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

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 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 [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 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^=−𝛽𝑆𝐼𝑁𝐼−𝑑𝑆−𝑝𝑆,_=𝛽𝑆𝑅𝑁𝐼−(𝑑+𝜃+𝛼)𝐼,_𝑆𝑛𝜏₊=𝜃𝐼−𝑑𝑅+𝑝𝑆,𝑡≠𝑛𝜏,𝐼𝑛𝜏₊=𝑆(𝑛𝜏)+𝑏₁𝑆(𝑛𝜏)+𝑏₁𝛾₁𝐼(𝑛𝜏)+𝑏₁𝛾₂𝑅(𝑛𝜏),𝑅𝑛𝜏₊=𝐼(𝑛𝜏)+𝑏₁1−𝛾₁𝐼(𝑛𝜏),=𝑅(𝑛𝜏)+𝑏₁1−𝛾₂𝑅(𝑛𝜏),𝑡=𝑛𝜏.(2.1) 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 𝛽≥𝛼.

From (2.1), we obtain 𝑁𝑛𝜏𝑁^₊=−𝑑𝑁−𝛼𝐼,𝑡≠𝑛𝜏,=1+𝑏₁𝑁(𝑛𝜏),𝑡=𝑛𝜏.(2.2) Let 𝑠=𝑆/𝑁,𝑖=𝐼/𝑁,𝑟=𝑅/𝑁, then systems (2.1) and (2.2) can be written as follows: 𝑠𝑖^=−𝑝𝑠+(𝛼−𝛽)𝑠𝑖,_𝑟_=(𝛽𝑠−𝜃−𝛼)𝑖+𝛼𝑖²,𝑠𝑛𝜏₊=𝑝𝑠+𝜃𝑖+𝛼𝑖𝑟,𝑡≠𝑛𝜏,=𝑠(𝑛𝜏)+𝑏¹𝛾₁1+𝑏₁𝑖(𝑛𝜏)+𝑏¹𝛾₂1+𝑏₁𝑖𝑛𝜏₊𝑟(𝑛𝜏),=1+𝑏¹1−𝛾₁𝑟𝑛𝜏₊1+𝑏₁𝑖(𝑛𝜏),=1+𝑏¹1−𝛾₂1+𝑏₁𝑟(𝑛𝜏),𝑡=𝑛𝜏.(2.3) 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:

𝑠𝑖^=−𝑝𝑠+(𝛼−𝛽)𝑠𝑖,_𝑠𝑛𝜏₊=(𝛽𝑠−𝜃−𝛼)𝑖+𝛼𝑖²,𝑡≠𝑛𝜏,=𝑏¹𝛾₂1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑠(𝑛𝜏)+𝑏¹𝛾₁1+𝑏₁−𝑏¹𝛾₂𝑖𝑛𝜏₊1+𝑏₁𝑖(𝑛𝜏),=1+𝑏¹1−𝛾₁1+𝑏₁𝑖(𝑛𝜏),𝑡=𝑛𝜏.(2.4)

From biological view, we easily see that the domain Ω={(𝑠,𝑖,𝑟)∶𝑠≥0,𝑖≥0,𝑟≥0,𝑠+𝑖+𝑟=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(𝑡)=0,𝑡≥0. Under this condition, the growth of susceptible individuals and the population must satisfy 𝑠𝑛𝜏₊𝑠^=−𝑝𝑠,𝑡≠𝑛𝜏,=𝑏¹𝛾₂1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑠(𝑛𝜏),𝑡=𝑛𝜏.(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(𝑡)=𝑠𝑛𝜏⁺𝑒^{−𝑝(𝑡−𝑛𝜏)},𝑛𝜏<𝑡≤(𝑛+1)𝜏.(2.7) 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⁺.(2.8) Equation (2.8) has a unique fixed point 𝑠^∗=𝑏₁𝛾₂𝑒^𝑝𝜏/((1+𝑏₁)𝑒^𝑝𝜏−(1+𝑏₁−𝑏₁𝛾₂)). The fixed point 𝑠^∗ is locally stable because 𝑑𝐹(𝑠(𝑛𝜏⁺))/𝑑𝑠∣_{𝑠(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)=𝑏¹𝛾₂𝑒^𝑝𝜏1+𝑏₁𝑒^𝑝𝜏−1+𝑏₁−𝑏₁𝛾₂𝑒^{−𝑝(𝑡−𝑛𝜏)},𝑖̃(𝑡)=0.(2.9)

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

The local stability of the infection-free periodic solution (𝑠(̃𝑡),𝑖(̃𝑡)) may be determined by considering the linearized SIR equation of system (2.4) about the known periodic solution (𝑠(̃𝑡),𝑖(̃𝑡)) by setting 𝑠(𝑡)=𝑠(̃𝑡)+𝑥(𝑡),𝑖(𝑡)=𝑖(̃𝑡)+𝑦(𝑡), where 𝑥(𝑡) and 𝑦(𝑡) are small perturbation. The variables 𝑥(𝑡) and 𝑦(𝑡) are described by the relation 𝑥𝑦((𝑡𝑡))=Φ(𝑡)𝑥𝑦((00)),(3.1) where the fundamental solution matrix Φ(𝑡)=𝜑_𝑖𝑗(𝑡)(𝑖,𝑗=1,2) satisfies 𝑑Φ(𝑡)𝑑𝑡=−0𝑝𝛽(𝑠𝛼̃(𝑡−)−𝛽)𝜃𝑠̃−(𝑡)𝛼Φ(𝑡),(3.2) 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⎞⎟⎟⎟⎠𝑥𝑦((𝑛𝑛𝜏𝜏)).(3.3) Hence, according to the Floquet theory, if all eigenvalues of 𝑀(𝜏)=⎛⎜⎜⎜⎝1−𝑏¹𝛾₂1+𝑏₁𝑏₁𝛾₁−𝛾₂1+𝑏₁01+𝑏¹−𝑏₁𝛾₁1+𝑏₁⎞⎟⎟⎟⎠Φ(𝜏)(3.4) are less than one, then the infection-free periodic solution (𝑠(̃𝑡),0) is locally stable. By calculating, we have Φ(𝑡)=1Φ¹²0Φ₂₂,(3.5) where Φ₂₂(𝑡)=exp(𝛽∫𝑠(̃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{𝛽∫^𝜏0𝑠(̃𝜎)𝑑𝜎−(𝜃+𝛼)𝜏}, if and only if 𝜇₂<1. Define threshold of model (2.4) as follows: 𝑅^∗=𝛽∫^𝜏0𝑠̃(𝜎)𝑑𝜎ln1+𝑏₁/1+𝑏₁−𝑏₁𝛾₁+(𝜃+𝛼)𝜏,(3.6) where 𝑠(̃𝑡) is the infection-free periodic solution. That is, the infection-free periodic solution (𝑠(̃𝑡),𝑖(̃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 (𝑠(̃𝑡),𝑖(̃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 (𝑠(̃𝑡),𝑖(̃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 (𝑠(̃𝑡),𝑖(̃𝑡)) of system (2.4) is global asymptotically stable.

Proof. Because of 𝛼≤𝛽, and 𝛾₁≤𝛾₂, we have 𝑠𝑛𝜏₊𝑠^≤−𝑝𝑠,𝑡≠𝑛𝜏,≤𝑏¹𝛾₂1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑠(𝑛𝜏),𝑡=𝑛𝜏.(3.7) By impulsive differential inequality, we see that 𝑠(𝑡)≤𝑠0⁺_{𝑜<𝑛𝜏<𝑡}1−𝑏¹𝛾₂1+𝑏₁exp^𝑡0+(−𝑝)𝑑𝜎_{𝑜<𝑛𝜏<𝑡}_{𝑛𝜏<𝑗𝜏<𝑡}1−𝑏¹𝛾₂1+𝑏₁exp^𝑡𝑛𝜏(−𝑝)𝑑𝜎𝑏¹𝛾₂=𝑠̃(𝑡)+𝑠0₊1+𝑏₁1−𝑏¹𝛾₂1+𝑏₁^[]𝑡/𝑇𝑒^−𝑝𝑡−𝑏¹𝛾₂/1+𝑏₁1−𝑏₁𝛾₂/(1+𝑏₁)^[]𝑡/𝑇𝑒^−𝑝𝑡1−𝑏₁𝛾₂/1+𝑏₁𝑒^−𝑝𝑇.(3.8) Since lim_𝑡→∞𝑠0⁺1−𝑏¹𝛾₂1+𝑏₁^[]𝑡/𝑇𝑒^−𝑝𝑡−𝑏¹𝛾₂/1+𝑏₁1−𝑏₁𝛾₂/1+𝑏₁^[]𝑡/𝑇𝑒^−𝑝𝑡1−𝑏₁𝛾₂/1+𝑏₁𝑒^−𝑝𝑇=0,(3.9) for any given 𝜖₁>0, there exists 𝑇₁>0, such that 𝑠(𝑡)<𝑠(̃𝑡)+𝜖₁, for all 𝑡>𝑇₁.

Introduce the new variable 𝑢=𝑠+𝑟, then 𝑢𝑛𝜏₊𝑢^=(−𝛽𝑠+𝛼𝑢+𝜃)(1−𝑢),𝑡≠𝑛𝜏,=𝑏¹𝛾₁1+𝑏₁+1−𝑏¹𝛾₁1+𝑏₁𝑢(𝑛𝜏),𝑡=𝑛𝜏.(3.10)

Consider the following comparison system with pulse: 𝑣𝑛𝜏₊𝑣^=−𝛽𝑠̃(𝑡)+𝛽𝜖₁+𝛼−𝜃𝑣−𝛼𝑣²+𝜃−𝛽𝑠̃(𝑡)+𝜖₁,𝑡≠𝑛𝜏,=𝑏¹𝛾₁1+𝑏₁+1−𝑏¹𝛾₁1+𝑏₁𝑣(𝑛𝜏),𝑡=𝑛𝜏.(3.11)

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𝑛𝜏₊𝑦^=−𝛽𝑠̃(𝑡)−𝛼−𝜃+𝛽𝜖₁𝑦−𝛼𝑦²,𝑡≠𝑛𝜏,=1−𝑏¹𝛾₁1+𝑏₁𝑦(𝑛𝜏),𝑡=𝑛𝜏.(3.12)

Let 𝑧=1/𝑦, then 𝑧𝑛𝜏₊𝑧^=−𝛽𝑠̃(𝑡)−𝛼−𝜃+𝛽𝜖₁𝑧−𝛼,𝑡≠𝑛𝜏,=1+1𝑏₁+−𝑏¹𝑏₁𝛾₁𝑧(𝑛𝜏),𝑡=𝑛𝜏.(3.13)

Let 𝑞(𝑡)=𝛽𝑠(̃𝑡)−𝛼−𝜃+𝛽𝜖₁, solving system (3.13) between pulses (𝑇₁+𝑛𝜏,𝑇₁+(𝑛+1)𝜏], we have 𝑧(𝑡)=𝑒^{−∫𝑡𝑇1+𝑛𝜏𝑞(𝜎)𝑑𝜎}𝛼^𝑡𝑇_1+𝑛𝜏𝑒^{∫𝑢𝑇1+𝑛𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢+1+𝑏¹1+𝑏₁−𝑏₁𝛾₁𝑧𝑇₁+𝑛𝜏,(3.14) when 𝑡=𝑇₁+(𝑛+1)𝜏, (3.14) can be written as follows: 𝑧𝑇₁+(𝑛+1)𝜏=𝑒^{−∫𝑇1𝑇1+(𝑛+1)𝜏+𝑛𝜏𝑞(𝜎)𝑑𝜎}𝛼^{𝑇1𝑇+(𝑛+1)𝜏}_1+𝑛𝜏𝑒^{∫𝑢𝑇1+𝑛𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢+1+𝑏¹1+𝑏₁−𝑏₁𝛾₁𝑧𝑇₁+𝑛𝜏.(3.15) On the other hand, solving system (3.13) between pulses (𝑇₁+(𝑛−1)𝜏,𝑇₁+𝑛𝜏], we obtain 𝑧(𝑡)=𝑒^{−∫𝑡𝑇1+(𝑛−1)𝜏𝑞(𝜎)𝑑𝜎}𝛼^𝑡𝑇_{1+(𝑛−1)𝜏}𝑒^{∫𝑢𝑇1+(𝑛−1)𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢+1+𝑏¹1+𝑏₁−𝑏₁𝛾₁𝑧𝑇₁+(𝑛−1)𝜏,(3.16) then 𝑧𝑇₁+𝑛𝜏=𝑒^{−∫𝑇1𝑇1+(𝑛+1)𝜏+𝑛𝜏𝑞(𝜎)𝑑𝜎}𝛼^{𝑇1𝑇+𝑛𝜏}_{1+(𝑛−1)𝜏}𝑒^{∫𝑢𝑇1+(𝑛−1)𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢+1+𝑏¹1+𝑏₁−𝑏₁𝛾₁𝑧𝑇₁+(𝑛−1)𝜏.(3.17)

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+1𝑏₁+−𝑏¹𝑏₁𝛾₁^𝑛_{1≤𝑘≤𝑛}1+1𝑏₁+−𝑏¹𝑏₁𝛾₁^{𝑘−𝑛−1}+𝑒^{∫𝑇1𝑇1+(𝑛−𝑘)𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢_{𝑇1𝑇+(𝑛−𝑘+1)𝜏1+(𝑛−𝑘)𝜏}𝑒^{∫𝑢𝑇1+(𝑛−𝑘)𝜏𝑞(𝜎)𝑑𝜎}𝑑𝑢+𝑧𝑇₁,(3.18) where 𝑒^{−∫0𝑛𝜏𝑞(𝜎)𝑑𝜎}1+1𝑏₁+−𝑏¹𝑏₁𝛾₁^𝑛=exp𝑛−𝛽^𝜏0𝑠̃(𝜎)𝑑𝜎+(𝜃+𝛼)𝜏−𝛽𝜖₁𝜏+ln1+𝑏¹1+𝑏₁−𝑏₁𝛾₁.(3.19)

The condition 𝑅^∗<1 implies that lim_𝑛→∞𝑧(𝑛𝜏)=∞, then lim_𝑡→∞𝑥(𝑡)=1. The comparison principle and the condition 𝑢(𝑡)<1 imply that lim_𝑡→∞𝑢(𝑡)=1, so we have lim_𝑡→∞𝑖(𝑡)=0.

Because we have proved that lim_𝑡→∞𝑖(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𝑛𝜏₊𝑠^≥−𝑝𝑠+(𝛼−𝛽)𝜖₂𝑠,𝑡≠𝑛𝜏,≥𝑏¹𝛾₁1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑠(𝑛𝜏),𝑡=𝑛𝜏.(3.20) Therefore 𝑠(𝑡)≥𝑠𝑇⁺²_{𝑇2<𝑛𝜏<𝑡}1−𝑏¹𝛾₂+1+𝑏₁exp^𝑡𝑇₂−𝑝+(𝛼−𝛽)𝜖₂𝑑𝜎_{𝑇2<𝑛𝜏<𝑡}_{𝑇2<𝑗𝜏<𝑡}1−𝑏¹𝛾₂1+𝑏₁exp^𝑡𝑛𝜏−𝑝+(𝛼−𝛽)𝜖₂𝑑𝜎𝑏¹𝛾₁=𝑠𝑇₊₂1+𝑏₁1−𝑏¹𝛾₂1+𝑏₁^{[𝑡/𝑇]−[𝑇2/𝑇]}+𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑡−𝑇₂𝛾₁/1+𝑏₁1−1−𝑏₁𝛾₂/1+𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑇exp−𝑝+(𝛼−𝛽)𝜖²−𝑏₁𝑡−𝑇𝑡𝑇𝛾₁/1+𝑏₁×11−−1𝑏−₁𝑏₁𝛾₂/1+𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑇𝛾₂1+𝑏₁^{([𝑡/𝑇]−[𝑇2/𝑇]−1)}exp−𝑝+(𝛼−𝛽)𝜖₂𝑇.(3.21) For any given 𝜖₂>0, we have lim_𝑡→∞𝑠𝑇⁺²1−𝑏¹𝛾₂1+𝑏₁^{[𝑡/𝑇]−[𝑇2/𝑇]}−𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑡−𝑇₂𝛾₁/1+𝑏₁×11−−1𝑏−₁𝑏₁𝛾₂/1+𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑇𝛾₂1+𝑏₁^{([𝑡/𝑇]−[𝑇2/𝑇]−1)}lim_𝑡→∞exp−𝑝+(𝛼−𝛽)𝜖₂(𝑇)=0,𝑏₁𝛾₁/1+𝑏₁1−1−𝑏₁𝛾₂/1+𝑏₁exp−𝑝+(𝛼−𝛽)𝜖₂𝑇exp−𝑝+(𝛼−𝛽)𝜖²𝑡−𝑇𝑡𝑇=𝑠̃(𝑡).(3.22) Therefore, for any given 𝜖₃>0, there exists 𝑇₃>0, when 𝑡>𝑇₃, then we have 𝑠(𝑡)≥𝑠̃(𝑡)−𝜖₃.(3.23) For any given 𝜖>0. Let 𝑇=max{𝑇₁,𝑇₂,𝑇₃}, then 𝑡>𝑇, then we have 𝑠̃(𝑡)−𝜖≥𝑠(𝑡)≥𝑠̃(𝑡)+𝜖,(3.24) that is lim_𝑡→∞𝑠(𝑡)=𝑠(̃𝑡).

Therefore the infection-free periodic solution (𝑠(̃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, lim_𝑡→∞inf𝑖(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𝑛𝜏₊𝑥^=−𝑔𝑥−ℎ,𝑡≠𝑛𝜏,=𝑏¹𝛾₁1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑥(𝑛𝜏),𝑡=𝑛𝜏,(4.1) has a unique positive 𝜏-periodic solution 𝑥̃(𝑡) for which 𝑥̃(0)>0,   𝑡∈𝑅₊, and 𝑥̃(𝑡) is global asymptotically stable in the sense that lim_𝑡→∞|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)−ℎ^𝑡0𝑊(𝑡,𝜎)𝑑𝜎+𝑏¹𝛾₁1+𝑏₁_{0<𝑛𝜏<𝑡}𝑊𝑡,𝑛𝜏⁺,(4.2) where 𝑊𝑡,𝑡₀=_{𝑡0≤𝑛𝜏<𝑡}1−𝑏¹𝛾₂1+𝑏₁𝑒^{−𝑔(𝑡−𝑡0)}.(4.3) 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)=(−ℎ∫^𝜏0𝑊(𝜏⁺,𝜎)𝑑𝜎+(𝑏₁𝛾₁/(1+𝑏₁))𝑊(𝜏,𝜏))/(1−𝑊(𝜏,0)). Next, we only need to prove that lim_𝑡→∞|𝑥(𝑡)−𝑥̃(𝑡)|=0.
Since ||𝑥(𝑡)−𝑥̃(𝑡)||=𝑊(𝑡,0)||𝑥(0)−𝑥̃(0)||,(4.4) the result is obtained if 𝑊(𝑡,0)→0 as 𝑡→∞. Suppose 𝑡∈(𝑛𝜏,(𝑛+1)𝜏], then 𝑊(𝑡,0)=_{0≤𝑗𝜏<𝑡}1−𝑏¹𝛾₂1+𝑏₁𝑒^−𝑔𝑡=1−𝑏¹𝛾₂1+𝑏₁^[𝑡/𝜏]𝑒^−𝑔𝑡.(4.5) Thus lim_𝑡→∞𝑊(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 lim_𝑡→∞sup𝑖(𝑡)>𝑐 for all solutions of (2.4).

Proof. Suppose that for every 𝜖>0, there is some solution with lim_𝑡→∞sup𝑖(𝑡)<𝜖. From the first equation of (2.4), we have 𝑠^=−𝑝𝑠+(𝛼−𝛽)𝑠𝑖≥−𝑝𝑠+(𝛼−𝛽)𝜖,𝑡≠𝑛𝜏.(4.6) Consider the following equation: 𝑤𝑛𝜏𝑤^₊=−𝑝𝑤+(𝛼−𝛽)𝜖,𝑡≠𝑛𝜏,=𝑏¹𝛾₁1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑤(𝑛𝜏),𝑡=𝑛𝜏.(4.7) By Lemma 4.1, we see that (4.7) has a unique positive 𝜏-periodic solution 𝑤(𝑡), and 𝑤(𝑡) is global asymptotically stable. Solving (4.7), we have 𝑤(𝑡)=𝑊(𝑡,0)𝑤(0)+(𝛼−𝛽)𝜖^𝑡0𝑊(𝑡,𝜎)𝑑𝜎+𝑏¹𝛾₁1+𝑏₁_{0<𝑛𝜏<𝑡}𝑤(𝑡)=(𝛼−𝛽)𝜖𝑊(𝑡,0)∫_𝜏0𝑊𝑡,𝑛𝜏⁺,(4.8)𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0+1−𝑏₁𝑊(𝑡,𝜎)𝑑𝜎𝛾₂1+𝑏₁𝑊1(−𝑡,0𝑊)𝑊(𝜏,(𝜏0,)𝜏)+𝑏¹𝛾₁1+𝑏₁_{0<𝑛𝜏<𝑡}𝑊𝑡,𝑛𝜏⁺,(4.9) and 𝑤(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𝑊1(−𝑡,0𝑊)𝑊(𝜏,(𝜏0,)𝜏)+𝑏¹𝛾₁1+𝑏₁_{0<𝑛𝜏<𝑡}𝑊𝑡,𝑛𝜏⁺.(4.10) and we have 𝑠̃(𝑡)−𝑤(𝑡)=(𝛽−𝛼)𝜖𝑊(𝑡,0)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0𝑊(𝑡,𝜎)𝑑𝜎.(4.11) Let Δ=(𝛽−𝛼)max_{0≤𝑡≤𝜏}𝑊(𝑡,0)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0𝑊(𝑡,𝜎)𝑑𝜎,(4.12) by (4.11), we can see that 𝑤(𝑡)≥𝑠̃(𝑡)−Δ𝜖.(4.13) By comparison theory, we obtain that 𝑖^≥−(𝜃+𝛼)𝑖+𝛽𝑖𝑤(𝑡)+𝛼𝑖².(4.14) Since 𝑤(𝑡) is global asymptotically stable, for above 𝜖, there exists 𝑇₄>0, such that 𝑤(𝑡)≥𝑤(𝑡)−𝜖, 𝑡>𝑇₄. From (4.13) and (4.14), we have that 𝑖^≥[𝛽𝑠̃(𝑡)−(𝜃+𝛼)−(1+Δ)𝛽𝜖]𝑖.(4.15) Consider the following equation: 𝑖𝑛𝜏₊𝑖^≥[𝛽𝑠̃(𝑡)−(𝜃+𝛼)−(1+Δ)𝛽𝜖]𝑖,𝑡≠𝑛𝜏,=1+𝑏¹1−𝛾₁1+𝑏₁𝑖(𝑛𝜏),𝑡=𝑛𝜏.(4.16) By impulsive differential inequality, for 𝑡∈(𝑇₄+𝑛𝜏,𝑇₄+(𝑛+1)𝜏], we see that 𝑖(𝑡)≥𝑖𝑇₄_{𝑇4<𝑗𝜏<𝑡}1+𝑏₁1−𝛾₁1+𝑏₁=𝑖𝑇₄exp^𝑡𝑇₄[𝛽𝑠̃(𝜎)−(𝜃+𝛼)−(1+Δ)𝛽𝜖]𝑑𝜎1+𝑏¹(1−𝛾₁)1+𝑏₁^𝑛+exp^{𝑇4𝑇+𝑛𝜏}₄[𝛽𝑠̃(𝜎)−(𝜃+𝛼)−(1+Δ)𝛽𝜖]𝑑𝜎_{𝑡𝑇4+𝑛𝜏}≥𝐶exp𝑛𝑅_∗[𝛽𝑠̃(𝜎)−(𝜃+𝛼)−(1+Δ)𝛽𝜖]𝑑𝜎−1ln1+𝑏¹1+𝑏₁1−𝛾₁+(𝜃+𝛼)𝜏−(1+Δ)𝛽𝜖𝜏,(4.17) 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,(4.18) 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), lim_𝑡→∞inf𝑖(𝑡)≥𝜌>0.

Proof. Let 0<𝜂≤121−𝑅1^∗𝑀(𝑠𝜃̃(𝑡+)𝛼),(4.19) where 𝑀=max_{0≤𝑡≤𝜏}𝑊(𝑡,0)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0𝑊𝑡,𝑡₀𝑊(𝑡,𝜎)𝑑𝜎,=_{𝑡0≤𝑛𝜏<𝑡}1+𝑏₁1−𝛾₁1+𝑏₁𝑒^{−𝑝𝑡−𝑡0}.(4.20) 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 𝑡≥𝑡₀. 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.(4.21) Next, we introduce the new variable 𝑉=𝑠+𝑖, and it follows from the first two equations of (2.4) that 𝑉𝑖^=−𝑝𝑉−(𝜃+𝛼)𝑖+(𝛼𝑉+𝑝)𝑖,_𝑉𝑛𝜏₊=−(𝜃+𝛼)𝑖+𝛽(𝑉−𝑖)𝑖+𝛼𝑖²,𝑡≠𝑛𝜏,=𝑏¹𝛾₂1+𝑏₁+1−𝑏¹𝛾₂𝑖𝑛𝜏₊1+𝑏₁𝑉(𝑛𝜏),=1+𝑏¹1−𝛾₁1+𝑏₁𝑖(𝑛𝜏),𝑡=𝑛𝜏.(4.22) If 𝑖(𝑡)≤𝜂, then 𝑉^≥−𝑝𝑉−(𝜃+𝛼)𝜂,𝑡≠𝑛𝜏.
Consider the following equation: 𝑥𝑛𝜏₊𝑥^=−𝑝𝑥−(𝜃+𝛼)𝜂,𝑡≠𝑛𝜏,=𝑏¹𝛾₂1+𝑏₁+1−𝑏¹𝛾₂1+𝑏₁𝑥(𝑛𝜏),𝑡=𝑛𝜏.(4.23)
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)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0+1−𝑏₁𝑊(𝑡,𝜎)𝑑𝜎𝛾₂1+𝑏₁𝑊(𝑡,0)𝑊𝜏⁺,𝜏⁺(1−𝑊(𝜏,0))+𝑏¹𝛾₂1+𝑏₁_{0<𝑛𝜏<𝑡}𝑊𝑡,𝑛𝜏⁺.(4.24) From (4.10) and (4.24), it is easy to see that 𝑥̃(𝑡)−𝑠̃(𝑡)≥−(𝜃+𝛼)𝜂𝑊(𝑡,0)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0𝑊(𝑡,𝜎)𝑑𝜎.(4.25) By 0<𝜂≤(1/2)(1−1/𝑅^∗)(𝑠(̃𝑡)/𝑀(𝜃+𝛼)), 𝑀=max_{0≤𝑡≤𝜏}{𝑊(𝑡,0)∫^𝜏0𝑊(𝜏⁺,𝜎)𝑑𝜎/(1−𝑊(𝜏,0))+∫^𝑡0𝑊(𝑡,𝜎)𝑑𝜎}, and (𝑊(𝑡,0)∫^𝜏0𝑊(𝜏⁺,𝜎)𝑑𝜎/(1−𝑊(𝜏,0))+∫^𝑡0𝑊(𝑡,𝜎)𝑑𝜎)(1/𝑀)≤1, we obtain −(𝜃+𝛼)𝜂𝑊(𝑡,0)∫^𝜏0𝑊𝜏⁺,𝜎𝑑𝜎1−𝑊(𝜏,0)+^𝑡0𝑊(𝑡,𝜎)𝑑𝜎≥−121−𝑅1^∗𝑠̃(𝑡),(4.26) namely, 𝑥̃(𝑡)≥121+𝑅1^∗𝑠̃(𝑡).(4.27) 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+𝑅1^∗𝑠̃(𝑡),∀𝑡>𝑡¹+𝑇₅.(4.28) From (4.28) and the second equation of (4.22), we can see that 𝑖^≥𝛽21+𝑅1^∗𝑠̃(𝑡)−(𝜃+𝛼)𝑖+(𝛼−𝛽)𝑖².(4.29) Consider the following equation: 𝑦^=𝛽21+𝑅1^∗𝑠̃(𝑡)−(𝜃+𝛼)𝑦+(𝛼−𝛽)𝑦²𝑦𝑛𝜏₊,𝑡≠𝑛𝜏,=1+𝑏¹1−𝛾₁1+𝑏₁𝑦(𝑛𝜏),𝑡=𝑛𝜏.(4.30) Let 𝑧=𝑦⁻¹, then we have 𝑧^=(𝜃+𝛼)−𝛽𝑧𝑛𝜏₊21+𝑅1^∗𝑠̃(𝑡)𝑧+(𝛽−𝛼),𝑡≠𝑛𝜏,=1+𝑏¹1+𝑏₁1−𝛾₁𝑧(𝑛𝜏),𝑡=𝑛𝜏.(4.31) By the same method of Lemma 4.1, we can get a conclusion that 𝑊𝑧(̃𝑡()𝜏=,0)()𝛽)−𝛼)(𝑊(𝑡,0)∫^𝜏0𝑊(𝜏⁺,𝜎)𝑑𝜎/(1−𝑊(𝜏,0))+∫^𝑡0𝑊(𝑡,𝜎)𝑑𝜎)+(1+𝑏₁)/(1+𝑏₁(1−𝛾₁))(𝑊(𝑡,0)𝑊(𝜏,𝜏)/(1− is global asymptotically stable. Thus system (4.30) has a unique positive 𝜏-periodic solution 𝑦(̃𝑡), and 𝑦(̃𝑡) is global asymptotically stable, lim_𝑡→∞||𝑦(𝑡)−𝑦̃(𝑡)||=0.(4.32) From (4.32) we see that there exists a positive constant 𝑇₆>0 such that 𝑦(𝑡)>𝜌≡12min^𝑡₁^{≤𝑡≤𝑡}₁^+𝜏𝑦̃(𝑡)>0,∀𝑡>𝑡₁+𝑇₆.(4.33) 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,(4.34) 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)≥𝜌≥𝜌 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.

(a)

(b)

(c)

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.

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.

(a)

(b)

(c)

Figure 2 

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 lim_𝑡→∞𝑖(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).