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SIR epidemic model with nonlinear pulse vaccination and lifelong immunity is proposed. Due to the limited medical resources, vaccine immunization rate is considered as a nonlinear saturation function. Firstly, by using stroboscopic map and fixed point theory of difference equations, the existence of disease-free periodic solution is discussed, and the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Moreover, by using the bifurcation theorem, sufficient condition for the existence of positive periodic solution is obtained by choosing impulsive vaccination period as a bifurcation parameter. Lastly, some simulations are given to validate the theoretical results.

Infectious disease is one of the greatest enemies of human health. According to the World Health Statistics Report 2013 [

This paper is organized as follows. In Section

Let the total population number of model (

System (

Solving the first equation of system (

From system (

According to Lemma

System (

Next we will discuss the stability of the periodic solution. Suppose that

The eigenvalues of the matrix

The disease-free periodic solution

Next we will prove the global attractivity of the disease-free periodic solution

The disease-free periodic solution

Let

Letting

Synthesizing Theorems

The disease-free periodic solution

In this section, we will discuss the existence of the positive periodic solution and the branch of the system (

Obviously, the threshold value

If the condition

In this paper, we have considered a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity. The whole dynamics of the model is investigated under nonlinear impulsive effect. Firstly, the existence of disease-free periodic solution is discussed by using stroboscopic map and fixed point theory of difference equations, the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Then, by choosing impulsive vaccination period as a bifurcation parameter, sufficient condition for the existence of positive periodic solution was obtained by using the bifurcation theorem. We have found that the dynamics of the model (

Next, we focus on the relations of the

The relations of

From Figure

To show the influence of restriction of medical resources on the model dynamics, we give some numerical simulations. Let

Let

Time series of

Time series of

Phase diagram of system (

Letting

Time series of

Time series of

Phase diagram of system (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (no. 11371230), Natural Sciences Fund of Shandong Province of China (no. ZR2012AM012), a Project for Higher Educational Science and Technology Program of Shandong Province of China (no. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.