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This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.

The outbreak of infectious diseases had not only caused the loss of billions of lives but also badly damaged the social economy in a short time, which brought much pain to human society [

In fact, many infectious diseases have different kinds of delays during their spreading process in the population, such as latent period delay [

The outline of this paper is as follows. In the next section, stability of the endemic equilibrium is analyzed and the critical value of the time delay at which a Hopf bifurcation occurs is obtained. In Section

By a direct computation, we know that if (I)

Let

Routh-Hurwitz criterion implies that

(

For

For the polynomial equation (

if

if

if

Thus, (

Thus, if the condition

For system (

Let

By the Riesz representation theorem, there exists a

Let

Next, we can obtain the coefficients

Then, we can get the following coefficients which determine the properties of the Hopf bifurcation:

For system (

In order to verify the efficiency of the obtained results in the paper, we carry out some numerical simulations in this section. By extracting some values from [

In addition, according to (

We generalize a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate in this paper by introducing the time delay due to the period that is used to cure the infectious population into the SEIS model considered in the literature [

The main results are given in terms of local stability and Hopf bifurcation. Stability of the endemic equilibrium is investigated by analyzing the corresponding characteristic equation. By choosing the time delay as a bifurcation parameter, sufficient conditions have been established for local existence of Hopf bifurcation at the endemic equilibrium. Then, with the help of the normal form theory and the center manifold theorem due to Hassard et al. [

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2015A144).