Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 490437 | 18 pages | https://doi.org/10.1155/2009/490437

The Dynamics of the Pulse Birth in an SIR Epidemic Model with Standard Incidence

Academic Editor: Antonia Vecchio
Received21 Oct 2008
Revised06 Feb 2009
Accepted29 Apr 2009
Published08 Jul 2009

Abstract

An SIR epidemic model with pulse birth and standard incidence is presented. The dynamics of the epidemic model is analyzed. The basic reproductive number π‘…βˆ— is defined. It is proved that the infection-free periodic solution is global asymptotically stable if π‘…βˆ—<1. The infection-free periodic solution is unstable and the disease is uniform persistent if π‘…βˆ—>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 [3]. 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 [4–6]. 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 [7]. Nokes and Swinton studied the control of childhood viral infections by pulse vaccination [8]. Jin studied the global stability of the disease-free periodic solution for SIR and SIRS models with pulse vaccination [9]. Stone et al. presented a theoretical examination of the pulse vaccination policy in the SIR epidemic model [10]. 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 [11]. d'Onofrio studied the use of pulse vaccination strategy to eradicate infectious disease for SIR and SEIR epidemic models [12–15]. Shi and Chen studied stage-structured impulsive SI model for pest management [16]. 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 [17–19].

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 π‘…βˆ— is defined. Local stability and the global asymptotically stable of the infection-free periodic solution are obtained when π‘…βˆ—<1. Section 4 concentrates on the uniform persistence of the infectious disease when π‘…βˆ—>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 π‘†ξ…žπ‘†=βˆ’π›½π‘πΌπΌβˆ’π‘‘π‘†βˆ’π‘π‘†,ξ…žπ‘†=π›½π‘π‘…πΌβˆ’(𝑑+πœƒ+𝛼)𝐼,ξ…žπ‘†ξ€·=πœƒπΌβˆ’π‘‘π‘…+𝑝𝑆,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑆(π‘›πœ)+𝑏1𝑆(π‘›πœ)+𝑏1𝛾1𝐼(π‘›πœ)+𝑏1𝛾2𝐼𝑅(π‘›πœ),π‘›πœ+ξ€Έ=𝐼(π‘›πœ)+𝑏1ξ€·1βˆ’π›Ύ1𝑅𝐼(π‘›πœ),π‘›πœ+ξ€Έ=𝑅(π‘›πœ)+𝑏1ξ€·1βˆ’π›Ύ2𝑅(π‘›πœ),𝑑=π‘›πœ.(2.1) The total population size is denoted by 𝑁, with 𝑁=𝑆+𝐼+𝑅. Here the parameters 𝛽, 𝑑, 𝑝, πœƒ, 𝛼, 𝑏1, 𝛾1, 𝛾2 are all positive constants. 𝛽 is adequate contact rate, 𝑑 is the per capita death rate, πœƒ 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 𝛽β‰₯𝛼.

From (2.1), we obtain π‘ξ…žπ‘ξ€·=βˆ’π‘‘π‘βˆ’π›ΌπΌ,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=ξ€·1+𝑏1𝑁(π‘›πœ),𝑑=π‘›πœ.(2.2) Let 𝑠=𝑆/𝑁,𝑖=𝐼/𝑁,π‘Ÿ=𝑅/𝑁, then systems (2.1) and (2.2) can be written as follows: π‘ ξ…žπ‘–=βˆ’π‘π‘ +(π›Όβˆ’π›½)𝑠𝑖,ξ…ž=(π›½π‘ βˆ’πœƒβˆ’π›Ό)𝑖+𝛼𝑖2,π‘Ÿξ…žπ‘ ξ€·=𝑝𝑠+πœƒπ‘–+π›Όπ‘–π‘Ÿ,π‘‘β‰ π‘›πœ,π‘›πœ+𝑏=𝑠(π‘›πœ)+1𝛾11+𝑏1𝑏𝑖(π‘›πœ)+1𝛾21+𝑏1π‘–ξ€·π‘Ÿ(π‘›πœ),π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έξ€·1+𝑏1ξ€Έπ‘Ÿξ€·π‘–(π‘›πœ),π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ2ξ€Έξ€·1+𝑏1ξ€Έπ‘Ÿ(π‘›πœ),𝑑=π‘›πœ.(2.3) The total population size is normalized to one. By virtue of the equation 𝑠(𝑑)+𝑖(𝑑)+π‘Ÿ(𝑑)=1, we ignore the third and the sixth equations of system (2.3) to study the two-dimensional system:

π‘ ξ…žπ‘–=βˆ’π‘π‘ +(π›Όβˆ’π›½)𝑠𝑖,ξ…ž=(π›½π‘ βˆ’πœƒβˆ’π›Ό)𝑖+𝛼𝑖2𝑠,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑏𝑠(π‘›πœ)+1𝛾11+𝑏1βˆ’π‘1𝛾21+𝑏1𝑖𝑖(π‘›πœ),π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1𝑖(π‘›πœ),𝑑=π‘›πœ.(2.4)

From biological view, we easily see that the domain Ξ©={(𝑠,𝑖,π‘Ÿ)βˆΆπ‘ β‰₯0,𝑖β‰₯0,π‘Ÿβ‰₯0,𝑠+𝑖+π‘Ÿ=1}(2.5) 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(𝑑)=0,𝑑β‰₯0. Under this condition, the growth of susceptible individuals and the population must satisfy π‘ ξ…žπ‘ ξ€·=βˆ’π‘π‘ ,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑠(π‘›πœ),𝑑=π‘›πœ.(2.6) Integrating the first equation in system (2.6) between pulses, it is easy to obtain the solution with initial value 𝑠(0+)=𝑠0, 𝑠(𝑑)=π‘ π‘›πœ+ξ€Έπ‘’βˆ’π‘(π‘‘βˆ’π‘›πœ),π‘›πœ<𝑑≀(𝑛+1)𝜏.(2.7) Equation (2.7) holds between pulses. At each successive pulse, it yields 𝑠(𝑛+1)𝜏+ξ€Έ=𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1ξ‚Άπ‘’βˆ’π‘πœπ‘ ξ€·π‘›πœ+𝑠=πΉπ‘›πœ+ξ€Έξ€Έ.(2.8) Equation (2.8) has a unique fixed point π‘ βˆ—=𝑏1𝛾2π‘’π‘πœ/((1+𝑏1)π‘’π‘πœβˆ’(1+𝑏1βˆ’π‘1𝛾2)). The fixed point π‘ βˆ— is locally stable because 𝑑𝐹(𝑠(π‘›πœ+))/π‘‘π‘ βˆ£π‘ (π‘›πœ+)=π‘ βˆ—=(1βˆ’π‘1𝛾2/(1+𝑏1))π‘’βˆ’π‘πœ<1, By substituting 𝑠(π‘›πœ+)=π‘ βˆ— to (2.7), we obtain the complete expression for the infection-free periodic solution over the nth time-interval π‘›πœ<𝑑≀(𝑛+1)𝜏, 𝑏̃𝑠(𝑑)=1𝛾2π‘’π‘πœξ€·1+𝑏1ξ€Έπ‘’π‘πœβˆ’ξ€·1+𝑏1βˆ’π‘1𝛾2ξ€Έπ‘’βˆ’π‘(π‘‘βˆ’π‘›πœ),̃𝑖(𝑑)=0.(2.9)

Therefore the system (2.4) has the infection-free periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)).

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 Μƒ(̃𝑠(𝑑),𝑖(𝑑)).

The local stability of the infection-free periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)) may be determined by considering the linearized SIR equation of system (2.4) about the known periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)) by setting ̃𝑠(𝑑)=̃𝑠(𝑑)+π‘₯(𝑑),𝑖(𝑑)=𝑖(𝑑)+𝑦(𝑑), where π‘₯(𝑑) and 𝑦(𝑑) are small perturbation. The variables π‘₯(𝑑) and 𝑦(𝑑) are described by the relation ξƒͺξƒͺπ‘₯(𝑑)𝑦(𝑑)=Ξ¦(𝑑)π‘₯(0)𝑦(0),(3.1) where the fundamental solution matrix Ξ¦(𝑑)=πœ‘π‘–π‘—(𝑑)(𝑖,𝑗=1,2) satisfies 𝑑Φ(𝑑)=ξƒͺπ‘‘π‘‘βˆ’π‘(π›Όβˆ’π›½)̃𝑠(𝑑)0𝛽̃𝑠(𝑑)βˆ’πœƒβˆ’π›ΌΞ¦(𝑑),(3.2) with Ξ¦(0)=𝐸, where 𝐸 is the identity matrix. The resetting of the equations of (2.4) becomes π‘₯ξ€·π‘›πœ+ξ€Έπ‘¦ξ€·π‘›πœ+ξ€Έξƒͺ=βŽ›βŽœβŽœβŽœβŽπ‘1βˆ’1𝛾21+𝑏1𝑏1𝛾1βˆ’π›Ύ2ξ€Έ1+𝑏101+𝑏1βˆ’π‘1𝛾11+𝑏1βŽžβŽŸβŽŸβŽŸβŽ ξƒ©π‘₯ξƒͺ(π‘›πœ)𝑦(π‘›πœ).(3.3) Hence, according to the Floquet theory, if all eigenvalues of βŽ›βŽœβŽœβŽœβŽπ‘π‘€(𝜏)=1βˆ’1𝛾21+𝑏1𝑏1𝛾1βˆ’π›Ύ2ξ€Έ1+𝑏101+𝑏1βˆ’π‘1𝛾11+𝑏1⎞⎟⎟⎟⎠Φ(𝜏)(3.4) are less than one, then the infection-free periodic solution (̃𝑠(𝑑),0) is locally stable. By calculating, we have =Φ(𝑑)1Ξ¦120Ξ¦22ξƒͺ,(3.5) where Ξ¦22∫(𝑑)=exp(𝛽̃𝑠(𝜎)π‘‘πœŽβˆ’(πœƒ+𝛼)𝑑).

The eigenvalues of 𝑀 denoted by πœ‡1,πœ‡2 are πœ‡1=(1βˆ’π‘1𝛾2/(1+𝑏1))π‘’βˆ’π‘πœ<1, and πœ‡2=((1+𝑏1βˆ’π‘1𝛾1)/(1+𝑏1∫))exp{π›½πœ0̃𝑠(𝜎)π‘‘πœŽβˆ’(πœƒ+𝛼)𝜏}, if and only if πœ‡2<1. Define threshold of model (2.4) as follows: π‘…βˆ—=π›½βˆ«πœ0̃𝑠(𝜎)π‘‘πœŽlnξ€·ξ€·1+𝑏1ξ€Έ/ξ€·1+𝑏1βˆ’π‘1𝛾1ξ€Έξ€Έ+(πœƒ+𝛼)𝜏,(3.6) where ̃𝑠(𝑑) is the infection-free periodic solution. That is, the infection-free periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)) is locally asymptotically stable if π‘…βˆ—<1. So we obtained following theorem.

Theorem 3.1. If π‘…βˆ—<1, then the infection-free periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)) 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 Μƒ(̃𝑠(𝑑),𝑖(𝑑)), we need to use to comparison theory and impulsive differential inequality [1, 2].

Theorem 3.2. If π‘…βˆ—<1, then the infection-free periodic solution Μƒ(̃𝑠(𝑑),𝑖(𝑑)) of system (2.4) is global asymptotically stable.

Proof. Because of 𝛼≀𝛽, and 𝛾1≀𝛾2, we have π‘ ξ…žπ‘ ξ€·β‰€βˆ’π‘π‘ ,π‘‘β‰ π‘›πœ,π‘›πœ+≀𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑠(π‘›πœ),𝑑=π‘›πœ.(3.7) By impulsive differential inequality, we see that ξ€·0𝑠(𝑑)≀𝑠+ξ€Έξ‘π‘œ<π‘›πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άξ‚»ξ€œexp𝑑0(ξ‚Ό+ξ“βˆ’π‘)π‘‘πœŽπ‘œ<π‘›πœ<π‘‘ξƒ―ξ‘π‘›πœ<π‘—πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άξ‚Έξ€œexpπ‘‘π‘›πœξ‚Ήπ‘(βˆ’π‘)π‘‘πœŽ1𝛾21+𝑏1ξƒ°ξ€·0=̃𝑠(𝑑)+𝑠+ξ€Έξ‚΅1βˆ’π‘1𝛾21+𝑏1ξ‚Ά[]𝑑/π‘‡π‘’βˆ’π‘π‘‘βˆ’ξ€·π‘1𝛾2/ξ€·1+𝑏1ξ€Έξ€Έξ€·1βˆ’π‘1𝛾2/(1+𝑏1)ξ€Έ[]𝑑/π‘‡π‘’βˆ’π‘π‘‘ξ€·1βˆ’π‘1𝛾2/ξ€·1+𝑏1π‘’ξ€Έξ€Έβˆ’π‘π‘‡.(3.8) Since limπ‘‘β†’βˆžξƒ―π‘ ξ€·0+𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά[]𝑑/π‘‡π‘’βˆ’π‘π‘‘βˆ’ξ€·π‘1𝛾2/ξ€·1+𝑏1ξ€Έξ€Έξ€·1βˆ’π‘1𝛾2/ξ€·1+𝑏1ξ€Έξ€Έ[]𝑑/π‘‡π‘’βˆ’π‘π‘‘ξ€·1βˆ’π‘1𝛾2/ξ€·1+𝑏1π‘’ξ€Έξ€Έβˆ’π‘π‘‡ξƒ°=0,(3.9) for any given πœ–1>0, there exists 𝑇1>0, such that 𝑠(𝑑)<̃𝑠(𝑑)+πœ–1, for all 𝑑>𝑇1.

Introduce the new variable 𝑒=𝑠+π‘Ÿ, then π‘’ξ…ž=𝑒(βˆ’π›½π‘ +𝛼𝑒+πœƒ)(1βˆ’π‘’),π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾11+𝑏1+𝑏1βˆ’1𝛾11+𝑏1𝑒(π‘›πœ),𝑑=π‘›πœ.(3.10)

Consider the following comparison system with pulse: π‘£ξ…žξ€·=βˆ’π›½Μƒπ‘ (𝑑)+π›½πœ–1ξ€Έ+π›Όβˆ’πœƒπ‘£βˆ’π›Όπ‘£2ξ€·+πœƒβˆ’π›½Μƒπ‘ (𝑑)+πœ–1𝑣,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾11+𝑏1+𝑏1βˆ’1𝛾11+𝑏1𝑣(π‘›πœ),𝑑=π‘›πœ.(3.11)

The first equation of (3.11) is Riccati equation. It is easy to see that 𝑣(𝑑)=1 is a solution of system (3.11). Let 𝑦=π‘£βˆ’1, then π‘¦ξ…žξ€·=βˆ’π›½Μƒπ‘ (𝑑)βˆ’π›Όβˆ’πœƒ+π›½πœ–1ξ€Έπ‘¦βˆ’π›Όπ‘¦2𝑦,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1βˆ’1𝛾11+𝑏1𝑦(π‘›πœ),𝑑=π‘›πœ.(3.12)

Let 𝑧=1/𝑦, then π‘§ξ…žξ€·=βˆ’π›½Μƒπ‘ (𝑑)βˆ’π›Όβˆ’πœƒ+π›½πœ–1ξ€Έπ‘§ξ€·π‘§βˆ’π›Ό,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=ξ‚΅1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑧(π‘›πœ),𝑑=π‘›πœ.(3.13)

Let π‘ž(𝑑)=𝛽̃𝑠(𝑑)βˆ’π›Όβˆ’πœƒ+π›½πœ–1, solving system (3.13) between pulses (𝑇1+π‘›πœ,𝑇1+(𝑛+1)𝜏], we have 𝑧(𝑑)=π‘’βˆ’βˆ«π‘‘π‘‡1+π‘›πœπ‘ž(𝜎)π‘‘πœŽξ‚Έπ›Όξ€œπ‘‘π‘‡1+π‘›πœπ‘’βˆ«π‘’π‘‡1+π‘›πœπ‘ž(𝜎)π‘‘πœŽπ‘‘π‘’+1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑧𝑇1ξ€Έξ‚Ή+π‘›πœ,(3.14) when 𝑑=𝑇1+(𝑛+1)𝜏, (3.14) can be written as follows: 𝑧𝑇1ξ€Έ+(𝑛+1)𝜏=π‘’βˆ’βˆ«π‘‡1𝑇1+(𝑛+1)𝜏+π‘›πœπ‘ž(𝜎)π‘‘πœŽξ‚Έπ›Όξ€œπ‘‡1𝑇+(𝑛+1)𝜏1+π‘›πœπ‘’βˆ«π‘’π‘‡1+π‘›πœπ‘ž(𝜎)π‘‘πœŽπ‘‘π‘’+1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑧𝑇1ξ€Έξ‚Ή+π‘›πœ.(3.15) On the other hand, solving system (3.13) between pulses (𝑇1+(π‘›βˆ’1)𝜏,𝑇1+π‘›πœ], we obtain 𝑧(𝑑)=π‘’βˆ’βˆ«π‘‘π‘‡1+(π‘›βˆ’1)πœπ‘ž(𝜎)π‘‘πœŽξ‚Έπ›Όξ€œπ‘‘π‘‡1+(π‘›βˆ’1)πœπ‘’βˆ«π‘’π‘‡1+(π‘›βˆ’1)πœπ‘ž(𝜎)π‘‘πœŽπ‘‘π‘’+1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑧𝑇1ξ€Έξ‚Ή+(π‘›βˆ’1)𝜏,(3.16) then 𝑧𝑇1ξ€Έ+π‘›πœ=π‘’βˆ’βˆ«π‘‡1𝑇1+(𝑛+1)𝜏+π‘›πœπ‘ž(𝜎)π‘‘πœŽξ‚Έπ›Όξ€œπ‘‡1𝑇+π‘›πœ1+(π‘›βˆ’1)πœπ‘’βˆ«π‘’π‘‡1+(π‘›βˆ’1)πœπ‘ž(𝜎)π‘‘πœŽπ‘‘π‘’+1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑧𝑇1ξ€Έξ‚Ή+(π‘›βˆ’1)𝜏.(3.17)

Similarly, we can get the expressions of 𝑧(𝑇1+(π‘›βˆ’1)𝜏),𝑧(𝑇1+(π‘›βˆ’2)𝜏),…,𝑧(𝑇1). Then using iterative technique step by step, 𝑧𝑇1ξ€Έ+π‘›πœ=π‘’βˆ’βˆ«0π‘›πœπ‘ž(𝜎)π‘‘πœŽξ‚΅1+𝑏11+𝑏1βˆ’π‘1𝛾1𝑛1β‰€π‘˜β‰€π‘›ξ‚΅1+𝑏11+𝑏1βˆ’π‘1𝛾1ξ‚Άπ‘˜βˆ’π‘›βˆ’1π‘’βˆ«π‘‡1𝑇1+(π‘›βˆ’π‘˜)πœπ‘ž(𝜎)π‘‘πœŽ+ξ€œπ‘‘π‘’π‘‡1𝑇+(π‘›βˆ’π‘˜+1)𝜏1+(π‘›βˆ’π‘˜)πœπ‘’βˆ«π‘’π‘‡1+(π‘›βˆ’π‘˜)πœπ‘ž(𝜎)π‘‘πœŽξ€·π‘‡π‘‘π‘’+𝑧1ξ€Έξ‚Ή,(3.18) where π‘’βˆ’βˆ«0π‘›πœπ‘ž(𝜎)π‘‘πœŽξ‚΅1+𝑏11+𝑏1βˆ’π‘1𝛾1ξ‚Άπ‘›ξ‚»π‘›ξ‚Έξ€œ=expβˆ’π›½πœ0̃𝑠(𝜎)π‘‘πœŽ+(πœƒ+𝛼)πœβˆ’π›½πœ–1𝜏+ln1+𝑏11+𝑏1βˆ’π‘1𝛾1ξ‚Ήξ‚Ό.(3.19)

The condition π‘…βˆ—<1 implies that limπ‘›β†’βˆžπ‘§(π‘›πœ)=∞, then limπ‘‘β†’βˆžπ‘₯(𝑑)=1. The comparison principle and the condition 𝑒(𝑑)<1 imply that limπ‘‘β†’βˆžπ‘’(𝑑)=1, so we have limπ‘‘β†’βˆžπ‘–(𝑑)=0.

Because we have proved that limπ‘‘β†’βˆžπ‘–(𝑑)=0 when π‘…βˆ—<1, for any given πœ–2>0, there exists 𝑇2>0, such that βˆ’πœ–2<𝑖(𝑑)<πœ–2, for all 𝑑>𝑇2.

When 𝑑>𝑇2, from system (2.4), we have π‘ ξ…žβ‰₯βˆ’π‘π‘ +(π›Όβˆ’π›½)πœ–2𝑠𝑠,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έβ‰₯𝑏1𝛾11+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑠(π‘›πœ),𝑑=π‘›πœ.(3.20) Therefore 𝑇𝑠(𝑑)β‰₯𝑠+2𝑇2<π‘›πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άξ‚»ξ€œexp𝑑𝑇2ξ€·βˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ‚Ό+ξ“π‘‘πœŽπ‘‡2<π‘›πœ<𝑑𝑇2<π‘—πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άξ‚»ξ€œexpπ‘‘π‘›πœξ€·βˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ‚Όπ‘π‘‘πœŽ1𝛾11+𝑏1𝑇=𝑠+2𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά[𝑑/𝑇]βˆ’[𝑇2/𝑇]ξ€·expβˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ€·π‘‘βˆ’π‘‡2ξ€Έ+𝑏1𝛾1/ξ€·1+𝑏1ξ€Έξ€·1βˆ’1βˆ’π‘1𝛾2/ξ€·1+𝑏1ξ€·ξ€Έξ€Έexpβˆ’π‘+(π›Όβˆ’π›½)πœ–2𝑇expβˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ‚€ξ‚ƒπ‘‘π‘‘βˆ’π‘‡ξ‚„π‘‡ξ‚βˆ’π‘1𝛾1/ξ€·1+𝑏1ξ€Έξ€·1βˆ’1βˆ’π‘1𝛾2/ξ€·1+𝑏1ξ€·ξ€Έξ€Έexpβˆ’π‘+(π›Όβˆ’π›½)πœ–2𝑇×𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά([𝑑/𝑇]βˆ’[𝑇2/𝑇]βˆ’1)ξ€·expβˆ’π‘+(π›Όβˆ’π›½)πœ–2𝑇.(3.21) For any given πœ–2>0, we have limπ‘‘β†’βˆžξƒ―π‘ ξ€·π‘‡+2𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά[𝑑/𝑇]βˆ’[𝑇2/𝑇]ξ€·expβˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ€·π‘‘βˆ’π‘‡2ξ€Έβˆ’π‘1𝛾1/ξ€·1+𝑏1ξ€Έξ€·1βˆ’1βˆ’π‘1𝛾2/ξ€·1+𝑏1ξ€·ξ€Έξ€Έexpβˆ’π‘+(π›Όβˆ’π›½)πœ–2𝑇×𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά([𝑑/𝑇]βˆ’[𝑇2/𝑇]βˆ’1)ξ€·expβˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξƒ°(𝑇)=0,limπ‘‘β†’βˆžπ‘1𝛾1/ξ€·1+𝑏1ξ€Έξ€·1βˆ’1βˆ’π‘1𝛾2/ξ€·1+𝑏1ξ€·ξ€Έξ€Έexpβˆ’π‘+(π›Όβˆ’π›½)πœ–2𝑇expβˆ’π‘+(π›Όβˆ’π›½)πœ–2ξ€Έξ‚€ξ‚ƒπ‘‘π‘‘βˆ’π‘‡ξ‚„π‘‡ξ‚=̃𝑠(𝑑).(3.22) Therefore, for any given πœ–3>0, there exists 𝑇3>0, when 𝑑>𝑇3, then we have 𝑠(𝑑)β‰₯̃𝑠(𝑑)βˆ’πœ–3.(3.23) For any given πœ–>0. Let 𝑇=max{𝑇1,𝑇2,𝑇3}, then 𝑑>𝑇, then we have ̃𝑠(𝑑)βˆ’πœ–β‰₯𝑠(𝑑)β‰₯̃𝑠(𝑑)+πœ–,(3.24) that is limπ‘‘β†’βˆžπ‘ (𝑑)=̃𝑠(𝑑).

Therefore the infection-free periodic solution (̃𝑠(𝑑),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, limπ‘‘β†’βˆžinf𝑖(𝑑)β‰₯𝜌>0 if π‘…βˆ—>1.

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

Lemma 4.1. For the following impulsive equation, π‘₯ξ…žπ‘₯ξ€·=βˆ’π‘”π‘₯βˆ’β„Ž,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾11+𝑏1+𝑏1βˆ’1𝛾21+𝑏1ξ‚Άπ‘₯(π‘›πœ),𝑑=π‘›πœ,(4.1) has a unique positive 𝜏-periodic solution Μƒπ‘₯(𝑑) for which Μƒπ‘₯(0)>0, β€‰β€‰π‘‘βˆˆπ‘…+, and Μƒπ‘₯(𝑑) is global asymptotically stable in the sense that limπ‘‘β†’βˆž|π‘₯(𝑑)βˆ’Μƒπ‘₯(𝑑)|=0, where π‘₯(𝑑) is any solution of system (2.2) with positive initial value π‘₯(0)>0 and 𝑔,β„Ž are positive constants.

Proof. Solving (4.1), we have ξ€œπ‘₯(𝑑)=π‘Š(𝑑,0)π‘₯(0)βˆ’β„Žπ‘‘0π‘π‘Š(𝑑,𝜎)π‘‘πœŽ+1𝛾11+𝑏10<π‘›πœ<π‘‘π‘Šξ€·π‘‘,π‘›πœ+ξ€Έ,(4.2) where π‘Šξ€·π‘‘,𝑑0ξ€Έ=𝑑0β‰€π‘›πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άπ‘’βˆ’π‘”(π‘‘βˆ’π‘‘0).(4.3) Since π‘Š(𝜏,0)=(1βˆ’π‘1𝛾2/(1+𝑏1))π‘’βˆ’π‘”πœ<1, (4.1) has a unique positive 𝜏-periodic solution Μƒπ‘₯(𝑑) with the initial value βˆ«Μƒπ‘₯(0)=(βˆ’β„Žπœ0π‘Š(𝜏+,𝜎)π‘‘πœŽ+(𝑏1𝛾1/(1+𝑏1))π‘Š(𝜏,𝜏))/(1βˆ’π‘Š(𝜏,0)). Next, we only need to prove that limπ‘‘β†’βˆž|π‘₯(𝑑)βˆ’Μƒπ‘₯(𝑑)|=0.
Since ||||||||π‘₯(𝑑)βˆ’Μƒπ‘₯(𝑑)=π‘Š(𝑑,0)π‘₯(0)βˆ’Μƒπ‘₯(0),(4.4) the result is obtained if π‘Š(𝑑,0)β†’0 as π‘‘β†’βˆž. Suppose π‘‘βˆˆ(π‘›πœ,(𝑛+1)𝜏], then ξ‘π‘Š(𝑑,0)=0β‰€π‘—πœ<𝑑𝑏1βˆ’1𝛾21+𝑏1ξ‚Άπ‘’βˆ’π‘”π‘‘=𝑏1βˆ’1𝛾21+𝑏1ξ‚Ά[𝑑/𝜏]π‘’βˆ’π‘”π‘‘.(4.5) Thus limπ‘‘β†’βˆžπ‘Š(𝑑,0)=0. The proof is complete.

Lemma 4.2. If π‘…βˆ—>1, then the disease uniformly weakly persists in the population, in the sense that there exists some 𝑐>0 such that limπ‘‘β†’βˆžsup𝑖(𝑑)>𝑐 for all solutions of (2.4).

Proof. Suppose that for every πœ–>0, there is some solution with limπ‘‘β†’βˆžsup𝑖(𝑑)<πœ–. From the first equation of (2.4), we have π‘ ξ…ž=βˆ’π‘π‘ +(π›Όβˆ’π›½)𝑠𝑖β‰₯βˆ’π‘π‘ +(π›Όβˆ’π›½)πœ–,π‘‘β‰ π‘›πœ.(4.6) Consider the following equation: π‘€ξ…žπ‘€ξ€·=βˆ’π‘π‘€+(π›Όβˆ’π›½)πœ–,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾11+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑀(π‘›πœ),𝑑=π‘›πœ.(4.7) By Lemma 4.1, we see that (4.7) has a unique positive 𝜏-periodic solution 𝑀(𝑑), and 𝑀(𝑑) is global asymptotically stable. Solving (4.7), we have ξ€œπ‘€(𝑑)=π‘Š(𝑑,0)𝑀(0)+(π›Όβˆ’π›½)πœ–π‘‘0π‘π‘Š(𝑑,𝜎)π‘‘πœŽ+1𝛾11+𝑏10<π‘›πœ<π‘‘π‘Šξ€·π‘‘,π‘›πœ+ξ€Έ,ξ‚ξƒ©βˆ«(4.8)𝑀(𝑑)=(π›Όβˆ’π›½)πœ–π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒͺ+ξ‚΅π‘π‘Š(𝑑,𝜎)π‘‘πœŽ1βˆ’1𝛾21+𝑏1ξ‚Άπ‘Š(𝑑,0)π‘Š(𝜏,𝜏)+𝑏1βˆ’π‘Š(𝜏,0)1𝛾11+𝑏10<π‘›πœ<π‘‘π‘Šξ€·π‘‘,π‘›πœ+ξ€Έ,(4.9) and 𝑀(𝑑) is global asymptotically stable. By (4.9), let 𝛼=𝛽, we obtain the periodic solution of (2.6) that 𝑏̃𝑠(𝑑)=1βˆ’1𝛾21+𝑏1ξ‚Άπ‘Š(𝑑,0)π‘Š(𝜏,𝜏)+𝑏1βˆ’π‘Š(𝜏,0)1𝛾11+𝑏10<π‘›πœ<π‘‘π‘Šξ€·π‘‘,π‘›πœ+ξ€Έ.(4.10) and we have ξ‚ξƒ©βˆ«Μƒπ‘ (𝑑)βˆ’π‘€(𝑑)=(π›½βˆ’π›Ό)πœ–π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒͺπ‘Š(𝑑,𝜎)π‘‘πœŽ.(4.11) Let Ξ”=(π›½βˆ’π›Ό)max0β‰€π‘‘β‰€πœξƒ―βˆ«π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒ°π‘Š(𝑑,𝜎)π‘‘πœŽ,(4.12) by (4.11), we can see that 𝑀(𝑑)β‰₯̃𝑠(𝑑)βˆ’Ξ”πœ–.(4.13) By comparison theory, we obtain that π‘–ξ…žβ‰₯βˆ’(πœƒ+𝛼)𝑖+𝛽𝑖𝑀(𝑑)+𝛼𝑖2.(4.14) Since 𝑀(𝑑) is global asymptotically stable, for above πœ–, there exists 𝑇4>0, such that 𝑀(𝑑)β‰₯𝑀(𝑑)βˆ’πœ–, 𝑑>𝑇4. From (4.13) and (4.14), we have that π‘–ξ…žβ‰₯[]𝛽̃𝑠(𝑑)βˆ’(πœƒ+𝛼)βˆ’(1+Ξ”)π›½πœ–π‘–.(4.15) Consider the following equation: π‘–ξ…žβ‰₯[]𝑖𝛽̃𝑠(𝑑)βˆ’(πœƒ+𝛼)βˆ’(1+Ξ”)π›½πœ–π‘–,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1𝑖(π‘›πœ),𝑑=π‘›πœ.(4.16) By impulsive differential inequality, for π‘‘βˆˆ(𝑇4+π‘›πœ,𝑇4+(𝑛+1)𝜏], we see that 𝑇𝑖(𝑑)β‰₯𝑖4𝑇4<π‘—πœ<𝑑1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1ξ‚»ξ€œexp𝑑𝑇4[]𝑇𝛽̃𝑠(𝜎)βˆ’(πœƒ+𝛼)βˆ’(1+Ξ”)π›½πœ–π‘‘πœŽ=𝑖4ξ€Έξ‚΅1+𝑏1(1βˆ’π›Ύ1)1+𝑏1ξ‚Άπ‘›ξ‚»ξ€œexp𝑇4𝑇+π‘›πœ4[]+ξ€œπ›½Μƒπ‘ (𝜎)βˆ’(πœƒ+𝛼)βˆ’(1+Ξ”)π›½πœ–π‘‘πœŽπ‘‘π‘‡4+π‘›πœ[]𝑛𝑅𝛽̃𝑠(𝜎)βˆ’(πœƒ+𝛼)βˆ’(1+Ξ”)π›½πœ–π‘‘πœŽβ‰₯𝐢expβˆ—ξ€Έξƒ©βˆ’1ln1+𝑏11+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έξƒͺ,+(πœƒ+𝛼)πœβˆ’(1+Ξ”)π›½πœ–πœξƒ­ξƒ°(4.17) where 𝐢=𝑖(𝑇4)exp{βˆ’[(πœƒ+𝛼)+(1+Ξ”)π›½πœ–]𝜏}. Taking 𝑅0<πœ–<βˆ—βˆ’1ξ€Έξ€Ίlnξ€·ξ€·1+𝑏1ξ€Έ/ξ€·1+𝑏1ξ€·1βˆ’π›Ύ1+ξ€»ξ€Έξ€Έξ€Έ(πœƒ+𝛼)𝜏(1+Ξ”)π›½πœ,(4.18) thus 𝑖(𝑑)β†’βˆž as π‘‘β†’βˆž, a contradiction to the fact that 𝑖(𝑑) is bounded. The proof is complete.

Theorem 4.3. If π‘…βˆ—>1, then the disease is uniformly persistent, that is, there exists a positive constant 𝜌 such that for every positive solution of (2.4), limπ‘‘β†’βˆžinf𝑖(𝑑)β‰₯𝜌>0.

Proof. Let 10<πœ‚β‰€2ξ‚€11βˆ’π‘…βˆ—ξ‚Μƒπ‘ (𝑑)𝑀(πœƒ+𝛼),(4.19) where 𝑀=max0β‰€π‘‘β‰€πœξƒ―βˆ«π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒ°,π‘Šξ€·π‘Š(𝑑,𝜎)π‘‘πœŽπ‘‘,𝑑0ξ€Έ=𝑑0β‰€π‘›πœ<𝑑1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1π‘’ξ€·βˆ’π‘π‘‘βˆ’π‘‘0ξ€Έ.(4.20) It can be obtained from Lemma 4.1 that for any positive solution of (2.4) there exists at least one 𝑑0>0 such that 𝑖(𝑑0)>πœ‚>0. Then, we are left to consider two case. The first case is 𝑖(𝑑)β‰₯πœ‚ for all large 𝑑β‰₯𝑑0. The second case is 𝑖(𝑑) oscillates about πœ‚ for large t. The conclusion of Theorem 4.3 is obvious in the first case since we can choose 𝜌=πœ‚. For the second case, let 𝑑1>𝑑0, and let 𝑑2>𝑑1 satisfy 𝑖𝑑1𝑑=𝑖2ξ€Έ=πœ‚,𝑖(𝑑)<πœ‚for𝑑1<𝑑<𝑑2.(4.21) Next, we introduce the new variable 𝑉=𝑠+𝑖, and it follows from the first two equations of (2.4) that π‘‰ξ…žπ‘–=βˆ’π‘π‘‰βˆ’(πœƒ+𝛼)𝑖+(𝛼𝑉+𝑝)𝑖,ξ…ž=βˆ’(πœƒ+𝛼)𝑖+𝛽(π‘‰βˆ’π‘–)𝑖+𝛼𝑖2𝑉,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1𝑖𝑉(π‘›πœ),π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1𝑖(π‘›πœ),𝑑=π‘›πœ.(4.22) If 𝑖(𝑑)β‰€πœ‚, then π‘‰ξ…žβ‰₯βˆ’π‘π‘‰βˆ’(πœƒ+𝛼)πœ‚,π‘‘β‰ π‘›πœ.
Consider the following equation: π‘₯ξ…žπ‘₯ξ€·=βˆ’π‘π‘₯βˆ’(πœƒ+𝛼)πœ‚,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=𝑏1𝛾21+𝑏1+𝑏1βˆ’1𝛾21+𝑏1ξ‚Άπ‘₯(π‘›πœ),𝑑=π‘›πœ.(4.23)
By Lemma 4.1, we see that (4.23) has a unique positive 𝜏-periodic solution Μƒπ‘₯(𝑑), and Μƒπ‘₯(𝑑) is global asymptotically stable. Solving (4.23), we have ξƒ©βˆ«Μƒπ‘₯(𝑑)=βˆ’(πœƒ+𝛼)πœ‚π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒͺ+ξ‚΅π‘π‘Š(𝑑,𝜎)π‘‘πœŽ1βˆ’1𝛾21+𝑏1ξ‚Άξ€·πœπ‘Š(𝑑,0)π‘Š+,𝜏+ξ€Έ+𝑏(1βˆ’π‘Š(𝜏,0))1𝛾21+𝑏10<π‘›πœ<π‘‘π‘Šξ€·π‘‘,π‘›πœ+ξ€Έ.(4.24) From (4.10) and (4.24), it is easy to see that ξƒ©βˆ«Μƒπ‘₯(𝑑)βˆ’Μƒπ‘ (𝑑)β‰₯βˆ’(πœƒ+𝛼)πœ‚π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒͺπ‘Š(𝑑,𝜎)π‘‘πœŽ.(4.25) By 0<πœ‚β‰€(1/2)(1βˆ’1/π‘…βˆ—)(̃𝑠(𝑑)/𝑀(πœƒ+𝛼)), 𝑀=max0β‰€π‘‘β‰€πœβˆ«{π‘Š(𝑑,0)𝜏0π‘Š(𝜏+∫,𝜎)π‘‘πœŽ/(1βˆ’π‘Š(𝜏,0))+𝑑0π‘Š(𝑑,𝜎)π‘‘πœŽ}, and ∫(π‘Š(𝑑,0)𝜏0π‘Š(𝜏+∫,𝜎)π‘‘πœŽ/(1βˆ’π‘Š(𝜏,0))+𝑑0π‘Š(𝑑,𝜎)π‘‘πœŽ)(1/𝑀)≀1, we obtain ξƒ©βˆ«βˆ’(πœƒ+𝛼)πœ‚π‘Š(𝑑,0)𝜏0π‘Šξ€·πœ+ξ€Έ,πœŽπ‘‘πœŽ+ξ€œ1βˆ’π‘Š(𝜏,0)𝑑0ξƒͺ1π‘Š(𝑑,𝜎)π‘‘πœŽβ‰₯βˆ’2ξ‚€11βˆ’π‘…βˆ—ξ‚Μƒπ‘ (𝑑),(4.26) namely, 1Μƒπ‘₯(𝑑)β‰₯2ξ‚€11+π‘…βˆ—ξ‚Μƒπ‘ (𝑑).(4.27) The comparison principle and the global asymptotically stable of Μƒπ‘₯(𝑑) imply that there exists a positive constant 𝑇5>0 such that 1𝑉(𝑑)β‰₯2ξ‚€11+π‘…βˆ—ξ‚Μƒπ‘ (𝑑),βˆ€π‘‘>𝑑1+𝑇5.(4.28) From (4.28) and the second equation of (4.22), we can see that π‘–ξ…žβ‰₯𝛽2ξ‚€11+π‘…βˆ—ξ‚ξ‚ΉΜƒπ‘ (𝑑)βˆ’(πœƒ+𝛼)𝑖+(π›Όβˆ’π›½)𝑖2.(4.29) Consider the following equation: π‘¦ξ…ž=𝛽2ξ‚€11+π‘…βˆ—ξ‚ξ‚ΉΜƒπ‘ (𝑑)βˆ’(πœƒ+𝛼)𝑦+(π›Όβˆ’π›½)𝑦2𝑦,π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=1+𝑏1ξ€·1βˆ’π›Ύ1ξ€Έ1+𝑏1𝑦(π‘›πœ),𝑑=π‘›πœ.(4.30) Let 𝑧=π‘¦βˆ’1, then we have π‘§ξ…ž=𝛽(πœƒ+𝛼)βˆ’2ξ‚€11+π‘…βˆ—ξ‚ξ‚Ήπ‘§ξ€·Μƒπ‘ (𝑑)𝑧+(π›½βˆ’π›Ό),π‘‘β‰ π‘›πœ,π‘›πœ+ξ€Έ=1+𝑏11+𝑏1ξ€·1βˆ’π›Ύ1𝑧(π‘›πœ),𝑑=π‘›πœ.(4.31) By the same method of Lemma 4.1, we can get a conclusion that βˆ«Μƒπ‘§(𝑑)=(π›½βˆ’π›Ό)(π‘Š(𝑑,0)𝜏0π‘Š(𝜏+∫,𝜎)π‘‘πœŽ/(1βˆ’π‘Š(𝜏,0))+𝑑0π‘Š(𝑑,𝜎)π‘‘πœŽ)+(1+𝑏1)/(1+𝑏1(1βˆ’π›Ύ1))(π‘Š(𝑑,0)π‘Š(𝜏,𝜏)/(1βˆ’π‘Š(𝜏,0))) is global asymptotically stable. Thus system (4.30) has a unique positive 𝜏-periodic solution ̃𝑦(𝑑), and ̃𝑦(𝑑) is global asymptotically stable, limπ‘‘β†’βˆž||||𝑦(𝑑)βˆ’Μƒπ‘¦(𝑑)=0.(4.32) From (4.32) we see that there exists a positive constant 𝑇6>0 such that 1𝑦(𝑑)>πœŒβ‰‘2min𝑑1≀𝑑≀𝑑1+πœΜƒπ‘¦(𝑑)>0,βˆ€π‘‘>𝑑1+𝑇6.(4.33) Let π‘‡βˆ—=max{𝑇5,𝑇6}, and define 𝜌=min{𝜌,πœ‚exp(βˆ’(πœƒ+𝛼)𝜏)}. If 𝑑2βˆ’π‘‘1<π‘‡βˆ—, from the second equation of (4.22), we have the inequality π‘–ξ…ž(𝑑)β‰₯βˆ’(πœƒ+𝛼)𝑖,(4.34) and the comparison principle implies that 𝑖(𝑑)β‰₯πœ‚exp{βˆ’(πœƒ+𝛼)(π‘‘βˆ’π‘‘1)}β‰₯πœ‚exp{βˆ’(πœƒ+𝛼)π‘‡βˆ—}, that is, 𝑖(𝑑)β‰₯𝜌 for all π‘‘βˆˆ(𝑑1,𝑑2).
If 𝑑2βˆ’π‘‘1>π‘‡βˆ—, we divide the interval [𝑑1,𝑑2] into two subintervals [𝑑1,𝑑1+π‘‡βˆ—] and [𝑑1+π‘‡βˆ—,𝑑2], 𝑖(𝑑)β‰₯𝜌 is obvious in the interval [𝑑1,𝑑1+π‘‡βˆ—]. In the interval [𝑑1+π‘‡βˆ—,𝑑2], we have the inequality (4.29) and (4.33). The comparison principle shows that 𝑖(𝑑)β‰₯𝑦(𝑑)β‰₯𝜌β‰₯𝜌 for π‘‘βˆˆ[𝑑1+π‘‡βˆ—,𝑑2]. The analysis above is the independent of the selection of interval [𝑑1,𝑑2], 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 π‘…βˆ—<1. The periodic infection-free solution is unstable if the basic reproductive number π‘…βˆ—>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 π‘…βˆ—<1. The parameters are chosen as 𝑝=0.03, 𝛽=0.8, 𝛼=0.002, πœƒ=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 π‘…βˆ—=0.9876. It suggests that the disease-free periodic solution is global asymptotically stable when π‘…βˆ—<1.

Figure 2 shows that the positive periodic solution is existence when π‘…βˆ—β†’1+, moreover, the positive periodic solution is global asymptotically stable. The parameters are chosen as 𝑝=0.005, 𝛽=0.8, 𝛼=0.002, πœƒ=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 π‘…βˆ—=1.0236, the other parameters are the same as Figure 1.

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 π‘…βˆ— is identified and is established as a sharp threshold parameter. If π‘…βˆ—<1, the infection-free periodic solution is global asymptotically stable which implies that the disease will extinct. If π‘…βˆ—>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 limπ‘‘β†’βˆžπ‘–(𝑑)=0 by traditional techniques. In this case, we made the conclusion by making use of the new variable 𝑣=𝑠+π‘Ÿ. 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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Copyright © 2009 Juping Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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