DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation49043710.1155/2009/490437490437Research ArticleThe Dynamics of the Pulse Birth in an SIR Epidemic Model with Standard IncidenceZhangJuping1, 2JinZhen1XueYakui1LiYouwen1VecchioAntonia1Department of Mathematics, School of ScienceNorth University of ChinaTaiyuan, Shanxi 030051Chinanuc.edu.cn2School of Mechatronic EngineeringNorth University of ChinaTaiyuan, Shanxi 030051Chinanuc.edu.cn20090706200920092110200806022009290420092009Copyright © 2009This 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.

An SIR epidemic model with pulse birth and standard incidence is presented. The dynamics of the epidemic model is analyzed. The basic reproductive number R is defined. It is proved that the infection-free periodic solution is global asymptotically stable if R<1. The infection-free periodic solution is unstable and the disease is uniform persistent if R>1. Our theoretical results are confirmed by numerical simulations.

1. Introduction

Every year billions of population suffer or die of various infectious disease. Mathematical models have become important tools in analyzing the spread and control of infectious diseases. Differential equation models have been used to study the dynamics of many diseases in wild animal population. Birth is one of the very important dynamic factors. Many models have invariably assumed that the host animals are born throughout the year, whereas it is often the case that births are seasonal or occur in regular pulse, such as the blue whale, polar bear, Orinoco crocodile, Yangtse alligator, and Giant panda. The dynamic factors of the population usually impact the spread of epidemic. Therefore, it is more reasonable to describe the natural phenomenon by means of the impulsive differential equation [1, 2].

Roberts and Kao established an SI epidemic model with pulse birth, and they found the periodic solutions and determined the criteria for their stability . In view of animal life histories which exhibit enormous diversity, some authors studied the model with stage structure and pulse birth for the dynamics in some species . Vaccination is an effective way to control the transmission of a disease. Mathematical modeling can contribute to the design and assessment of the vaccination strategies. Many infectious diseases always take on strongly infectivity during a period of the year; therefore, seasonal preventing is an effective and practicable way to control infectious disease . Nokes and Swinton studied the control of childhood viral infections by pulse vaccination . Jin studied the global stability of the disease-free periodic solution for SIR and SIRS models with pulse vaccination . Stone et al. presented a theoretical examination of the pulse vaccination policy in the SIR epidemic model . They found a disease-free periodic solution and studied the local stability of this solution. Fuhrman et al. studied asymptotic behavior of an SI epidemic model with pulse removal . d'Onofrio studied the use of pulse vaccination strategy to eradicate infectious disease for SIR and SEIR epidemic models . Shi and Chen studied stage-structured impulsive SI model for pest management . And the incidence of a disease is the number of new cases per unit time and plays an important role in the study of mathematical epidemiology. Many works have focused on the epidemic models with bilinear incidence whereas Anderson and May and De Jong et al. pointed out that the epidemic models with standard incidence provide a more natural description for humankind and gregarious animals .

The purpose of this paper is to study the dynamical behavior of an SIR model with pluse birth and standard incidence. We suppose that a mass vaccination program is introduced, under which newborn animals are vaccinated at a constant rate p (0<p<1) and vaccination confers lifelong immunity. Immunity is not conferred at birth, and thus all newborns are susceptible.This paper is organized as follows. In the next section, we present an SIR model with pulse birth and standard incidence and obtain the existence of the infection-free periodic solution. In Section 3, the basic reproductive number R* is defined. Local stability and the global asymptotically stable of the infection-free periodic solution are obtained when R*<1. Section 4 concentrates on the uniform persistence of the infectious disease when R*>1. Numerical simulation is given in Section 5.

2. The SIR Model with Pulse Birth

In our study, we analyze the dynamics of the SIR model of a population of susceptible (S), infective (I), and recovered (R) with immunity individuals. Immunity is not conferred at birth, and thus all newborns are susceptible. Vaccination gives lifelong immunity to pS susceptible who are, as a consequence, transferred to the recovered class (R) of the population. Using the impulsive differential equation, we have S=-βSNI-dS-pS,I=βSNI-(d+θ+α)I,R=θI-dR+pS,tnτ,S(nτ+)=S(nτ)+b1S(nτ)+b1γ1I(nτ)+b1γ2R(nτ),I(nτ+)=I(nτ)+b1(1-γ1)I(nτ),R(nτ+)=R(nτ)+b1(1-γ2)R(nτ),t=nτ. The total population size is denoted by N, with N=S+I+R. Here the parameters β, d, p, θ, α, b1, γ1, γ2 are all positive constants. β is adequate contact rate, d is the per capita death rate, θ is the removed rate, and p is vaccination, a fraction of the entire susceptible population. b1 is the proportion of the offspring of population. To some diseases, all the offspring of susceptible parents are still susceptible individuals, but it is different to the recovered. Because individual differences cause different immune response, a fraction γ2 (0<γ2<1) of their offspring are the susceptible; the rest are immunity (e.g., when Giant panda gives the breast to her baby, the immunity of Giant panda baby is obtained. But if Giant panda baby did not eat breast, their immunity to disease is very poor. They are vulnerable to suffering from respiratory and digestive disease. Therefore, they become the susceptible.). Similarly, a fraction γ1 (0<γ1<1) of the infectious offspring are susceptible, and the rest are infectious. Due to the effect of the diseases to the infectious, the ratio of the susceptible in their offspring is relatively low. So we assume the fraction γ1<γ2. α represents the death rate due to disease. From biological view, we assume βα.

From (2.1), we obtain N=-dN-αI,tnτ,N(nτ+)=(1+b1)N(nτ),t=nτ. Let s=S/N,i=I/N,r=R/N, then systems (2.1) and (2.2) can be written as follows: s=-ps+(α-β)si,i=(βs-θ-α)i+αi2,r=ps+θi+αir,tnτ,s(nτ+)=s(nτ)+b1γ11+b1i(nτ)+b1γ21+b1r(nτ),i(nτ+)=1+b1(1-γ1)(1+b1)i(nτ),r(nτ+)=1+b1(1-γ2)(1+b1)r(nτ),t=nτ. The total population size is normalized to one. By virtue of the equation s(t)+i(t)+r(t)=1, we ignore the third and the sixth equations of system (2.3) to study the two-dimensional system:

s=-ps+(α-β)si,i=(βs-θ-α)i+αi2,tnτ,s(nτ+)=b1γ21+b1+(1-b1γ21+b1)s(nτ)+(b1γ11+b1-b1γ21+b1)i(nτ),i(nτ+)=1+b1(1-γ1)1+b1i(nτ),t=nτ.

From biological view, we easily see that the domain Ω={(s,i,r):s0,i0,r0,s+i+r=1} is the positive invariant set of system (2.3).

We first demonstrate the existence of infection-free periodic solution of system (2.4), in which infectious individuals are entirely absent from the population permanently, that is, i(t)=0,t0. Under this condition, the growth of susceptible individuals and the population must satisfy s=-ps,tnτ,s(nτ+)=b1γ21+b1+(1-b1γ21+b1)s(nτ),t=nτ. Integrating the first equation in system (2.6) between pulses, it is easy to obtain the solution with initial value s(0+)=s0, s(t)=s(nτ+)e-p(t-nτ),  nτ<t(n+1)τ. Equation (2.7) holds between pulses. At each successive pulse, it yields s((n+1)τ+)=b1γ21+b1+(1-b1γ21+b1)e-pτs(nτ+)=F(s(nτ+)). Equation (2.8) has a unique fixed point s*=b1γ2epτ/((1+b1)epτ-(1+b1-b1γ2)). The fixed point s* is locally stable because dF(s(nτ+))/dss(nτ+)=s*=(1-b1γ2/(1+b1))e-pτ<1, By substituting s(nτ+)=s* to (2.7), we obtain the complete expression for the infection-free periodic solution over the nth time-interval nτ<t(n+1)τ, s̃(t)=b1γ2epτ(1+b1)epτ-(1+b1-b1γ2)e-p(t-nτ),  ĩ(t)=0.

Therefore the system (2.4) has the infection-free periodic solution (s̃(t),ĩ(t)).

3. The Stability of the Infection-Free Periodic Solution

In this section,we will prove the local and global asymptotically stable of the infection-free periodic solution (s̃(t),ĩ(t)).

The local stability of the infection-free periodic solution (s̃(t),ĩ(t)) may be determined by considering the linearized SIR equation of system (2.4) about the known periodic solution (s̃(t),ĩ(t)) by setting s(t)=s̃(t)+x(t),i(t)=ĩ(t)+y(t), where x(t) and y(t) are small perturbation. The variables x(t) and y(t) are described by the relation (x(t)y(t))=Φ(t)(x(0)y(0)), where the fundamental solution matrix Φ(t)=φij(t)(i,j=1,2) satisfies dΦ(t)dt=(-p(α-β)s̃(t)0βs̃(t)-θ-α)Φ(t), with Φ(0)=E, where E is the identity matrix. The resetting of the equations of (2.4) becomes (x(nτ+)y(nτ+))=(1-b1γ21+b1b1(γ1-γ2)1+b101+b1-b1γ11+b1)(x(nτ)y(nτ)). Hence, according to the Floquet theory, if all eigenvalues of M(τ)=(1-b1γ21+b1b1(γ1-γ2)1+b101+b1-b1γ11+b1)Φ(τ) are less than one, then the infection-free periodic solution (s̃(t),0) is locally stable. By calculating, we have Φ(t)=(1Φ120Φ22), where Φ22(t)=exp(βs̃(σ)dσ-(θ+α)t).

The eigenvalues of M denoted by μ1,μ2 are μ1=(1-b1γ2/(1+b1))e-pτ<1, and μ2=((1+b1-b1γ1)/(1+b1))exp{β0τs̃(σ)dσ-(θ+α)τ}, if and only if μ2<1. Define threshold of model (2.4) as follows: R*=β0τs̃(σ)dσln((1+b1)/(1+b1-b1γ1))+(θ+α)τ, where s̃(t) is the infection-free periodic solution. That is, the infection-free periodic solution (s̃(t),ĩ(t)) is locally asymptotically stable if R*<1. So we obtained following theorem.

Theorem 3.1.

If R*<1, then the infection-free periodic solution (s̃(t),ĩ(t)) of system (2.4) is locally asymptotically stable.

Now we give the global asymptotically stable of the infection-free periodic solution. In order to prove the global stability of the infection-free periodic solution (s̃(t),ĩ(t)), we need to use to comparison theory and impulsive differential inequality [1, 2].

Theorem 3.2.

If R*<1, then the infection-free periodic solution (s̃(t),ĩ(t)) of system (2.4) is global asymptotically stable.

Proof.

Because of αβ, and γ1γ2, we have s-ps,tnτ,s(nτ+)b1γ21+b1+(1-b1γ21+b1)s(nτ),t=nτ. By impulsive differential inequality, we see that s(t)s(0+)o<nτ<t(1-b1γ21+b1)exp{0t(-p)dσ}+o<nτ<t{nτ<jτ<t(1-b1γ21+b1)exp[nτt(-p)dσ]b1γ21+b1}=s̃(t)+s(0+)(1-b1γ21+b1)[t/T]e-pt-(b1γ2/(1+b1))(1-b1γ2/(1+b1))[t/T]e-pt(1-b1γ2/(1+b1))e-pT.   Since limt{s(0+)(1-b1γ21+b1)[t/T]e-pt-(b1γ2/(1+b1))(1-b1γ2/(1+b1))[t/T]e-pt(1-b1γ2/(1+b1))e-pT}=0, for any given ϵ1>0, there exists T1>0, such that s(t)<s̃(t)+ϵ1, for all t>T1.

Introduce the new variable u=s+r, then u=(-βs+αu+θ)(1-u),tnτ,u(nτ+)=b1γ11+b1+(1-b1γ11+b1)u(nτ),t=nτ.

Consider the following comparison system with pulse: v=-(βs̃(t)+βϵ1+α-θ)v-αv2+θ-β(s̃(t)+ϵ1),tnτ,v(nτ+)=b1γ11+b1+(1-b1γ11+b1)v(nτ),t=nτ.

The first equation of (3.11) is Riccati equation. It is easy to see that v(t)=1 is a solution of system (3.11). Let y=v-1, then y=-(βs̃(t)-α-θ+βϵ1)y-αy2,tnτ,y(nτ+)=(1-b1γ11+b1)y(nτ),t=nτ.

Let z=1/y, then z=-(βs̃(t)-α-θ+βϵ1)z-α,tnτ,z(nτ+)=(1+b11+b1-b1γ1)z(nτ),t=nτ.

Let q(t)=βs̃(t)-α-θ+βϵ1, solving system (3.13) between pulses (T1+nτ,T1+(n+1)τ], we have z(t)=e-T1+nτtq(σ)dσ[αT1+nτteT1+nτuq(σ)dσdu+1+b11+b1-b1γ1z(T1+nτ)], when t=T1+(n+1)τ, (3.14) can be written as follows: z(T1+(n+1)τ)=e-T1+nτT1+(n+1)τq(σ)dσ[αT1+nτT1+(n+1)τeT1+nτuq(σ)dσdu+1+b11+b1-b1γ1z(T1+nτ)]. On the other hand, solving system (3.13) between pulses (T1+(n-1)τ,T1+nτ], we obtain z(t)=e-T1+(n-1)τtq(σ)dσ[αT1+(n-1)τteT1+(n-1)τuq(σ)dσdu+1+b11+b1-b1γ1z(T1+(n-1)τ)], then z(T1+nτ)=e-T1+nτT1+(n+1)τq(σ)dσ[αT1+(n-1)τT1+nτeT1+(n-1)τuq(σ)dσdu+1+b11+b1-b1γ1z(T1+(n-1)τ)].

Similarly, we can get the expressions of z(T1+(n-1)τ),z(T1+(n-2)τ),,z(T1). Then using iterative technique step by step, z(T1+nτ)=e-0nτq(σ)dσ(1+b11+b1-b1γ1)n[1kn(1+b11+b1-b1γ1)k-n-1eT1T1+(n-k)τq(σ)dσdu+T1+(n-k)τT1+(n-k+1)τeT1+(n-k)τuq(σ)dσdu+z(T1)], where e-0nτq(σ)dσ(1+b11+b1-b1γ1)n=exp{n[-β0τs̃(σ)dσ+(θ+α)τ-βϵ1τ+ln1+b11+b1-b1γ1]}.

The condition R*<1 implies that limnz(nτ)=, then limtx(t)=1. The comparison principle and the condition u(t)<1 imply that limtu(t)=1, so we have limti(t)=0.

Because we have proved that limti(t)=0 when R*<1, for any given ϵ2>0, there exists T2>0, such that -ϵ2<i(t)<ϵ2, for all t>T2.

When t>T2, from system (2.4), we have s-ps+(α-β)ϵ2s,tnτ,s(nτ+)b1γ11+b1+(1-b1γ21+b1)s(nτ),t=nτ. Therefore s(t)s(T2+)T2<nτ<t(1-b1γ21+b1)exp{T2t(-p+(α-β)ϵ2)dσ}+T2<nτ<t{T2<jτ<t(1-b1γ21+b1)exp{nτt(-p+(α-β)ϵ2)dσ}b1γ11+b1}=s(T2+)(1-b1γ21+b1)[t/T]-[T2/T]exp(-p+(α-β)ϵ2)(t-T2)+b1γ1/(1+b1)1-(1-b1γ2/(1+b1))exp(-p+(α-β)ϵ2)Texp(-p+(α-β)ϵ2)(t-[tT]T)-b1γ1/(1+b1)1-(1-b1γ2/(1+b1))exp(-p+(α-β)ϵ2)T×(1-b1γ21+b1)([t/T]-[T2/T]-1)exp(-p+(α-β)ϵ2)T. For any given ϵ2>0, we have limt{s(T2+)(1-b1γ21+b1)[t/T]-[T2/T]exp(-p+(α-β)ϵ2)(t-T2)-b1γ1/(1+b1)1-(1-b1γ2/(1+b1))exp(-p+(α-β)ϵ2)T×(1-b1γ21+b1)([t/T]-[T2/T]-1)exp(-p+(α-β)ϵ2)(T)}=0,limtb1γ1/(1+b1)1-(1-b1γ2/(1+b1))exp(-p+(α-β)ϵ2)Texp(-p+(α-β)ϵ2)(t-[tT]T)=s̃(t). Therefore, for any given ϵ3>0, there exists T3>0, when t>T3, then we have s(t)s̃(t)-ϵ3. For any given ϵ>0. Let T=max{T1,T2,T3}, then t>T, then we have s̃(t)-ϵs(t)s̃(t)+ϵ, that is limts(t)=s̃(t).

Therefore the infection-free periodic solution (s̃(t),0) is global asymptotically stable.

4. The Uniform Persistence of the Infectious Disease

In this section, we will discuss the uniform persistence of the infectious disease, that is, limtinfi(t)ρ>0 if R*>1.

To discuss the uniform persistence, we need the following lemma.

Lemma 4.1.

For the following impulsive equation, x=-gx-h,tnτ,x(nτ+)=b1γ11+b1+(1-b1γ21+b1)x(nτ),t=nτ, has a unique positive τ-periodic solution x̃(t) for which x̃(0)>0,   tR+, and x̃(t) is global asymptotically stable in the sense that limt|x(t)-x̃(t)|=0, where x(t) is any solution of system (2.2) with positive initial value x(0)>0 and g,h are positive constants.

Proof.

Solving (4.1), we have x(t)=W(t,0)x(0)-h0tW(t,σ)dσ+b1γ11+b10<nτ<tW(t,nτ+), where W(t,t0)=t0nτ<t(1-b1γ21+b1)e-g(t-t0). Since W(τ,0)=(1-b1γ2/(1+b1))e-gτ<1, (4.1) has a unique positive τ-periodic solution x̃(t) with the initial value x̃(0)=(-h0τW(τ+,σ)dσ+(b1γ1/(1+b1))W(τ,τ))/(1-W(τ,0)). Next, we only need to prove that limt|x(t)-x̃(t)|=0.

Since |x(t)-x̃(t)|=W(t,0)|x(0)-x̃(0)|, the result is obtained if W(t,0)0 as t. Suppose t(nτ,(n+1)τ], then W(t,0)=0jτ<t(1-b1γ21+b1)e-gt=(1-b1γ21+b1)[t/τ]e-gt. Thus limtW(t,0)=0. The proof is complete.

Lemma 4.2.

If R*>1, then the disease uniformly weakly persists in the population, in the sense that there exists some c>0 such that limtsupi(t)>c for all solutions of (2.4).

Proof.

Suppose that for every ϵ>0, there is some solution with limtsupi(t)<ϵ. From the first equation of (2.4), we have s=-ps+(α-β)si-ps+(α-β)ϵ,tnτ. Consider the following equation: w=-pw+(α-β)ϵ,tnτ,w(nτ+)=b1γ11+b1+(1-b1γ21+b1)w(nτ),t=nτ. By Lemma 4.1, we see that (4.7) has a unique positive τ-periodic solution w̃(t), and w̃(t) is global asymptotically stable. Solving (4.7), we have w(t)=W(t,0)w(0)+(α-β)ϵ0tW(t,σ)dσ+b1γ11+b10<nτ<tW(t,nτ+),w̃(t)=(α-β)ϵ(W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ)+(1-b1γ21+b1)W(t,0)W(τ,τ)1-W(τ,0)+b1γ11+b10<nτ<tW(t,nτ+), and w̃(t) is global asymptotically stable. By (4.9), let α=β, we obtain the periodic solution of (2.6) that s̃(t)=(1-b1γ21+b1)W(t,0)W(τ,τ)1-W(τ,0)+b1γ11+b10<nτ<tW(t,nτ+). and we have s̃(t)-w̃(t)=(β-α)ϵ(W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ). Let Δ=(β-α)max0tτ{W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ}, by (4.11), we can see that w̃(t)s̃(t)-Δϵ. By comparison theory, we obtain that i-(θ+α)i+βiw(t)+αi2. Since w̃(t) is global asymptotically stable, for above ϵ, there exists T4>0, such that w(t)w̃(t)-ϵ, t>T4. From (4.13) and (4.14), we have that i[βs̃(t)-(θ+α)-(1+Δ)βϵ]i. Consider the following equation: i[βs̃(t)-(θ+α)-(1+Δ)βϵ]i,tnτ,i(nτ+)=1+b1(1-γ1)1+b1i(nτ),t=nτ. By impulsive differential inequality, for t(T4+nτ,T4+(n+1)τ], we see that i(t)i(T4)T4<jτ<t1+b1(1-γ1)1+b1exp{T4t[βs̃(σ)-(θ+α)-(1+Δ)βϵ]dσ}=i(T4)(1+b1(1-γ1)1+b1)nexp{T4T4+nτ[βs̃(σ)-(θ+α)-(1+Δ)βϵ]dσ+T4+nτt[βs̃(σ)-(θ+α)-(1+Δ)βϵ]dσ}Cexp{n[(R*-1)(ln1+b11+b1(1-γ1)+(θ+α)τ)-(1+Δ)βϵτ]}, where C=i(T4)exp{-[(θ+α)+(1+Δ)βϵ]τ}. Taking 0<ϵ<(R*-1)[ln((1+b1)/(1+b1(1-γ1)))+(θ+α)τ](1+Δ)βτ, thus i(t) as t, a contradiction to the fact that i(t) is bounded. The proof is complete.

Theorem 4.3.

If R*>1, then the disease is uniformly persistent, that is, there exists a positive constant ρ such that for every positive solution of (2.4), limtinfi(t)ρ>0.

Proof.

Let 0<η12(1-1R*)s̃(t)M(θ+α), where M=max0tτ{W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ},W(t,t0)=t0nτ<t1+b1(1-γ1)1+b1e-p(t-t0). It can be obtained from Lemma 4.1 that for any positive solution of (2.4) there exists at least one t0>0 such that i(t0)>η>0. Then, we are left to consider two case. The first case is i(t)η for all large tt0. The second case is i(t) oscillates about η for large t. The conclusion of Theorem 4.3 is obvious in the first case since we can choose ρ=η. For the second case, let t1>t0, and let t2>t1 satisfy i(t1)=i(t2)=η,i(t)<ηfort1<t<t2. Next, we introduce the new variable V=s+i, and it follows from the first two equations of (2.4) that V=-pV-(θ+α)i+(αV+p)i,i=-(θ+α)i+β(V-i)i+αi2,tnτ,V(nτ+)=b1γ21+b1+(1-b1γ21+b1)V(nτ),i(nτ+)=1+b1(1-γ1)1+b1i(nτ),t=nτ. If i(t)η, then V-pV-(θ+α)η,tnτ.

Consider the following equation: x=-px-(θ+α)η,tnτ,x(nτ+)=b1γ21+b1+(1-b1γ21+b1)x(nτ),t=nτ.

By Lemma 4.1, we see that (4.23) has a unique positive τ-periodic solution x̃(t), and x̃(t) is global asymptotically stable. Solving (4.23), we have x̃(t)=-(θ+α)η(W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ)+(1-b1γ21+b1)W(t,0)W(τ+,τ+)(1-W(τ,0))+b1γ21+b10<nτ<tW(t,nτ+). From (4.10) and (4.24), it is easy to see that x̃(t)-s̃(t)-(θ+α)η(W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ). By 0<η(1/2)(1-1/R*)(s̃(t)/M(θ+α)), M=max0tτ{W(t,0)0τW(τ+,σ)dσ/(1-W(τ,0))+0tW(t,σ)dσ}, and (W(t,0)0τW(τ+,σ)dσ/(1-W(τ,0))+0tW(t,σ)dσ)(1/M)1, we obtain -(θ+α)η(W(t,0)0τW(τ+,σ)dσ1-W(τ,0)+0tW(t,σ)dσ)-12(1-1R*)s̃(t), namely, x̃(t)12(1+1R*)s̃(t). The comparison principle and the global asymptotically stable of x̃(t) imply that there exists a positive constant T5>0 such that V(t)12(1+1R*)s̃(t),t>t1+T5. From (4.28) and the second equation of (4.22), we can see that i[β2(1+1R*)s̃(t)-(θ+α)]i+(α-β)i2. Consider the following equation: y=[β2(1+1R*)s̃(t)-(θ+α)]y+(α-β)y2,tnτ,y(nτ+)=1+b1(1-γ1)1+b1y(nτ),t=nτ. Let z=y-1, then we have z=[(θ+α)-β2(1+1R*)s̃(t)]z+(β-α),tnτ,z(nτ+)=1+b11+b1(1-γ1)z(nτ),t=nτ. By the same method of Lemma 4.1, we can get a conclusion that z̃(t)=(β-α)(W(t,0)0τW(τ+,σ)dσ/(1-W(τ,0))+0tW(t,σ)dσ)+(1+b1)/(1+b1(1-γ1))(W(t,0)W(τ,τ)/(1-W(τ,0))) is global asymptotically stable. Thus system (4.30) has a unique positive τ-periodic solution ỹ(t), and ỹ(t) is global asymptotically stable, limt|y(t)-ỹ(t)|=0. From (4.32) we see that there exists a positive constant T6>0 such that y(t)>ρ12mint1tt1+τỹ(t)>0,  t>t1+T6. Let T*=max{T5,T6}, and define ρ=min{ρ,ηexp(-(θ+α)τ)}. If t2-t1<T*, from the second equation of (4.22), we have the inequality i(t)-(θ+α)i, and the comparison principle implies that i(t)ηexp{-(θ+α)(t-t1)}ηexp{-(θ+α)T*}, that is, i(t)ρ for all t(t1,t2).

If t2-t1>T*, we divide the interval [t1,t2] into two subintervals [t1,t1+T*] and [t1+T*,t2], i(t)ρ is obvious in the interval [t1,t1+T*]. In the interval [t1+T*,t2], we have the inequality (4.29) and (4.33). The comparison principle shows that i(t)y(t)ρρ for t[t1+T*,t2]. The analysis above is the independent of the selection of interval [t1,t2], and the choice of ρ is the independent of the selection of interval independent of any positive solution of (2.4). The persistence is uniform to all positive solution. The proof is complete.

5. Numerical Simulation

For the birth pulses of SIR model with standard incidence, we know that the periodic infection-free solution is global asymptotically stable if the basic reproductive number R*<1. The periodic infection-free solution is unstable if the basic reproductive number R*>1, in this case, the disease will be uniform persistent. Here we do computer simulation to give a geometric impression on our results. In all simulation unit was set to unity (scaled to unity).

In Figure 1, we show the case report with the outcome of the system (2.4) when the basic reproductive number R*<1. The parameters are chosen as p=0.03, β=0.8, α=0.002, θ=0.2, b1=0.4, γ1=0.86,γ2=0.9, and τ=40. The three Figures 1(a), 1(b), and 1(c) in have the same initial value as s(0)=0.6296, i(0)=0.006. We fixed p=0.03 and changed parameter τ. Figures 1(a), 1(b), and 1(c) show the solutions for τ=40 and R*=0.9876. It suggests that the disease-free periodic solution is global asymptotically stable when R*<1.

The time series and the orbits of the system (2.4) with R*<1. (a) and (b) show the time series for susceptible and infective, respectively. (c) shows the orbits s-i plane.

Figure 2 shows that the positive periodic solution is existence when R*1+, moreover, the positive periodic solution is global asymptotically stable. The parameters are chosen as p=0.005, β=0.8, α=0.002, θ=0.2, b1=0.4, γ1=0.32,γ2=0.9, and τ=8. Here we choose the initial value of (2.4) s(0)=0.3080, i(0)=0.006. In Figures 2(a), 2(b) and 2(c) with τ=15 and R*=1.0236, the other parameters are the same as Figure 1.

The time series and the orbits of the system (2.4) with R*>1. (a) and (b) show the time series for susceptible and infective, respectively. (c) shows the orbits s-i plane.

6. Discussion

In this paper, we have investigated the dynamic behaviors of the classical SIR model. A distinguishing feature of the SIR model considered here is that the epidemic incidence is standard form instead of bilinear form as usual. The basic reproductive number R* is identified and is established as a sharp threshold parameter. If R*<1, the infection-free periodic solution is global asymptotically stable which implies that the disease will extinct. If R*>1, the disease will have uniform persistence and lead to epidemic disease eventually. Our theoretical results are confirmed by numerical results.

When we are modeling the transmission of some infectious diseases with pulse birth, the introduction of the standard incidence can make the model more realistic, whereas it raises hardness of the problem at the same time. For example, we attempted to achieve the global stability of infection-free periodic solution in Section 3, and we found it is impossible to prove limti(t)=0 by traditional techniques. In this case, we made the conclusion by making use of the new variable v=s+r. The SIR epidemic model with pulse birth is one of the simple and important epidemic models.

At the same time, the paper assumes the susceptible, infectious, and recovered have the same birth rate. But by the effect of the infectious diseases to the fertility of the infected, we can also assume that the susceptible and recovered have the same birth rate, which is higher than the infectious birth rate. Furthermore, we can assume that the infectious has a lower fertility than the susceptible and recovered due to the effect of the disease. So a distinguishing feature of the model considered here is that the susceptible, infectious, and recovered have different birth rates, which makes the model more realistic. For the above models we could get the similar condition for the stability of the infection-free periodic solution.

Acknowledgments

This work is supported by the National Sciences Foundation of China ( 60771026), Program for New Century Excellent Talents in University (NECT050271), and Science Foundation of Shanxi Province ( 2009011005-1).

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