We study an epidemic model with a nonlinear incidence rate which describes the psychological effect of certain serious diseases on the community when the ratio of the number of infectives to that of the susceptibles is getting larger. The model has set up a challenging issue regarding its dynamics near the origin since it is not well defined there. By carrying out a global analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

In recent years, attempts have been made to develop realistic mathematical models for the transmission dynamics of infectious diseases. In modeling of communicable diseases, the incidence function has been considered to play a key role in ensuring that the models indeed give reasonable qualitative description of the transmission dynamics of the diseases [

Let

Note that the infectious force

To describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals is very large, Xiao and Ruan [

Instead of the infectious force given by (

This paper is organized as follows: in Section

Assuming that the infectious force takes the form (

The plane

Summing up the three equations in (

In the following, we consider the existence of equilibria of system (

if

if

In the following section, we will study the properties of these equilibria and perform a global qualitative analysis of model (

It is clear that the limit set of system (

To be concise in notations, rescale (

Denote

Note that the equilibrium

Note that we are interested in the dynamics of system (

In the following, we will study the properties of the equilibria and perform a global qualitative analysis of model (

The equilibrium

First of all, we introduce the polar coordinates

To determine if there exists an orbit of system (

There exist

Since

In fact, the only orbit tending to

First,we give the following result regarding the nonexistence of periodic orbits in system (

System (

Consider system (

The following theorem shows the properties of the disease-free equilibrium

The disease-free equilibrium

a global asymptotically stable node if

a saddle if

global asymptotically stable if

The Jacobian matrix of system (

If

If

If

Suppose

The Jacobian matrix of system (

Summarizing Theorems

Let

If

If

If

Topological structure of system (

Topological structure of system (

Topological structure of system (

Several nonlinear incidence rates have been proposed by researchers, see, for example, Capasso and Serio [

In this paper, different from the classical nonlinear incident rate, we assume that the infectious force is a function of the ratio of the number of the infectives to that of the susceptibles which takes the form

Research is supported by the National Natural Science Foundation of China (10871129) and the Educational Committee Foundation of Shanghai (09YZ208).