Abstract
The purpose of this work is to investigate the dynamic behaviors of the SIRS epidemic model with nonlinear incident rate under regime switching. We establish the existence of a unique positive solution of our system. Furthermore, we obtain the conditions for the extinction of diseases, and we show the existence of the stationary distribution for our stochastic SIRS model under regime switching. Numerical simulations are employed to illustrate our theoretical analysis.
1. Introduction
Several of mathematicians have developed various epidemic models to prevent and control the spread of transmissible diseases in the community.
The classical SIR model presented by Kermack and McKendrick [1] has played an important role in mathematical epidemiology. The SIR model are used to study the disease spread between three groups of population to know the susceptible S, the infective I, and the recovered R.
In this work, we introduce a switched stochastic SIRS epidemic model with specific functional response. Then, we consider the following deterministic SIRS epidemic model with specific functional response:where denotes the number of susceptible individuals, denotes the number of infective individuals, and represents the number of removed individuals. is the recruitment rate of the population, is the natural death rate of the population, is the rate at which recovered individuals loss immunity and return to the susceptible class, denotes the natural recovery rate of the infectious individuals, and denotes the disease inducing death rate. The infection transmission process in (1) is modeled by the specific functional response , where is the transmission coefficient between compartments S and I, and , , are the saturation factors measuring the psychological or inhibitory effect. In addition, this functional response generalizes many common types existing in the literature such as the Crowley–Martin functional response introduced in [2] and used in [3] when and the Beddington–DeAngelis functional response proposed in [4] and used in [5] when .
Environmental fluctuations have been indicated to play an important role in the propagation of disease [6, 7]. In effect, disease infestation is highly stochastic, and stochastic noise can raise the probability of disease extinction in the early phase of epidemics. By running an ODE system, we can get only a certain sample solution, whereas by running an SDE system, we can obtain the stochastic distribution of disease dynamics [8]. Lately, dynamic modeling of infectious diseases based on stochastic differential equations (SDE) has received considerable attention from experts and academics [9–11]. The SIRS epidemic model with white noise is expressed bywhere are independent Brownian motions and are their intensities. Besides environment white noise, in this paper, we will also consider another noise, namely, telegraph noise ([12–15]). The latter can be described as a switching between two or more regimes of environment, which differ in terms of factors such as nutrition, climatic characteristics, or sociocultural factors. The switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed. The regime switching can hence be modeled by a finite-state Markov chain taking values in a finite-state space . The stochastic system (1) with regime switching can be described by the following model:
Throughout this paper, we 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). Let be a right-continuous Markov chain on the probability space taking values in a finite-state space with the generator given, for , by
Here, is the transition rate from to and if , while
Suppose that the Markov chain is independent of the Brownian motion and it is irreducible. Under this condition, the Markov chain has a unique stationary (probability) distribution , which can be determined by solving the linear equation , subject to , and , . Thereafter, for any vector , let and .
The rest of the paper is organized as follows. In Section 2, we show that there exists a unique global positive solution of system (3). In Section 3, we give sufficient conditions for the extinction of the disease. In Section 4, sufficient conditions for the existence of the ergodic stationary distribution are established for model (3). Finally, numerical simulations are carried out to support the theoretical results.
2. Existence and Uniqueness of the Global Positive Solution
In this section, we will prove that model (3) has a unique global positive solution. We also denote
Thus, we established the following theorem.
Theorem 1. For any given initial value, , there is a unique positive solution of model (3) on and the solution will remain in with probability 1, namely, for all almost surely.
Proof. Since the coefficients of system (3) are locally Lipshitz continuous, for any initial value , there exists a unique local solution on , where is the explosion time. We need to show that this solution is global almost surely that is, . Let be sufficiently large such that every component of lies within the interval . For each integer , define a sequence of stopping times byWe set , where denotes the empty set. Obviously, is increasing as . Set with . Now, we need to show . If this statement is violated, then there exist and such thatHence, there is an integer such thatDefine a function asBy Itô’s formula, we havewherewhich implies thatHence,Integrating both sides of the above inequality from 0 to , and taking the expectations, we getSet for and by (4), we have for each . For every , we haveThen, we obtainwhere is the indicator function of . Letting leads to the contradiction . So, we must therefore have . This completes the proof.
3. Extinction
Our goal in this section is to study the extinction and give the extinction threshold of system (3). Then, the following theorem gives a sufficient condition for extinction of the disease.
Theorem 2. If , then the disease tends to zero exponentially with probability one, i.e.,
Proof. Applying Itô’s formula, we can getIntegrating (19) from 0 to and then dividing by into both sides leads towhere is a local martingale defined by , whose quadratic variation is Making use of the strong law of large numbers for martingales ([16]) yieldsTaking the superior limit on both sides of (20) and applying the ergodicity of Markov chain , we getwhich implies that . This completes the proof of theorem.
4. Existence of Ergodic Stationary Distribution
In this section, we shall discuss sufficient conditions for the existence of an ergodic stationary distribution to model (3). The following lemma gives a criterion for positive recurrence in terms of lyapunov function [17].
Let is the diffusion Markov process and satisfy the following equationwhere , satisfying . For each , and for any twice continuously differentiable function , the operator can be defined by
Lemma 1. If the following conditions are satisfied:(1) for any .(2)For each , is symmetric and satisfies with some constant for all .(3)There exists a nonempty open set with compact closure, satisfying that, for each , there is a nonnegative function : such that is twice continuously differential and that for some ,
Then, of system (23) is positive recurrent and ergodic. That is to say, there exists a unique stationary distribution such that for any Borel measurable function satisfyingwe have
Theorem 3. Assume that , then for any initial value , the solution of system (3) admits a unique ergodic stationary distribution.
Proof. In order to prove Theorem 3, it is sufficient to prove conditions (1), (2) and (3) in Lemma 1. Assumption for any , in Section 1 implies that condition (1) in Lemma 1 is satisfied. To verify condition (2), consider the bounded open subsetwhere is a sufficiently large number. Then, . We have , where , . Then, is positive semi-definite, and since is a nonsingular matrix, we deduce that is positive definite. Hence,in addition, we have for all ,It is easy to see that and are two continuous functions of , and . Therefore, and , which implies thatwhere . Then, condition (2) in Lemma 1 is verified.
Now, we verify condition . Define a function byIn addition, is a continuous function on with respect to . So there is a unique minimum value point of in the interior of ; then, we define a nonnegative -function : as follows:where and is a sufficiently large number, , , and satisfies the following condition:Applying Itô’s formula to leads towhere . Since the generator matrix is irreducible, then for , there exists solution of the Poisson system ([18]):where denotes the column vector with all its entries equal to 1. Then,Next, we calculate and We haveThe differential operator acting on the function yieldsDefining the following compact setwhere is a sufficiently small number. In the set , we can choose sufficiently small such that the following conditions hold:Next, we can divide into the following six domains:Clearly, . Next, we will prove that
Case 1. If , we have thatwhich together with implies that
Case 2. If , we haveCombining with , it can be achieved that
Case 3. If , we getwhich follows from .
Case 4. If , it follows thatwhich together with (44) implies that
Case 5. If , we obtainwhich follows from (44). Hence,
Case 6. If , we haveIn view of (44), we arrive atTherefore, we have proof thatThus, condition 3 in Lemma 1 has been satisfied, and system (3) has a unique stationary distribution and ergodicity holds. This completes the proof.
Remark 1. Assume the condition holds. Disease I goes to extinction exponentially with probability one, Theorem 2, and if there is a unique ergodic stationary distribution (⋅,⋅) of system (3), which implies that disease I persists Theorem 3. Then, the number can be considered as a threshold to identifying the stochastic extinction and persistence of system (3).
5. Simulations
Numerical solutions of stochastic differential equations are very important in the study of real examples of epidemic. In this section, we present some numerical results to illustrate the theoretical one. For numerical simulations of the SDEMS model (3), we use the Euler–Maruyama (EM) method ([19]).
Let be a right-continuous Markov chain taking values on the state space with the generator
Obviously, the Markov chain has a unique stationary distribution . Given a step size , the Markov chain can be simulated by computing the one-step transition probability matrix ([20]), and the transition probability matrix is given by
Figure 1 shows a result of one simulation run of the Markov chain .
Example 1. To illustrate Theorem 2, we choose the parameter values in system (3) as follows:Simple computations result,as a consequence result of Theorem 2. Disease I dies out exponentially with probability one. Figure 2 confirms this.
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Example 2. For this example, we haveBy calculating, we findThen according to Theorem 3, the solution (S(t), I(t), and R(t)) of system (3) with any initial value (S(0), I(0), and R(0)) = (0.7, 0.6, and 0) has a unique stationary distribution, and it has the ergodic property, that is, the epidemic disease is permanent. Figures 3–5 confirm this.
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6. Conclusion
This article discusses the dynamic behavior of a SIRS epidemic model with a regime switching and nonlinear incidence rate. We obtain sufficient conditions for the extinction of system (3) if . We prove the stochastic system (3) under regime switching has a unique stationary distribution which is ergodic and positive recurrent by using the Lyapunov function method. In future works, it is interesting to study the effect of Lévy noise and a color noise (telegraph noise) in the stochastic SIRS epidemic model (2). We will investigate this case in our future works.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.