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 coefficientswhere 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 equationwith 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 Thenobviously, 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 thatDefine a -function : as follows: the nonnegativity of this function can be obtained from Applying Itô’s formula yields wherewhere is a positive constant.
So we haveIntegrating (24) from 0 to and taking expectations on both sides yieldLet ; from inequality (25) we can see that We haveBy (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 : whereand ; is a sufficiently large positive constant and satisfies the following conditions: wherewhereNext 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 whereNote that and hold; then Define the -periodic function satisfying SoApplying Itô’s formula, we can also have where ThereforeNow, 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 thatwhere , , , , 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 thatwhere