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We investigate the dynamics of a nonautonomous stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. By constructing suitable stochastic Lyapunov functions and using Has’minskii theory, we prove that there exists at least one nontrivial positive periodic solution of the system. Moreover, the sufficient conditions for extinction of the disease are obtained by using the theory of nonautonomous stochastic differential equations. Finally, numerical simulations are utilized to illustrate our theoretical analysis.

The SIS (Susceptible-Infected-Susceptible) model is a basic biological mathematical model describing susceptible and infected epidemic process and is first introduced by Kermack and McKendrick [

In the real world, population systems and epidemic systems are inevitably infected by some uncertain environmental disturbances. Hence, many authors have introduced stochastic interferences into differential systems, and the stochastic dynamics of such systems were widely investigated (see [

However, many infectious diseases of human fluctuate over time and often show the seasonal morbidity. Therefore, the existence of periodic solutions of some nonautonomous epidemic models was explored [

Based on the discussion above, in this paper, we consider a nonautonomous stochastic SIS model with periodic coefficients

To the best of our knowledge, there are only few works on research of nonautonomous stochastic epidemic models with nonlinear saturated incidence rate and double epidemic hypothesis. Therefore, based on an autonomous stochastic epidemic model, we propose a nonautonomous stochastic model and investigate the existence of stochastic periodic solution and the extinction of the two epidemic diseases.

This paper is organized as follows. In Section

Throughout this paper, let

For simplicity, some notations are given first. If

Here we present some basic theory in stochastic differential equations which are introduced in [

In general, consider the

A stochastic process

It is shown in [

Suppose that coefficients of (

Let

Then

In this section, we prove that system (

For any initial value

From system (

Since the coefficients of system (

Let

So we have

Let

In this section, we verify that system (

When

Define a

By Itô’s formula, we obtain

Next we will prove that

According to (

In view of (

By (

Together with (

By virtue of (

Clearly, one can see from (

In this section, we investigate the conditions for the extinction of the two infectious diseases of system (

Let

Let

Then if

Applying Itô’s formula to system (

Taking the superior limit of both sides of (

Theorem

Now we introduce some numerical simulations examples which illustrate our theoretical results.

In model (

The solution

Choose the parameters in model (

The solution

Choose the parameters in model (

Note that

The solution

This paper explores the existence of nontrivial positive T-periodic solution of a nonautonomous stochastic SIS epidemic model with nonlinear growth rate and double epidemic hypothesis. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of nontrivial positive

If

If

If

Some interesting questions deserve further investigation. On the one hand, we may explore some realistic but complex models, such as considering the effects of impulsive or delay perturbations on system (

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (11371230), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province, the SDUST Research Fund (2014TDJH102), and Shandong Provincial Natural Science Foundation, China (ZR2015AQ001).