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Applied Mathematics, 2011, 2, 57-63 doi:10.4236/am.2011.21007 Published Online January 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay Youquan Luo, Shujing Gao, Shuixian Yan Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, China E-mail: gaosjmath@126.com Received August 1, 2010; revised November 10, 2010; accepted November 15, 2010 Abstract In this paper, an impulsive epidemic model with time delay is proposed, which susceptible population is di- vided into two groups: high risk susceptibles and non-high risk susceptibles. We introduce two thresholds R1, R2 and demonstrate that the disease will be extinct if 11R and persistent if 21R. Our results show that larger pulse vaccination rates or a shorter the period of pulsing will lead to the eradication of the disease. The conclusions are confirmed by numerical simulations. Keywords: High Risk Susceptible, Non-High Risk Susceptible, Pulse Vaccination, Extinction 1. Introduction Infectious diseases have tremendous influence on human life. Every year, millions of people die of various infec- tious diseases. Controlling infectious diseases has been an increasingly complex issue in recent years [1]. Over the last fifty years, many scholars have payed great at- tention to construct mathematical models to describe the spread of infectious diseases. See the literatures [2-8], the books [9-11] and the references therein. In the clas- sical epidemiological models, a population of total size N is divided into S (susceptible numbers), I (infective num- bers), or S, I and R (recovered numbers) or S, E (exposed numbers), I and R, and corresponding epidemiological models such as SI, SIS, SIR, SIRS, SI ER and SEIRS are constructed. All these models are extensions of the SIR model elaborated by Kermack and McKendrick in 1927 [12]. Anderson and May [5,9] discussed the spreading nature of biological viruses, parasites etc. leading to in- fectious diseases in human population through several epidemic models. Cooke and Driessche [8] investigated an SEIRS model with the latent period and the immune period. The consideration of the latent period and the immune period gave rise to models with the incorpora- tion of delays and integral equation formulations. However, owing to the physical health status, age and other factors, susceptibles population show different in- fective to a infectious disease. In this paper, we divide the susceptible population into two groups: nonhigh risk susceptibles (S1) and high risk su scep tibles (S2), such that individuals in each group have homogeneous susceptibil- ity, but the susceptibilities of individuals from different groups are distinct. In this paper, we propose a new SIR epidemic model, which two noninteracting susceptible subclasses, the nonlinear incidence p SI , time delay and pulse vaccination are considered. The main purpose of this paper is to study the dynamical behavior of the model and establish sufficient condition s that the disease will be extinct or not. The organization of this paper is as follows. In the next section, we construct a delayed and impulsive SIR epidemic model with two noninteracting susceptible sub- classes. In Section 3, using the discrete dynamical sys- tem determined by the stroboscopic map, we establish sufficient conditions for the global attractivity of infec- tion-free periodic solution. And the sufficient conditions for the permanence of the model are obtained in section 4. Finally, we present some numerical simulations to illustrate our results. 2. Model Formulation and Preliminaries Gao etc. [13] proposed a delayed SIR epidemic model with pulse vacci nat i on: Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 58 ()()() (), ()() ()() ()(),, ()() ()(), () (1)(), () (),, ( )()(), StStIt St I tStIteStIt IttnTnN Rte StItRt St St I tIt tnTnN RtRt St (1) In model (1), the authors assumed that the birth rate ( ) is equal to the death rate, and use a bilinear inci- dence rate. Motivated by [13], in this paper, we assume that there are two cases noninteracting susceptible sub- classes. We denote the density of the susceptible indi- viduals that belong to different subclasses, the infected individuals, and the recovered individuals in the popula- tion by S1, S2, I and R, respectively, that is, the total va- riable population 12 .NSS IR Moreover, if the nonlinear incidence (1) p IS p , different constant recruitments and death rates are incorporated into model (1). Then the corresponding model is investigated: 2 12 1 1 11 1111 222 22 12122 122 111 222 , ,, , , 1, 1, , p p ppdp dp dp dp StStItSt StS tItSttnTnN ItStItStIteSt IteSt ItdIt RteSt IteSt ItRt St St St St It It Rt 112 2 , , tnTnN RtStSt (2) where , 1,2 ii are the recruitment rate into th e i-th susceptible class, respectively parameters , 1,2 ii , d and are the death rates of the susceptible, infected and recovered individuals, and , 1,2 ii are the contact rates, is the infectious period. The term , 1,2 i dp ieSt Iti reflects the fact that an individual has recovered from the infective compart- ments and are still alive after infectious period . , 1,2 ii are the vaccination rates, and T is the period of pulsing. Adding all the equations in model (2), the total varia- ble population size is given by the differential equation 1211 22.NtStSt dItRt and we have 12 .Nt hNt where 12 min,, ,hd . It follows that 12 limsup . tNt h Note that the first three equations of system (2) do not depend on the forth equation. Thus, we restrict our atten- tion to the following reduced system: 2 12 1 11 1111 222 22 12122 111 222 , ,, , 1, 1, , , p p ppdp dp StStItSt StS tItStt nTnN ItStItStIteSt IteSt ItdIt St St StStt nTnN It It (3) The initial conditions of (3) are 112 23 ,,,,0,SttSttItt fort (4) where 123 ,, , T P C and PC is the space of all piecewise functions 3 :,0 R with points of discontinuity at nT nN of the first kind and Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 59 which are continuous from the left, i.e., 0nT nT , and 33 123 ,,0, 1,2,3. i RxxxRxi Define a subset of 3 R 3 12 12 12 ,, 0 StStIt R St St Ith From biological considerations, we discuss system (3) in the closed set. It is easy to verify that is positively invariant with respect to system (3). 3. Global Attractivity of Disease-Free Periodic Solution To prove our main results, we state some notations and lemmas which will be essential to our proofs. Lemma 1 (see [6]) Consider the following impulsive differential equation ,, 1,, utabuttnT utut t nT where 0, 0,01ab . Then above system exists a unique positive periodic solution given by *,1, bt nT aa utuefornT tnT bb which is globally asymptotically stable, where 11 . 11 bT bT e a ube Definition 1 (see [14]). Let 3 :VRRR , then V is said to belong to class 0 V if i) V is continuous in 3 ,1nT nTR and for each 3 X R , ,, lim ,, tynTXVtyVnT X exists. ii) V is locally Lipschitzian in X. Lemma 2 (see [14]). Let 0 VV . Assume that ,,,,, ,,,, n DV txgtVtxtnT VtxVtxt nTt where : g RR R is con tinuous in 1,nTTRn and for ,uR nN, ,, lim ,, tynT uVtyVnTu exists, : nRR is non-decreasing. Let rt be the maximal solution of the scalar impulsive differential equation 0 ,, , ,, 0, n ugtutnT uu tt T u tttn u existing on 0, . Then 00 0,Vxu implies that ,,0Vtx rttt , where x t is any solution of (3). In the following we shall demonstrate that the dis- ease-free periodic solution ** 12 ,,0SStt is global attractive. We firstly show the existence of the dis- ease-free periodic solution, in which the infectious indi- viduals are entirely absent from the population perma- nently, i.e. 0It for all 0t. Under this condition, the growth of the i-th 1, 2i susceptible individuals must satisfy 1 ,, ,. iiii ii St t tt StnT SStnT According to Lemma 1, we know that the periodic so- lution of the system *111 for1 ,1,2 i i tnT ii iT ii e Se nT tnTi t is globally asymptotically stable. Therefore system (3) has a unique di sease -f ree periodic solut i on ** 12 ,,0SStt. Denote 12 12 12 12 12 2 1 11 1111 . p p TT TT i ee ee Rd (5) Theorem 1 If 11R , then the disease-free periodic solution ** 12 ,,0SStt of system (3) is globally at- tractive. Proof. Since 11R , we can choose 0 suffi- ciently small such that 1 1 2 2 1 11 2 222 1 11 1. 11 p T T i p T T e e ed e (6) From the first equation and the second equation of system (3), we have , 1,2 iiii SSitt . Then we consider the following impulsive comparison system Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 60 ,, 1,, iiii ii ututt nT ututt nT (7) According to Lemma 1, we obtain the periodic solu- tion of system (7) *1, 11 for1,1,2 , i i tnT ii iT ii e ue nT tnTi t which is globally asymptotically stable. By the compari- son theorem [14], we have that there exists 1 nZ such that for 1 1,nTtnT nn *111 ,1,2. i i T ii ii T ii i e Se t Si tu (8) Furthermore, from the third equation of system (3), we get 112 2 pp t I SSItd for tnT and 1 nn. From (6) and (8), we have 0 I t , then lim 0 ttI , i.e., for any sufficiently small 10 , there exists an integer 21 nn such that 1 It , for all 2 tnT. From the first equation and the second equation of system (3), we have for 2 tnT 11,( 1,2). p iiii ii SS itSt Then we consider the following impulsive comparison systems 11,, 1,. p iiiiii iii vSvtnTtt tnTtvvt (9) From Lemma 1, we obtain the periodic solution of system (9) 11 11 * 11 1,1,1,2, 11 p ii i p ii i StnT ii ipST ii ii te vnTtnTi Se which is globally asymptotically stable. In view of the comparison theorem [14], there exists an integer 32 nn such that for 3 1, ,nTtnTn n 11 11 * 11 ()1, 1,2 11 p ii i p ii i ST ii ii pST ii ii e St vi S t e (10) Since and 1 are sufficiently small, from (8) and (10), we know that * lim,1,2 . i tSStit Hence, disease-free periodic solution ** 12 ,,0SStt of system (3) is globally attractive. The proof is com- pleted. Next, we give some accounts of the Theorem 1 for a well biological meaning. By simple calculation, from (5) we get 1 1 1 2 2 2 111 1 111 122 1 222 10, 11 10, 11 p T pT pp T p T pT pp T e Rpe de e Rpe de (11) and 1 2 1 2 1 2 11 111 22 112 2 11 12 12 11 0. 11 11 p p TT pp TT ppp p TT ee Rpep e Td d ee (12) Theorem 1 determines the global attravtivity of the disease-free periodic solution of system (3) in for the case 11R. Its epidemiology implies that the dis- ease will die out. From (11) and (12), we can see that larger pulse vaccination rates or a shorter period of im- mune vaccination will make for the disease eradication. 4. Permanence In this section, we state the disease is endemic if the in- fectious population p ersists above a certain positive lev el for sufficiently large time. The endemicity of the disease can be well captured and studied through the notation of uniform persistence. Definition 2. System (3) is said to be uniformly per- sistent if there exist positive constants 0 ii Mm 1,2,3i (both are independent of the initial values), such that every solution 12 ,,tSStIt with posi- tive initial condition s of system (3) satisfies 1112 2233 ,,.mS MmSMmIttt M Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 61 Denote 12 12 12 12 12 12 2 11 11 1111 . pp TT dd TT ee ee ee Rd Theorem 2. If 21R, then system (3) is uniformly persistent. Proof. Let 12 ,,StStIt be any solution of (3) with initial conditions (4), then it is easy to see that 1212 12 12 ,,,forall0.SSIt h ttt hh We are left to prove there exist positive constants 123 ,,mmm such that 112 23 ,,,ttSmSmItm for all sufficiently large t. Firstly, from the first and second equations of system (3), we have 12 ,1,2. p iiiii SSi h tt Considering the following comparison equations 12 , ,1 , , i i p ii ii ii u h u uttt nT utt t nT According to Lemma 1 and the comparison theorem, we know that for any sufficiently small 0 , there exists a 0 t such that for 0 tt, 12 12 * 12 1. 11 p ii p ii T h ii iii i pT h ii i e Suu m e tt h t Now, we shall prove there exist a 30msuch that 3 It m for all sufficiently large t. For convenience, we prove it through the following two steps: Step I. Since 21R, there exist sufficiently small *0 I m and 0 such that 112 2 11, pd pd eed (13) where * 12 * 12 * 12 1, 1,2. 11 p iIi p iIi mT h ii ipmT h iIi i ei me h For * I m, we can claim that there exists a 10t such that * 1 I I tm. Otherwise, * I It m for all 0t. It follows from the first and second equations of (3) that 1* 12 ,1,2. p iiiIii Sm htitS Considering the following impulsive comparison sys- tems 1* 12 ,, 1,, p ii Iii ii i i tvmv htt nT tttvnTv Similarly, we know that there exists 20t, such that for 2 tt 1* 12 1* 12 * 1* 12 .1 11 p iIi p iIi mT h ii iii i pmT h iIi i te Svv me h tt (14) Further, the third equations of system (3) can be re- written as 112 2 11 pp dp ISISI eSt It ttttt 22 112 2 11 11 dp pdpd p t d t eSt ItdI SI eSI e d dIeSId dt t tt tt t Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 62 22 p t d t d eSId dt Define 11 22 . p t d t p t d t VIe SId eSId tt For 2 tt, the derivative of Vt along the solution of system (3) is 112 2 112 2 11 11 pdpd pd pd VSeSedI eed t tI tt t From (13), we have 0Vt for 2 tt, which im- plies that Vt as t. This contradicts the boundedness of ()Vt. Hence, there exists a 10t such that * 1 I I tm. Step II. According to step I, for any positive solution 12 , , ttSSIt of (3), we are left to consider two cases. First, If * I It m for all 1 tt, then our result is proved. Second, if * I It m for some 1 tt,we can choose constants 0 and 021 ,Tmaxtt (0 T is sufficiently large) such that * I It m, * 0 I I Tm, * 0 I I Tm and 112 2 , tSSt for 00 ,tTT . Thus, there exists a 0gg such that for 00 ,tTT g *. 2 I It m (15) In this case, we shall discuss three possible cases in term of the sizes of , g and . Case 1. If g , then it is obvious that *2 I It m, for 00 ,tTT . Case 2. If g , then from the third equation of (3), we can deduce 112 2, tdt pp t I SSIedt (16) From (14), (15) and (16), we have 0 0112 200 *** 112 2 ,for , 2 Tg dt pp T pp d II ISSIedtTgT me t gm Set *** 3min ,0 2II m mm , for 00 ,tTT , we have 3 It m. Case 3. If g , we will discuss the following two cases, respectively. Case 3.1. For 00 ,tTT , it is easy to obtain ** I It m. Case 3.2. For 00 ,tT T . We claim that ** I It m. Otherwise, there exists a 00 ,tT T , such that ** I It m for 0,tT t , and ** I I tm. From (13), (14) and (16), we have ** ** 112 21122, 1d tdt pp pp I I t e I tSSIedm m d which is contradictory to ** I I tm. Hence, the claim holds true. According to the arbitrary of 0 T, we can obtain that 3 It m holds for all 0 tT. The proof is completed. 5. Numerical Simulations In this section, we give some numerical simulations to illustrate the effects of different probability on popula- tion. In system (3), 10.25, 20.2, 10.03, 20.1, 10.03, 20.05, 2,p0.1,d2, 3T. Time series are drawn in Figure 1(a) and Figure 1(b) with initial values 120.5sin,tt 2t 20.8cos,t 310.5sin, 2,0ttt for 30 puls- ing cycles. If we take 10.45, 20.90 , then 1 R 0.9956 . By Theorem 1, we know that the disease will disappear (see Figure 1(a)). If we let 10.10, 2 0.20 , then 21.4685R . According to Theorem 2, we know that the disease will be permanent (see Figure 1(b)). (a) Y. Q. LUO ET AL. Copyright © 2011 SciRes. AM 63 (b) Figure 1. Two figures show that movement paths of S1, S2 and I as functions of time t. (a) Disease will be extinct with R1 = 0.9956 and θ1 = 0.45, θ2 = 0.9; (b) Disease will be per- sistent with R2 = 1.4685 and θ1 = 0.1, θ2 = 0.2. 6. Acknowledgements The research of Shujing Gao has been supported by The Natural Science Foundation of China (10971037) and The National Key Technologies R & D Program of Chi- na (2008BAI68B01). 7. References [1] X. B. Zhang, H. F. Huo, X. K. Sun and Q. Fu, “The Dif- ferential Susceptibility SIR Epidemic Model with Time Delay and Pulse Vaccination,” Journal of Applied Mathe- matics and Computing, Vol. 34, No. 1-2, 2009, pp. 287- 298. [2] Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, “Pulse Mass Measles Vaccination across Age Cohorts,” Pro- ceedings of the National Academy of Sciences of the United States of America, Vol. 90, No. 24, 1993, pp. 11698-11702. doi:10.1073/pnas.90.24.11698 [3] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—II: The Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 138, No. 834, 1932, pp. 55-83. doi:10.1098/rspa. 1932.0171 [4] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—III: Further Stu- dies of the Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 141, No. 843, 1933, pp. 94-122. doi:10.1098/rspa.1933.0106 [5] R. M. Anderson and R. M. May, “Population Biology of Infectious Disease: Part I,” Nature, Vol. 280, 1979, pp. 361-367. doi:10.1038/280361a0 [6] S. Gao, L. Chen and J. J. Nieto, “Angela Torres, Analysis of a Delayed Epidemic Model with Pulse Vaccination and Saturation Incidence,” Vaccine, Vol. 24, No. 35-36, 2006, pp. 6037-6045. doi:10.1016/j.vaccine.2006.05.018 [7] C. 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