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A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain time

As we know, in the SIR model, the population is divided into three classes,

However, the time period of immunity for many infectious diseases is short, or even they have no immunity. Furthermore, these diseases often lead to some other more dangerous diseases. For examples, Chagas disease, hepatitis C, gonorrhea, and other sexually transmitted diseases may advance through several infective stages and have different ability to transmit these infections in different stages of infection. Their infectivity usually depends on the parasite or viral loads in infected individuals or vectors [

Motivated by the above discussions, we introduce a new epidemiological model in this paper. In this model, the population is divided into three classes,

Our model to be considered takes the following form:

It is easy to see that system (

For system (

if

if

the equilibrium

The rest of this paper is organized as follows. In Section

When

When

Taking the initial value

Taking

When

When

If

In this case, we neglect the time from infectious disease

In order to examine local stability of an equilibrium, we should compute the eigenvalues of the linearized operator for system (

Consider disease-free equilibrium

The local stability of disease-free equilibrium

If

If

If

Now, the local stability of the endemic equilibrium

If

To study the global stability of an equilibrium, we first present two lemmas.

There exists a positively invariant set

If the initial value

By (

By adding the first three equations of system (

In summary, if the initial value

It is clear that the limit set of system (

The limit set of system (

One has the following result regarding the nonexistence of periodic orbits in system (

System (

Consider system (

When

If

If

(i) It is easy to check the conclusion by Theorem

Clearly,

(ii) By Theorem

In this section, we consider the system (

By this method, we linearize the system (

It is easy to see that

By discussing (

When

If

If

If

(i) By implicit function theorem for complex variables, we know that the root of (

If

(ii) We set

(iii) If

Now, the local stability of the endemic equilibrium

If

By implicit function theorem for complex variables, we know that the root of (

Now, we examine the sign of

By (

Now that

In what follows, we consider the Hopf bifurcation of the disease-free equilibrium

If

When

Suppose that

Now, we consider the Hopf bifurcation of the endemic equilibrium

By Theorem

If

By Theorem

We know that (

Assume that (

We set

Assume that the condition of Lemma

We consider the transversal conditions [

Suppose that

Since

By Lemma

In this paper, we propose the SID model with time delays. Our purpose is to comprehend the number of disease

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

The authors are very grateful to the anonymous referees for careful reading and helpful suggestions which led to the improvement of their original paper. This work was partially supported by the National Natural Science Foundation of China, Program for Changjiang Scholars and Innovative Research Team in University (IRT1226), and Research Fund for the Doctoral Program of Higher Education of China (no. 20124410110001).