Complexity
Volume 2017 (2017), Article ID 4861391, 14 pages
https://doi.org/10.1155/2017/4861391

Research Article

Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis

Haokun Qi,¹ Lidan Liu,¹ and Xinzhu Meng1,2

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
²State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Xinzhu Meng; mxz721106@sdust.edu.cn

Received 25 August 2017; Accepted 11 October 2017; Published 8 November 2017

Academic Editor: Benito M. Chen-Charpentier

Copyright © 2017 Haokun Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1]. The SIS model is defined in that individuals start off susceptible, at some stage catch the disease, and after a short infectious period become susceptible again [2]. Therefore, some deterministic SIS epidemic models have been studied by many authors [3–10]. Recently, the authors of [11–13] investigated the epidemic model with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. For example, the deterministic SIS epidemic model with nonlinear saturated incidence rate and double epidemic hypothesis can be expressed as follows [11]:

where , , and represent the number of susceptibles and infected individuals with viruses and at time , respectively. The parameters in model (1) have the following meanings: is the total input susceptible population size, represents the natural death rate of , , and , represents the disease transmission coefficient between compartments and , and are the recovery rates of the two diseases, and and are mortality rates due to diseases, respectively. Functions and represent two different types of saturated incidence rates for the two epidemic diseases and . All parameter values are nonnegative.

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 [14–28]). Moreover, numerous scholars have investigated some stochastic epidemic models (see [29–34]). For example, in [11, 30] they obtained thresholds of the stochastic system which determines the extinction and persistence of the epidemic. Zhang et al. [29] proved that there is a unique ergodic stationary distribution of his model. We assume that environment fluctuations will manifest themselves mainly as fluctuations in the saturated response rate, so that , where is a standard Brownian motion with intensity . Therefore, a stochastic model is described by [11]

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 [35–37]. Recently, many scholars focused on nonautonomous stochastic periodic systems. With the development of stochastic differential equations and application of Has’minskii theory, the existence of stochastic periodic solution has been studied [23, 38, 39]. In [23], Zhang et al. considered a nonautonomous stochastic Lotka-Volterra predator-prey model with impulsive effects; they got thresholds for stochastic persistence and extinction of the system. Authors of [38–40] investigated periodic solution of a stochastic nonautonomous epidemic model.

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

where the parameter functions , , , , , , are positive, nonconstant, and continuous periodic functions with positive period .

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 2, we give some definitions and known results. In Section 3, we prove that system (3) has a unique global positive solution. In Section 4, we present sufficient condition for the existence a nontrivial positive periodic solution of system (3). In Section 5, we obtain the sufficient conditions of system (3) for extinction of the two epidemic diseases. In Section 6, we carry out a series of numerical simulations to illustrate our theoretical findings.

2. Preliminaries

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). The function is defined on this complete probability space.

For simplicity, some notations are given first. If is an integrable function defined on , define If is a bounded function on , define and

Here we present some basic theory in stochastic differential equations which are introduced in [41].

In general, consider the -dimensional stochastic differential equation

with initial value . stands for a -dimensional standard Brownian motion defined on the complete probability space . Denote by the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in . The differential operator of (4) is defined by [41]

If acts on a function , then

where , , and In view of Itô’s formula, if , then

Definition 1 (see [42]). A stochastic process is said to be periodic with period if for every finite sequence of numbers the joint distribution of random variables is independent of , where .

It is shown in [42] that a Markov process is -periodic if and only if its transition probability function is -periodic and the function satisfies the equation

Consider the following equation:

Lemma 2 (see [42]). Suppose that coefficients of (9) are -periodic in and satisfy the condition

in every cylinder , where is a constant. And suppose further that there exists a function in which is -periodic in and satisfies the following conditions:

as . outside some compact set, where the operator is given by

Then there exists a solution of (9) which is a -periodic Markov process.

Lemma 3 (see [2], strong law of large numbers). Let be a real-valued continuous local martingale vanishing at .
Then

and also

3. Existence and Uniqueness of the Global Positive Solution

In this section, we prove that system (3) has a unique global positive solution.

Theorem 4. For any initial value , there is a unique positive solution of (3) on and the solution will remain in with probability one.

Proof. From system (3), we can get

Then

obviously, we have

Since the coefficients of system (3) satisfy the local Lipschitz conditions, then for any given initial value , there is a unique local solution on , where is the explosion time. To demonstrate that this solution is global, we only need to prove that a.s.
Let be sufficiently large for any initial value , and lying within the interval . For each integer , define the following stopping time:

where we set (as usual denotes the empty set). Clearly, is increasing as . Let ; hence a.s. Next, we only need to verify a.s. If this statement is false, then there exist two constants and such that

Thus there is an integer such that

Define a -function : as follows:

the nonnegativity of this function can be obtained from

Applying Itô’s formula yields

where

where is a positive constant.
So we have

Integrating (24) from 0 to and taking expectations on both sides yield

Let ; from inequality (25) we can see that We have

By (25) and (26), one has

where is the indicator function of .
Let ; we have

So we obtain The proof is completed.

4. Existence of Nontrivial -Periodic Solution

In this section, we verify that system (3) admits at least one nontrivial positive -periodic solution. Define

Theorem 5. When and hold, if , then there exists a nontrivial positive T-periodic solution of system (3).

Proof. Define a -function :

where

and ; is a sufficiently large positive constant and satisfies the following conditions:

where

where

Next we prove that condition in Lemma 2 holds. It is easy to check that is a -periodic function in and satisfies

where and is a sufficiently large number.
By Itô’s formula, we obtain

where

Note that and hold; then

Define the -periodic function satisfying

So

Applying Itô’s formula, we can also have

where

Therefore

Now, we are in the position to construct a compact subset such that in Lemma 2 holds. Define the following bounded closed set:

where is a sufficiently small number. In the set , we can choose sufficiently small such that

where , , , , and are positive constants which can be found from the following inequations (52), (54), (56), (59), and (61), respectively. For the sake of convenience, we divide into six domains,

Next we will prove that on , which is equivalent to proving it on the above six domains.
Case  1. If , one can see that

where

According to (45), we have for all .
Case  2. If , one can get that

where

In view of (46), we can obtain that for all .
Case  3. If , we have

where

By (47), we can conclude that for all .
Case  4. If , one can derive that

Together with (48), we can deduce that for all .
Case  5. If , it follows that

where

By virtue of (49), we can deduce that for all .
Case  6. If , we obtain

where

It follows from (50) that for all .
Clearly, one can see from (52), (54), (56), (58), (59), and (61) that, for a sufficiently small ,

Hence in Lemma 2 is satisfied. This completes the proof of Theorem 5.

5. Extinction

In this section, we investigate the conditions for the extinction of the two infectious diseases of system (3).

Let

Theorem 6. Let be a solution of system (3) with initial value .
Then if

or

hold, the two infectious diseases of system (3) go to extinction a.s.; that is,

Proof. Applying Itô’s formula to system (3), we have

Case (i). Integrating (68) from 0 to and dividing on both sides, we obtain

Case (ii). Integrating (68) from 0 to first and then dividing by on both sides yield

where , which is a local continuous martingale with . By Lemma 3, we have

Taking the limit superior of both sides of (69) leads to

which implies
Taking the superior limit of both sides of (70) leads to

which implies This completes the proof.

Remark 7. Theorem 6 shows that the two diseases will die out if the white noise disturbance is large or the white noise disturbance is not large and . When , the two infectious diseases of system (3) die out almost surely; that is to say, large white noise stochastic disturbance can lead to the two epidemics being extinct.

6. Numerical Simulations

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

Example 8. In model (3), let

Note that , , and hold; that is, the conditions of Theorem 5 hold. Hence, system (3) has a positive periodic solution with . Figure 1(a) shows the periodicity of the nonautonomous stochastic model (3) with and . Figure 1(b) shows that solution of the nonautonomous stochastic model (3) with the initial value tends to a periodic orbit in the sense of joint distribution.

(a)

(b)

Figure 1: The solution to the nonautonomous stochastic model (3).

Example 9. Choose the parameters in model (3) as follows:

Note that Therefore, conditions (i) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 2(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent. Figure 2(b) shows that the two diseases will die out under the large white noise disturbance of model (3).

(a)

(b)

Figure 2: The solution to the nonautonomous stochastic model (3).

Example 10. Choose the parameters in model (3) as follows:

Note that , , and . That is, conditions (ii) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 3(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent without the white noises. Figure 3(b) shows that the two diseases will die out under a small white noise disturbance of model (3).

(a)

(b)

Figure 3: The solution to the nonautonomous stochastic model (3).

7. Discussion and Conclusions

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 -periodic solution of system (3). Furthermore, the sufficient conditions for the extinction of the two diseases are obtained. Our results are given as follows:

(1)If , , and hold, the SIS model has at least one nontrivial positive -periodic solution, where

(2)If , the two infectious diseases go extinct.(3)If and , the two infectious diseases also go extinct, where

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 (3). On the other hand, we can concern the dynamics of a nonautonomous stochastic SIS epidemic model with two infectious diseases driven by Lévy jumps. What is more, we can also investigate the nonautonomous stochastic SIS epidemic model with two infectious diseases by a continuous time Markov chain. We will investigate these cases in our future work.

Conflicts of Interest

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

Acknowledgments

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).

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