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An SIR epidemic model with pulse birth and
standard incidence is presented. The dynamics of the epidemic model
is analyzed. The basic reproductive number

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 [

Roberts and Kao established an SI epidemic model with pulse birth, and they found the periodic solutions and determined the criteria for their stability [

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

In our study, we analyze the dynamics of the SIR model of a population of susceptible (

From (

From biological view, we easily see that the domain

We first demonstrate the existence of infection-free periodic solution of system (

Therefore the system (

In this section,we will prove the local and global asymptotically stable of the infection-free periodic solution

The local stability of the infection-free periodic solution

The eigenvalues of

If

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

If

Because of

Introduce the new variable

Consider the following comparison system with pulse:

The first equation of (

Let

Let

Similarly, we can get the expressions of

The condition

Because we have proved that

When

Therefore the infection-free periodic solution

In this section, we will discuss the uniform persistence of the infectious disease, that is,

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

For the following impulsive equation,

Solving (

Since

If

Suppose that for every

If

Let

Consider the following equation:

By Lemma

If

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

In Figure

The time series and the orbits of the system (

Figure

The time series and the orbits of the system (

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

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

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.

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).