEquations vol 188 no 1 pp 135ndash163 2003
[34] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1986
[35] S Liu Y Pei C Li and L Chen ldquoThree kinds of TVS in aSIR epidemic model with saturated infectious force and verticaltransmissionrdquo Applied Mathematical Modelling vol 33 no 4pp 1923ndash1932 2009
[36] W R Derrick and P van denDriessche ldquoA disease transmissionmodel in a nonconstant populationrdquo Journal of MathematicalBiology vol 31 no 5 pp 495ndash512 1993
[37] A Korobeinikov and P K Maini ldquoNon-linear incidence andstability of infectious disease modelsrdquo Mathematical Medicineand Biology vol 22 no 2 pp 113ndash128 2005
[38] S M Moghadas and M E Alexander ldquoBifurcations of anepidemic model with non-linear incidence and infection-dependent removal raterdquo Mathematical Medicine and Biologyvol 23 no 3 pp 231ndash254 2006
[39] X Ji Y Pei and C Li ldquoTwo patterns of recruitment in anepidemic model with difference in immunity of individualsrdquoNonlinear Analysis Real World Applications vol 11 no 3 pp2078ndash2090 2010
[40] P van den Driessche ldquoTime delay in epidemic modelsrdquo inMathematical Approaches for Emerging and Reemerging Infec-tious Diseases An Introduction C Castillo-Chavez S BlowerP van den Driessche D Kirschner and A Yakubu Eds vol126 of The IMA Volumes in Mathematics and its Applicationspp 119ndash128 Springer New York NY USA 2002
[41] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[42] W Ma Y Takeuchi T Hara and E Beretta ldquoPermanence of anSIR epidemic model with distributed time delaysrdquo The TohokuMathematical Journal vol 54 no 4 pp 581ndash591 2002
[43] R Naresh A Tripathi J M Tchuenche and D SharmaldquoStability analysis of a time delayed SIR epidemic modelwith nonlinear incidence raterdquo Computers amp Mathematics withApplications vol 58 no 2 pp 348ndash359 2009
[44] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences vol 57 Springer New York NYUSA 2011
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
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
