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 š‘…āˆ— is defined. It is proved that the infection-free periodic solution is global asymptotically stable if š‘…āˆ—<1. The infection-free periodic solution is unstable and the disease is uniform persistent if š‘…āˆ—>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 [4ā€“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 [12ā€“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 [17ā€“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 š‘ (0<š‘<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 š‘…āˆ— is defined. Local stability and the global asymptotically stable of the infection-free periodic solution are obtained when š‘…āˆ—<1. Section 4 concentrates on the uniform persistence of the infectious disease when š‘…āˆ—>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 (š‘†), infective (š¼), and recovered (š‘…) with immunity individuals. Immunity is not conferred at birth, and thus all newborns are susceptible. Vaccination gives lifelong immunity to š‘š‘† susceptible who are, as a consequence, transferred to the recovered class (š‘…) of the population. Using the impulsive differential equation, we have š‘†ī…žš‘†=āˆ’š›½š‘š¼š¼āˆ’š‘‘š‘†āˆ’š‘š‘†,ī…žš‘†=š›½š‘š‘…š¼āˆ’(š‘‘+šœƒ+š›¼)š¼,ī…žš‘†ī€·=šœƒš¼āˆ’š‘‘š‘…+š‘š‘†,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘†(š‘›šœ)+š‘1š‘†(š‘›šœ)+š‘1š›¾1š¼(š‘›šœ)+š‘1š›¾2š¼ī€·š‘…(š‘›šœ),š‘›šœ+ī€ø=š¼(š‘›šœ)+š‘1ī€·1āˆ’š›¾1ī€øš‘…ī€·š¼(š‘›šœ),š‘›šœ+ī€ø=š‘…(š‘›šœ)+š‘1ī€·1āˆ’š›¾2ī€øš‘…(š‘›šœ),š‘”=š‘›šœ.(2.1) The total population size is denoted by š‘, with š‘=š‘†+š¼+š‘…. Here the parameters š›½, š‘‘, š‘, šœƒ, š›¼, š‘1, š›¾1, š›¾2 are all positive constants. š›½ is adequate contact rate, š‘‘ is the per capita death rate, šœƒ is the removed rate, and š‘ is vaccination, a fraction of the entire susceptible population. š‘1 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 š›¾2 (0<š›¾2<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 š›¾1 (0<š›¾1<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 š›¾1<š›¾2. š›¼ represents the death rate due to disease. From biological view, we assume š›½ā‰„š›¼.

From (2.1), we obtain š‘ī…žš‘ī€·=āˆ’š‘‘š‘āˆ’š›¼š¼,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=ī€·1+š‘1ī€øš‘(š‘›šœ),š‘”=š‘›šœ.(2.2) Let š‘ =š‘†/š‘,š‘–=š¼/š‘,š‘Ÿ=š‘…/š‘, then systems (2.1) and (2.2) can be written as follows: š‘ ī…žš‘–=āˆ’š‘š‘ +(š›¼āˆ’š›½)š‘ š‘–,ī…ž=(š›½š‘ āˆ’šœƒāˆ’š›¼)š‘–+š›¼š‘–2,š‘Ÿī…žš‘ ī€·=š‘š‘ +šœƒš‘–+š›¼š‘–š‘Ÿ,š‘”ā‰ š‘›šœ,š‘›šœ+ī€øš‘=š‘ (š‘›šœ)+1š›¾11+š‘1š‘š‘–(š‘›šœ)+1š›¾21+š‘1š‘–ī€·š‘Ÿ(š‘›šœ),š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾1ī€øī€·1+š‘1ī€øš‘Ÿī€·š‘–(š‘›šœ),š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾2ī€øī€·1+š‘1ī€øš‘Ÿ(š‘›šœ),š‘”=š‘›šœ.(2.3) The total population size is normalized to one. By virtue of the equation š‘ (š‘”)+š‘–(š‘”)+š‘Ÿ(š‘”)=1, we ignore the third and the sixth equations of system (2.3) to study the two-dimensional system:

š‘ ī…žš‘–=āˆ’š‘š‘ +(š›¼āˆ’š›½)š‘ š‘–,ī…ž=(š›½š‘ āˆ’šœƒāˆ’š›¼)š‘–+š›¼š‘–2š‘ ī€·,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶ī‚µš‘š‘ (š‘›šœ)+1š›¾11+š‘1āˆ’š‘1š›¾21+š‘1ī‚¶š‘–ī€·š‘–(š‘›šœ),š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1š‘–(š‘›šœ),š‘”=š‘›šœ.(2.4)

From biological view, we easily see that the domain Ī©={(š‘ ,š‘–,š‘Ÿ)āˆ¶š‘ ā‰„0,š‘–ā‰„0,š‘Ÿā‰„0,š‘ +š‘–+š‘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, š‘–(š‘”)=0,š‘”ā‰„0. Under this condition, the growth of susceptible individuals and the population must satisfy š‘ ī…žš‘ ī€·=āˆ’š‘š‘ ,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘ (š‘›šœ),š‘”=š‘›šœ.(2.6) Integrating the first equation in system (2.6) between pulses, it is easy to obtain the solution with initial value š‘ (0+)=š‘ 0, ī€·š‘ (š‘”)=š‘ š‘›šœ+ī€øš‘’āˆ’š‘(š‘”āˆ’š‘›šœ),š‘›šœ<š‘”ā‰¤(š‘›+1)šœ.(2.7) Equation (2.7) holds between pulses. At each successive pulse, it yields š‘ ī€·(š‘›+1)šœ+ī€ø=š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘’āˆ’š‘šœš‘ ī€·š‘›šœ+ī€øī€·š‘ ī€·=š¹š‘›šœ+ī€øī€ø.(2.8) Equation (2.8) has a unique fixed point š‘ āˆ—=š‘1š›¾2š‘’š‘šœ/((1+š‘1)š‘’š‘šœāˆ’(1+š‘1āˆ’š‘1š›¾2)). The fixed point š‘ āˆ— is locally stable because š‘‘š¹(š‘ (š‘›šœ+))/š‘‘š‘ āˆ£š‘ (š‘›šœ+)=š‘ āˆ—=(1āˆ’š‘1š›¾2/(1+š‘1))š‘’āˆ’š‘šœ<1, By substituting š‘ (š‘›šœ+)=š‘ āˆ— to (2.7), we obtain the complete expression for the infection-free periodic solution over the nth time-interval š‘›šœ<š‘”ā‰¤(š‘›+1)šœ, š‘Ģƒš‘ (š‘”)=1š›¾2š‘’š‘šœī€·1+š‘1ī€øš‘’š‘šœāˆ’ī€·1+š‘1āˆ’š‘1š›¾2ī€øš‘’āˆ’š‘(š‘”āˆ’š‘›šœ),Ģƒš‘–(š‘”)=0.(2.9)

Therefore the system (2.4) has the infection-free periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)).

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 Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)).

The local stability of the infection-free periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)) may be determined by considering the linearized SIR equation of system (2.4) about the known periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)) by setting Ģƒš‘ (š‘”)=Ģƒš‘ (š‘”)+š‘„(š‘”),š‘–(š‘”)=š‘–(š‘”)+š‘¦(š‘”), where š‘„(š‘”) and š‘¦(š‘”) are small perturbation. The variables š‘„(š‘”) and š‘¦(š‘”) are described by the relation īƒ©īƒŖīƒ©īƒŖš‘„(š‘”)š‘¦(š‘”)=Ī¦(š‘”)š‘„(0)š‘¦(0),(3.1) where the fundamental solution matrix Ī¦(š‘”)=šœ‘š‘–š‘—(š‘”)(š‘–,š‘—=1,2) satisfies š‘‘Ī¦(š‘”)=īƒ©īƒŖš‘‘š‘”āˆ’š‘(š›¼āˆ’š›½)Ģƒš‘ (š‘”)0š›½Ģƒš‘ (š‘”)āˆ’šœƒāˆ’š›¼Ī¦(š‘”),(3.2) with Ī¦(0)=šø, where šø is the identity matrix. The resetting of the equations of (2.4) becomes īƒ©š‘„ī€·š‘›šœ+ī€øš‘¦ī€·š‘›šœ+ī€øīƒŖ=āŽ›āŽœāŽœāŽœāŽš‘1āˆ’1š›¾21+š‘1š‘1ī€·š›¾1āˆ’š›¾2ī€ø1+š‘101+š‘1āˆ’š‘1š›¾11+š‘1āŽžāŽŸāŽŸāŽŸāŽ īƒ©š‘„īƒŖ(š‘›šœ)š‘¦(š‘›šœ).(3.3) Hence, according to the Floquet theory, if all eigenvalues of āŽ›āŽœāŽœāŽœāŽš‘š‘€(šœ)=1āˆ’1š›¾21+š‘1š‘1ī€·š›¾1āˆ’š›¾2ī€ø1+š‘101+š‘1āˆ’š‘1š›¾11+š‘1āŽžāŽŸāŽŸāŽŸāŽ Ī¦(šœ)(3.4) are less than one, then the infection-free periodic solution (Ģƒš‘ (š‘”),0) is locally stable. By calculating, we have =īƒ©Ī¦(š‘”)1Ī¦120Ī¦22īƒŖ,(3.5) where Ī¦22āˆ«(š‘”)=exp(š›½Ģƒš‘ (šœŽ)š‘‘šœŽāˆ’(šœƒ+š›¼)š‘”).

The eigenvalues of š‘€ denoted by šœ‡1,šœ‡2 are šœ‡1=(1āˆ’š‘1š›¾2/(1+š‘1))š‘’āˆ’š‘šœ<1, and šœ‡2=((1+š‘1āˆ’š‘1š›¾1)/(1+š‘1āˆ«))exp{š›½šœ0Ģƒš‘ (šœŽ)š‘‘šœŽāˆ’(šœƒ+š›¼)šœ}, if and only if šœ‡2<1. Define threshold of model (2.4) as follows: š‘…āˆ—=š›½āˆ«šœ0Ģƒš‘ (šœŽ)š‘‘šœŽlnī€·ī€·1+š‘1ī€ø/ī€·1+š‘1āˆ’š‘1š›¾1ī€øī€ø+(šœƒ+š›¼)šœ,(3.6) where Ģƒš‘ (š‘”) is the infection-free periodic solution. That is, the infection-free periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)) is locally asymptotically stable if š‘…āˆ—<1. So we obtained following theorem.

Theorem 3.1. If š‘…āˆ—<1, then the infection-free periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)) 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 Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)), we need to use to comparison theory and impulsive differential inequality [1, 2].

Theorem 3.2. If š‘…āˆ—<1, then the infection-free periodic solution Ģƒ(Ģƒš‘ (š‘”),š‘–(š‘”)) of system (2.4) is global asymptotically stable.

Proof. Because of š›¼ā‰¤š›½, and š›¾1ā‰¤š›¾2, we have š‘ ī…žš‘ ī€·ā‰¤āˆ’š‘š‘ ,š‘”ā‰ š‘›šœ,š‘›šœ+ī€øā‰¤š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘ (š‘›šœ),š‘”=š‘›šœ.(3.7) By impulsive differential inequality, we see that ī€·0š‘ (š‘”)ā‰¤š‘ +ī€øī‘š‘œ<š‘›šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶ī‚»ī€œexpš‘”0(ī‚¼+ī“āˆ’š‘)š‘‘šœŽš‘œ<š‘›šœ<š‘”īƒÆī‘š‘›šœ<š‘—šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶ī‚øī€œexpš‘”š‘›šœī‚¹š‘(āˆ’š‘)š‘‘šœŽ1š›¾21+š‘1īƒ°ī€·0=Ģƒš‘ (š‘”)+š‘ +ī€øī‚µ1āˆ’š‘1š›¾21+š‘1ī‚¶[]š‘”/š‘‡š‘’āˆ’š‘š‘”āˆ’ī€·š‘1š›¾2/ī€·1+š‘1ī€øī€øī€·1āˆ’š‘1š›¾2/(1+š‘1)ī€ø[]š‘”/š‘‡š‘’āˆ’š‘š‘”ī€·1āˆ’š‘1š›¾2/ī€·1+š‘1š‘’ī€øī€øāˆ’š‘š‘‡.(3.8) Since limš‘”ā†’āˆžīƒÆš‘ ī€·0+ī€øī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶[]š‘”/š‘‡š‘’āˆ’š‘š‘”āˆ’ī€·š‘1š›¾2/ī€·1+š‘1ī€øī€øī€·1āˆ’š‘1š›¾2/ī€·1+š‘1ī€øī€ø[]š‘”/š‘‡š‘’āˆ’š‘š‘”ī€·1āˆ’š‘1š›¾2/ī€·1+š‘1š‘’ī€øī€øāˆ’š‘š‘‡īƒ°=0,(3.9) for any given šœ–1>0, there exists š‘‡1>0, such that š‘ (š‘”)<Ģƒš‘ (š‘”)+šœ–1, for all š‘”>š‘‡1.

Introduce the new variable š‘¢=š‘ +š‘Ÿ, then š‘¢ī…ž=š‘¢ī€·(āˆ’š›½š‘ +š›¼š‘¢+šœƒ)(1āˆ’š‘¢),š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾11+š‘1+ī‚µš‘1āˆ’1š›¾11+š‘1ī‚¶š‘¢(š‘›šœ),š‘”=š‘›šœ.(3.10)

Consider the following comparison system with pulse: š‘£ī…žī€·=āˆ’š›½Ģƒš‘ (š‘”)+š›½šœ–1ī€ø+š›¼āˆ’šœƒš‘£āˆ’š›¼š‘£2ī€·+šœƒāˆ’š›½Ģƒš‘ (š‘”)+šœ–1ī€øš‘£ī€·,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾11+š‘1+ī‚µš‘1āˆ’1š›¾11+š‘1ī‚¶š‘£(š‘›šœ),š‘”=š‘›šœ.(3.11)

The first equation of (3.11) is Riccati equation. It is easy to see that š‘£(š‘”)=1 is a solution of system (3.11). Let š‘¦=š‘£āˆ’1, then š‘¦ī…žī€·=āˆ’š›½Ģƒš‘ (š‘”)āˆ’š›¼āˆ’šœƒ+š›½šœ–1ī€øš‘¦āˆ’š›¼š‘¦2š‘¦ī€·,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=ī‚µš‘1āˆ’1š›¾11+š‘1ī‚¶š‘¦(š‘›šœ),š‘”=š‘›šœ.(3.12)

Let š‘§=1/š‘¦, then š‘§ī…žī€·=āˆ’š›½Ģƒš‘ (š‘”)āˆ’š›¼āˆ’šœƒ+š›½šœ–1ī€øš‘§ī€·š‘§āˆ’š›¼,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=ī‚µ1+š‘11+š‘1āˆ’š‘1š›¾1ī‚¶š‘§(š‘›šœ),š‘”=š‘›šœ.(3.13)

Let š‘ž(š‘”)=š›½Ģƒš‘ (š‘”)āˆ’š›¼āˆ’šœƒ+š›½šœ–1, solving system (3.13) between pulses (š‘‡1+š‘›šœ,š‘‡1+(š‘›+1)šœ], we have š‘§(š‘”)=š‘’āˆ’āˆ«š‘”š‘‡1+š‘›šœš‘ž(šœŽ)š‘‘šœŽī‚øš›¼ī€œš‘”š‘‡1+š‘›šœš‘’āˆ«š‘¢š‘‡1+š‘›šœš‘ž(šœŽ)š‘‘šœŽš‘‘š‘¢+1+š‘11+š‘1āˆ’š‘1š›¾1š‘§ī€·š‘‡1ī€øī‚¹+š‘›šœ,(3.14) when š‘”=š‘‡1+(š‘›+1)šœ, (3.14) can be written as follows: š‘§ī€·š‘‡1ī€ø+(š‘›+1)šœ=š‘’āˆ’āˆ«š‘‡1š‘‡1+(š‘›+1)šœ+š‘›šœš‘ž(šœŽ)š‘‘šœŽī‚øš›¼ī€œš‘‡1š‘‡+(š‘›+1)šœ1+š‘›šœš‘’āˆ«š‘¢š‘‡1+š‘›šœš‘ž(šœŽ)š‘‘šœŽš‘‘š‘¢+1+š‘11+š‘1āˆ’š‘1š›¾1š‘§ī€·š‘‡1ī€øī‚¹+š‘›šœ.(3.15) On the other hand, solving system (3.13) between pulses (š‘‡1+(š‘›āˆ’1)šœ,š‘‡1+š‘›šœ], we obtain š‘§(š‘”)=š‘’āˆ’āˆ«š‘”š‘‡1+(š‘›āˆ’1)šœš‘ž(šœŽ)š‘‘šœŽī‚øš›¼ī€œš‘”š‘‡1+(š‘›āˆ’1)šœš‘’āˆ«š‘¢š‘‡1+(š‘›āˆ’1)šœš‘ž(šœŽ)š‘‘šœŽš‘‘š‘¢+1+š‘11+š‘1āˆ’š‘1š›¾1š‘§ī€·š‘‡1ī€øī‚¹+(š‘›āˆ’1)šœ,(3.16) then š‘§ī€·š‘‡1ī€ø+š‘›šœ=š‘’āˆ’āˆ«š‘‡1š‘‡1+(š‘›+1)šœ+š‘›šœš‘ž(šœŽ)š‘‘šœŽī‚øš›¼ī€œš‘‡1š‘‡+š‘›šœ1+(š‘›āˆ’1)šœš‘’āˆ«š‘¢š‘‡1+(š‘›āˆ’1)šœš‘ž(šœŽ)š‘‘šœŽš‘‘š‘¢+1+š‘11+š‘1āˆ’š‘1š›¾1š‘§ī€·š‘‡1ī€øī‚¹+(š‘›āˆ’1)šœ.(3.17)

Similarly, we can get the expressions of š‘§(š‘‡1+(š‘›āˆ’1)šœ),š‘§(š‘‡1+(š‘›āˆ’2)šœ),ā€¦,š‘§(š‘‡1). Then using iterative technique step by step, š‘§ī€·š‘‡1ī€ø+š‘›šœ=š‘’āˆ’āˆ«0š‘›šœš‘ž(šœŽ)š‘‘šœŽī‚µ1+š‘11+š‘1āˆ’š‘1š›¾1ī‚¶š‘›īƒ¬ī“1ā‰¤š‘˜ā‰¤š‘›ī‚µ1+š‘11+š‘1āˆ’š‘1š›¾1ī‚¶š‘˜āˆ’š‘›āˆ’1š‘’āˆ«š‘‡1š‘‡1+(š‘›āˆ’š‘˜)šœš‘ž(šœŽ)š‘‘šœŽ+ī€œš‘‘š‘¢š‘‡1š‘‡+(š‘›āˆ’š‘˜+1)šœ1+(š‘›āˆ’š‘˜)šœš‘’āˆ«š‘¢š‘‡1+(š‘›āˆ’š‘˜)šœš‘ž(šœŽ)š‘‘šœŽī€·š‘‡š‘‘š‘¢+š‘§1ī€øī‚¹,(3.18) where š‘’āˆ’āˆ«0š‘›šœš‘ž(šœŽ)š‘‘šœŽī‚µ1+š‘11+š‘1āˆ’š‘1š›¾1ī‚¶š‘›ī‚»š‘›ī‚øī€œ=expāˆ’š›½šœ0Ģƒš‘ (šœŽ)š‘‘šœŽ+(šœƒ+š›¼)šœāˆ’š›½šœ–1šœ+ln1+š‘11+š‘1āˆ’š‘1š›¾1ī‚¹ī‚¼.(3.19)

The condition š‘…āˆ—<1 implies that limš‘›ā†’āˆžš‘§(š‘›šœ)=āˆž, then limš‘”ā†’āˆžš‘„(š‘”)=1. The comparison principle and the condition š‘¢(š‘”)<1 imply that limš‘”ā†’āˆžš‘¢(š‘”)=1, so we have limš‘”ā†’āˆžš‘–(š‘”)=0.

Because we have proved that limš‘”ā†’āˆžš‘–(š‘”)=0 when š‘…āˆ—<1, for any given šœ–2>0, there exists š‘‡2>0, such that āˆ’šœ–2<š‘–(š‘”)<šœ–2, for all š‘”>š‘‡2.

When š‘”>š‘‡2, from system (2.4), we have š‘ ī…žā‰„āˆ’š‘š‘ +(š›¼āˆ’š›½)šœ–2š‘ ī€·š‘ ,š‘”ā‰ š‘›šœ,š‘›šœ+ī€øā‰„š‘1š›¾11+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘ (š‘›šœ),š‘”=š‘›šœ.(3.20) Therefore ī€·š‘‡š‘ (š‘”)ā‰„š‘ +2ī€øī‘š‘‡2<š‘›šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶ī‚»ī€œexpš‘”š‘‡2ī€·āˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī‚¼+ī“š‘‘šœŽš‘‡2<š‘›šœ<š‘”īƒÆī‘š‘‡2<š‘—šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶ī‚»ī€œexpš‘”š‘›šœī€·āˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī‚¼š‘š‘‘šœŽ1š›¾11+š‘1īƒ°ī€·š‘‡=š‘ +2ī€øī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶[š‘”/š‘‡]āˆ’[š‘‡2/š‘‡]ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī€·š‘”āˆ’š‘‡2ī€ø+š‘1š›¾1/ī€·1+š‘1ī€øī€·1āˆ’1āˆ’š‘1š›¾2/ī€·1+š‘1ī€·ī€øī€øexpāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øš‘‡ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī‚€ī‚ƒš‘”š‘”āˆ’š‘‡ī‚„š‘‡ī‚āˆ’š‘1š›¾1/ī€·1+š‘1ī€øī€·1āˆ’1āˆ’š‘1š›¾2/ī€·1+š‘1ī€·ī€øī€øexpāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øš‘‡Ć—ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶([š‘”/š‘‡]āˆ’[š‘‡2/š‘‡]āˆ’1)ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øš‘‡.(3.21) For any given šœ–2>0, we have limš‘”ā†’āˆžīƒÆš‘ ī€·š‘‡+2ī€øī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶[š‘”/š‘‡]āˆ’[š‘‡2/š‘‡]ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī€·š‘”āˆ’š‘‡2ī€øāˆ’š‘1š›¾1/ī€·1+š‘1ī€øī€·1āˆ’1āˆ’š‘1š›¾2/ī€·1+š‘1ī€·ī€øī€øexpāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øš‘‡Ć—ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶([š‘”/š‘‡]āˆ’[š‘‡2/š‘‡]āˆ’1)ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øīƒ°(š‘‡)=0,limš‘”ā†’āˆžš‘1š›¾1/ī€·1+š‘1ī€øī€·1āˆ’1āˆ’š‘1š›¾2/ī€·1+š‘1ī€·ī€øī€øexpāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øš‘‡ī€·expāˆ’š‘+(š›¼āˆ’š›½)šœ–2ī€øī‚€ī‚ƒš‘”š‘”āˆ’š‘‡ī‚„š‘‡ī‚=Ģƒš‘ (š‘”).(3.22) Therefore, for any given šœ–3>0, there exists š‘‡3>0, when š‘”>š‘‡3, then we have š‘ (š‘”)ā‰„Ģƒš‘ (š‘”)āˆ’šœ–3.(3.23) For any given šœ–>0. Let š‘‡=max{š‘‡1,š‘‡2,š‘‡3}, then š‘”>š‘‡, then we have Ģƒš‘ (š‘”)āˆ’šœ–ā‰„š‘ (š‘”)ā‰„Ģƒš‘ (š‘”)+šœ–,(3.24) that is limš‘”ā†’āˆžš‘ (š‘”)=Ģƒš‘ (š‘”).

Therefore the infection-free periodic solution (Ģƒš‘ (š‘”),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, limš‘”ā†’āˆžinfš‘–(š‘”)ā‰„šœŒ>0 if š‘…āˆ—>1.

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

Lemma 4.1. For the following impulsive equation, š‘„ī…žš‘„ī€·=āˆ’š‘”š‘„āˆ’ā„Ž,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾11+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘„(š‘›šœ),š‘”=š‘›šœ,(4.1) has a unique positive šœ-periodic solution Ģƒš‘„(š‘”) for which Ģƒš‘„(0)>0, ā€‰ā€‰š‘”āˆˆš‘…+, and Ģƒš‘„(š‘”) is global asymptotically stable in the sense that limš‘”ā†’āˆž|š‘„(š‘”)āˆ’Ģƒš‘„(š‘”)|=0, where š‘„(š‘”) is any solution of system (2.2) with positive initial value š‘„(0)>0 and š‘”,ā„Ž are positive constants.

Proof. Solving (4.1), we have ī€œš‘„(š‘”)=š‘Š(š‘”,0)š‘„(0)āˆ’ā„Žš‘”0š‘š‘Š(š‘”,šœŽ)š‘‘šœŽ+1š›¾11+š‘1ī“0<š‘›šœ<š‘”š‘Šī€·š‘”,š‘›šœ+ī€ø,(4.2) where š‘Šī€·š‘”,š‘”0ī€ø=ī‘š‘”0ā‰¤š‘›šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘’āˆ’š‘”(š‘”āˆ’š‘”0).(4.3) Since š‘Š(šœ,0)=(1āˆ’š‘1š›¾2/(1+š‘1))š‘’āˆ’š‘”šœ<1, (4.1) has a unique positive šœ-periodic solution Ģƒš‘„(š‘”) with the initial value āˆ«Ģƒš‘„(0)=(āˆ’ā„Žšœ0š‘Š(šœ+,šœŽ)š‘‘šœŽ+(š‘1š›¾1/(1+š‘1))š‘Š(šœ,šœ))/(1āˆ’š‘Š(šœ,0)). Next, we only need to prove that limš‘”ā†’āˆž|š‘„(š‘”)āˆ’Ģƒš‘„(š‘”)|=0.
Since ||||||||š‘„(š‘”)āˆ’Ģƒš‘„(š‘”)=š‘Š(š‘”,0)š‘„(0)āˆ’Ģƒš‘„(0),(4.4) the result is obtained if š‘Š(š‘”,0)ā†’0 as š‘”ā†’āˆž. Suppose š‘”āˆˆ(š‘›šœ,(š‘›+1)šœ], then ī‘š‘Š(š‘”,0)=0ā‰¤š‘—šœ<š‘”ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘’āˆ’š‘”š‘”=ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶[š‘”/šœ]š‘’āˆ’š‘”š‘”.(4.5) Thus limš‘”ā†’āˆžš‘Š(š‘”,0)=0. The proof is complete.

Lemma 4.2. If š‘…āˆ—>1, then the disease uniformly weakly persists in the population, in the sense that there exists some š‘>0 such that limš‘”ā†’āˆžsupš‘–(š‘”)>š‘ for all solutions of (2.4).

Proof. Suppose that for every šœ–>0, there is some solution with limš‘”ā†’āˆžsupš‘–(š‘”)<šœ–. From the first equation of (2.4), we have š‘ ī…ž=āˆ’š‘š‘ +(š›¼āˆ’š›½)š‘ š‘–ā‰„āˆ’š‘š‘ +(š›¼āˆ’š›½)šœ–,š‘”ā‰ š‘›šœ.(4.6) Consider the following equation: š‘¤ī…žš‘¤ī€·=āˆ’š‘š‘¤+(š›¼āˆ’š›½)šœ–,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾11+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘¤(š‘›šœ),š‘”=š‘›šœ.(4.7) By Lemma 4.1, we see that (4.7) has a unique positive šœ-periodic solution ī‚š‘¤(š‘”), and ī‚š‘¤(š‘”) is global asymptotically stable. Solving (4.7), we have ī€œš‘¤(š‘”)=š‘Š(š‘”,0)š‘¤(0)+(š›¼āˆ’š›½)šœ–š‘”0š‘š‘Š(š‘”,šœŽ)š‘‘šœŽ+1š›¾11+š‘1ī“0<š‘›šœ<š‘”š‘Šī€·š‘”,š‘›šœ+ī€ø,ī‚īƒ©āˆ«(4.8)š‘¤(š‘”)=(š›¼āˆ’š›½)šœ–š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒŖ+ī‚µš‘š‘Š(š‘”,šœŽ)š‘‘šœŽ1āˆ’1š›¾21+š‘1ī‚¶š‘Š(š‘”,0)š‘Š(šœ,šœ)+š‘1āˆ’š‘Š(šœ,0)1š›¾11+š‘1ī“0<š‘›šœ<š‘”š‘Šī€·š‘”,š‘›šœ+ī€ø,(4.9) and ī‚š‘¤(š‘”) is global asymptotically stable. By (4.9), let š›¼=š›½, we obtain the periodic solution of (2.6) that ī‚µš‘Ģƒš‘ (š‘”)=1āˆ’1š›¾21+š‘1ī‚¶š‘Š(š‘”,0)š‘Š(šœ,šœ)+š‘1āˆ’š‘Š(šœ,0)1š›¾11+š‘1ī“0<š‘›šœ<š‘”š‘Šī€·š‘”,š‘›šœ+ī€ø.(4.10) and we have ī‚īƒ©āˆ«Ģƒš‘ (š‘”)āˆ’š‘¤(š‘”)=(š›½āˆ’š›¼)šœ–š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒŖš‘Š(š‘”,šœŽ)š‘‘šœŽ.(4.11) Let Ī”=(š›½āˆ’š›¼)max0ā‰¤š‘”ā‰¤šœīƒÆāˆ«š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒ°š‘Š(š‘”,šœŽ)š‘‘šœŽ,(4.12) by (4.11), we can see that ī‚š‘¤(š‘”)ā‰„Ģƒš‘ (š‘”)āˆ’Ī”šœ–.(4.13) By comparison theory, we obtain that š‘–ī…žā‰„āˆ’(šœƒ+š›¼)š‘–+š›½š‘–š‘¤(š‘”)+š›¼š‘–2.(4.14) Since ī‚š‘¤(š‘”) is global asymptotically stable, for above šœ–, there exists š‘‡4>0, such that ī‚š‘¤(š‘”)ā‰„š‘¤(š‘”)āˆ’šœ–, š‘”>š‘‡4. From (4.13) and (4.14), we have that š‘–ī…žā‰„[]š›½Ģƒš‘ (š‘”)āˆ’(šœƒ+š›¼)āˆ’(1+Ī”)š›½šœ–š‘–.(4.15) Consider the following equation: š‘–ī…žā‰„[]š‘–ī€·š›½Ģƒš‘ (š‘”)āˆ’(šœƒ+š›¼)āˆ’(1+Ī”)š›½šœ–š‘–,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1š‘–(š‘›šœ),š‘”=š‘›šœ.(4.16) By impulsive differential inequality, for š‘”āˆˆ(š‘‡4+š‘›šœ,š‘‡4+(š‘›+1)šœ], we see that ī€·š‘‡š‘–(š‘”)ā‰„š‘–4ī€øī‘š‘‡4<š‘—šœ<š‘”1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1ī‚»ī€œexpš‘”š‘‡4[]ī‚¼ī€·š‘‡š›½Ģƒš‘ (šœŽ)āˆ’(šœƒ+š›¼)āˆ’(1+Ī”)š›½šœ–š‘‘šœŽ=š‘–4ī€øī‚µ1+š‘1(1āˆ’š›¾1)1+š‘1ī‚¶š‘›ī‚»ī€œexpš‘‡4š‘‡+š‘›šœ4[]+ī€œš›½Ģƒš‘ (šœŽ)āˆ’(šœƒ+š›¼)āˆ’(1+Ī”)š›½šœ–š‘‘šœŽš‘”š‘‡4+š‘›šœ[]ī‚¼īƒÆš‘›īƒ¬ī€·š‘…š›½Ģƒš‘ (šœŽ)āˆ’(šœƒ+š›¼)āˆ’(1+Ī”)š›½šœ–š‘‘šœŽā‰„š¶expāˆ—ī€øīƒ©āˆ’1ln1+š‘11+š‘1ī€·1āˆ’š›¾1ī€øīƒŖ,+(šœƒ+š›¼)šœāˆ’(1+Ī”)š›½šœ–šœīƒ­īƒ°(4.17) where š¶=š‘–(š‘‡4)exp{āˆ’[(šœƒ+š›¼)+(1+Ī”)š›½šœ–]šœ}. Taking ī€·š‘…0<šœ–<āˆ—āˆ’1ī€øī€ŗlnī€·ī€·1+š‘1ī€ø/ī€·1+š‘1ī€·1āˆ’š›¾1+ī€»ī€øī€øī€ø(šœƒ+š›¼)šœ(1+Ī”)š›½šœ,(4.18) thus š‘–(š‘”)ā†’āˆž as š‘”ā†’āˆž, a contradiction to the fact that š‘–(š‘”) is bounded. The proof is complete.

Theorem 4.3. If š‘…āˆ—>1, then the disease is uniformly persistent, that is, there exists a positive constant šœŒ such that for every positive solution of (2.4), limš‘”ā†’āˆžinfš‘–(š‘”)ā‰„šœŒ>0.

Proof. Let 10<šœ‚ā‰¤2ī‚€11āˆ’š‘…āˆ—ī‚Ģƒš‘ (š‘”)š‘€(šœƒ+š›¼),(4.19) where š‘€=max0ā‰¤š‘”ā‰¤šœīƒÆāˆ«š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒ°,š‘Šī€·š‘Š(š‘”,šœŽ)š‘‘šœŽš‘”,š‘”0ī€ø=ī‘š‘”0ā‰¤š‘›šœ<š‘”1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1š‘’ī€·āˆ’š‘š‘”āˆ’š‘”0ī€ø.(4.20) It can be obtained from Lemma 4.1 that for any positive solution of (2.4) there exists at least one š‘”0>0 such that š‘–(š‘”0)>šœ‚>0. Then, we are left to consider two case. The first case is š‘–(š‘”)ā‰„šœ‚ for all large š‘”ā‰„š‘”0. The second case is š‘–(š‘”) oscillates about šœ‚ for large t. The conclusion of Theorem 4.3 is obvious in the first case since we can choose šœŒ=šœ‚. For the second case, let š‘”1>š‘”0, and let š‘”2>š‘”1 satisfy š‘–ī€·š‘”1ī€øī€·š‘”=š‘–2ī€ø=šœ‚,š‘–(š‘”)<šœ‚forš‘”1<š‘”<š‘”2.(4.21) Next, we introduce the new variable š‘‰=š‘ +š‘–, and it follows from the first two equations of (2.4) that š‘‰ī…žš‘–=āˆ’š‘š‘‰āˆ’(šœƒ+š›¼)š‘–+(š›¼š‘‰+š‘)š‘–,ī…ž=āˆ’(šœƒ+š›¼)š‘–+š›½(š‘‰āˆ’š‘–)š‘–+š›¼š‘–2š‘‰ī€·,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘–ī€·š‘‰(š‘›šœ),š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1š‘–(š‘›šœ),š‘”=š‘›šœ.(4.22) If š‘–(š‘”)ā‰¤šœ‚, then š‘‰ī…žā‰„āˆ’š‘š‘‰āˆ’(šœƒ+š›¼)šœ‚,š‘”ā‰ š‘›šœ.
Consider the following equation: š‘„ī…žš‘„ī€·=āˆ’š‘š‘„āˆ’(šœƒ+š›¼)šœ‚,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=š‘1š›¾21+š‘1+ī‚µš‘1āˆ’1š›¾21+š‘1ī‚¶š‘„(š‘›šœ),š‘”=š‘›šœ.(4.23)
By Lemma 4.1, we see that (4.23) has a unique positive šœ-periodic solution Ģƒš‘„(š‘”), and Ģƒš‘„(š‘”) is global asymptotically stable. Solving (4.23), we have īƒ©āˆ«Ģƒš‘„(š‘”)=āˆ’(šœƒ+š›¼)šœ‚š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒŖ+ī‚µš‘š‘Š(š‘”,šœŽ)š‘‘šœŽ1āˆ’1š›¾21+š‘1ī‚¶ī€·šœš‘Š(š‘”,0)š‘Š+,šœ+ī€ø+š‘(1āˆ’š‘Š(šœ,0))1š›¾21+š‘1ī“0<š‘›šœ<š‘”š‘Šī€·š‘”,š‘›šœ+ī€ø.(4.24) From (4.10) and (4.24), it is easy to see that īƒ©āˆ«Ģƒš‘„(š‘”)āˆ’Ģƒš‘ (š‘”)ā‰„āˆ’(šœƒ+š›¼)šœ‚š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒŖš‘Š(š‘”,šœŽ)š‘‘šœŽ.(4.25) By 0<šœ‚ā‰¤(1/2)(1āˆ’1/š‘…āˆ—)(Ģƒš‘ (š‘”)/š‘€(šœƒ+š›¼)), š‘€=max0ā‰¤š‘”ā‰¤šœāˆ«{š‘Š(š‘”,0)šœ0š‘Š(šœ+āˆ«,šœŽ)š‘‘šœŽ/(1āˆ’š‘Š(šœ,0))+š‘”0š‘Š(š‘”,šœŽ)š‘‘šœŽ}, and āˆ«(š‘Š(š‘”,0)šœ0š‘Š(šœ+āˆ«,šœŽ)š‘‘šœŽ/(1āˆ’š‘Š(šœ,0))+š‘”0š‘Š(š‘”,šœŽ)š‘‘šœŽ)(1/š‘€)ā‰¤1, we obtain īƒ©āˆ«āˆ’(šœƒ+š›¼)šœ‚š‘Š(š‘”,0)šœ0š‘Šī€·šœ+ī€ø,šœŽš‘‘šœŽ+ī€œ1āˆ’š‘Š(šœ,0)š‘”0īƒŖ1š‘Š(š‘”,šœŽ)š‘‘šœŽā‰„āˆ’2ī‚€11āˆ’š‘…āˆ—ī‚Ģƒš‘ (š‘”),(4.26) namely, 1Ģƒš‘„(š‘”)ā‰„2ī‚€11+š‘…āˆ—ī‚Ģƒš‘ (š‘”).(4.27) The comparison principle and the global asymptotically stable of Ģƒš‘„(š‘”) imply that there exists a positive constant š‘‡5>0 such that 1š‘‰(š‘”)ā‰„2ī‚€11+š‘…āˆ—ī‚Ģƒš‘ (š‘”),āˆ€š‘”>š‘”1+š‘‡5.(4.28) From (4.28) and the second equation of (4.22), we can see that š‘–ī…žā‰„ī‚øš›½2ī‚€11+š‘…āˆ—ī‚ī‚¹Ģƒš‘ (š‘”)āˆ’(šœƒ+š›¼)š‘–+(š›¼āˆ’š›½)š‘–2.(4.29) Consider the following equation: š‘¦ī…ž=ī‚øš›½2ī‚€11+š‘…āˆ—ī‚ī‚¹Ģƒš‘ (š‘”)āˆ’(šœƒ+š›¼)š‘¦+(š›¼āˆ’š›½)š‘¦2š‘¦ī€·,š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=1+š‘1ī€·1āˆ’š›¾1ī€ø1+š‘1š‘¦(š‘›šœ),š‘”=š‘›šœ.(4.30) Let š‘§=š‘¦āˆ’1, then we have š‘§ī…ž=ī‚øš›½(šœƒ+š›¼)āˆ’2ī‚€11+š‘…āˆ—ī‚ī‚¹š‘§ī€·Ģƒš‘ (š‘”)š‘§+(š›½āˆ’š›¼),š‘”ā‰ š‘›šœ,š‘›šœ+ī€ø=1+š‘11+š‘1ī€·1āˆ’š›¾1ī€øš‘§(š‘›šœ),š‘”=š‘›šœ.(4.31) By the same method of Lemma 4.1, we can get a conclusion that āˆ«Ģƒš‘§(š‘”)=(š›½āˆ’š›¼)(š‘Š(š‘”,0)šœ0š‘Š(šœ+āˆ«,šœŽ)š‘‘šœŽ/(1āˆ’š‘Š(šœ,0))+š‘”0š‘Š(š‘”,šœŽ)š‘‘šœŽ)+(1+š‘1)/(1+š‘1(1āˆ’š›¾1))(š‘Š(š‘”,0)š‘Š(šœ,šœ)/(1āˆ’š‘Š(šœ,0))) is global asymptotically stable. Thus system (4.30) has a unique positive šœ-periodic solution Ģƒš‘¦(š‘”), and Ģƒš‘¦(š‘”) is global asymptotically stable, limš‘”ā†’āˆž||||š‘¦(š‘”)āˆ’Ģƒš‘¦(š‘”)=0.(4.32) From (4.32) we see that there exists a positive constant š‘‡6>0 such that 1š‘¦(š‘”)>šœŒā‰”2minš‘”1ā‰¤š‘”ā‰¤š‘”1+šœĢƒš‘¦(š‘”)>0,āˆ€š‘”>š‘”1+š‘‡6.(4.33) Let š‘‡āˆ—=max{š‘‡5,š‘‡6}, and define šœŒ=min{šœŒ,šœ‚exp(āˆ’(šœƒ+š›¼)šœ)}. If š‘”2āˆ’š‘”1<š‘‡āˆ—, from the second equation of (4.22), we have the inequality š‘–ī…ž(š‘”)ā‰„āˆ’(šœƒ+š›¼)š‘–,(4.34) and the comparison principle implies that š‘–(š‘”)ā‰„šœ‚exp{āˆ’(šœƒ+š›¼)(š‘”āˆ’š‘”1)}ā‰„šœ‚exp{āˆ’(šœƒ+š›¼)š‘‡āˆ—}, that is, š‘–(š‘”)ā‰„šœŒ for all š‘”āˆˆ(š‘”1,š‘”2).
If š‘”2āˆ’š‘”1>š‘‡āˆ—, we divide the interval [š‘”1,š‘”2] into two subintervals [š‘”1,š‘”1+š‘‡āˆ—] and [š‘”1+š‘‡āˆ—,š‘”2], š‘–(š‘”)ā‰„šœŒ is obvious in the interval [š‘”1,š‘”1+š‘‡āˆ—]. In the interval [š‘”1+š‘‡āˆ—,š‘”2], we have the inequality (4.29) and (4.33). The comparison principle shows that š‘–(š‘”)ā‰„š‘¦(š‘”)ā‰„šœŒā‰„šœŒ for š‘”āˆˆ[š‘”1+š‘‡āˆ—,š‘”2]. The analysis above is the independent of the selection of interval [š‘”1,š‘”2], and the choice of šœŒ 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 š‘…āˆ—<1. The periodic infection-free solution is unstable if the basic reproductive number š‘…āˆ—>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 š‘…āˆ—<1. The parameters are chosen as š‘=0.03, š›½=0.8, š›¼=0.002, šœƒ=0.2, š‘1=0.4, š›¾1=0.86,š›¾2=0.9, and šœ=40. The three Figures 1(a), 1(b), and 1(c) in have the same initial value as š‘ (0)=0.6296, š‘–(0)=0.006. We fixed š‘=0.03 and changed parameter šœ. Figures 1(a), 1(b), and 1(c) show the solutions for šœ=40 and š‘…āˆ—=0.9876. It suggests that the disease-free periodic solution is global asymptotically stable when š‘…āˆ—<1.

Figure 2 shows that the positive periodic solution is existence when š‘…āˆ—ā†’1+, moreover, the positive periodic solution is global asymptotically stable. The parameters are chosen as š‘=0.005, š›½=0.8, š›¼=0.002, šœƒ=0.2, š‘1=0.4, š›¾1=0.32,š›¾2=0.9, and šœ=8. Here we choose the initial value of (2.4) š‘ (0)=0.3080, š‘–(0)=0.006. In Figures 2(a), 2(b) and 2(c) with šœ=15 and š‘…āˆ—=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 š‘…āˆ— is identified and is established as a sharp threshold parameter. If š‘…āˆ—<1, the infection-free periodic solution is global asymptotically stable which implies that the disease will extinct. If š‘…āˆ—>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 limš‘”ā†’āˆžš‘–(š‘”)=0 by traditional techniques. In this case, we made the conclusion by making use of the new variable š‘£=š‘ +š‘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 (ā€‰60771026), Program for New Century Excellent Talents in University (NECT050271), and Science Foundation of Shanxi Province (ā€‰2009011005-1).