Abstract
The article researches a stochastic hepatitis B epidemic model with saturated incidence rate, which is perturbed by both white noise and colored noise. Firstly, we obtain a significant criterion which relies on environmental noises. By means of Lyapunov function approach, we show that there is a stationary distribution if . Its condition implies that when white noise is small, in the stochastic model, there exists a stochastic positive equilibrium state without changing the basic properties of its corresponding deterministic model. Secondly, we derive sufficient criteria for extinction of the disease. Finally, we propose a definition of the solution to an impulsive stochastic functional differential equation with Markovian switching (ISFDM).
1. Introduction
Hepatitis B virus is a severe infectious disease that has emerged as one of the greatest threats to human health in the 21st century. An estimated 350 million people worldwide have been infected with hepatitis B virus [1]. The mathematical model to describe hepatitis B virus transmission and its dynamics has been extensively explored, which provides some effective suggestions for further study on the progression and its control [2–5]. Recently, Khan et al. [6] investigated a hepatitis B epidemic model with saturated incidence rate: with , and . In model (1), the birth rate is denoted by . The transmission rate of hepatitis B is given by , while and , respectively, demonstrated the natural and disease-induced death rates. Recovery rate is denoted by , while the vaccination and saturation rates are and , respectively. According to the theory in [6], model (1) always has the disease-free equilibrium , where the components are defined as , and . If , is globally asymptotically stable. If , is unstable and there exists an endemic equilibrium which is globally asymptotically stable, where .
In fact, epidemic models are inherently subject to a continuous spectrum of disturbances [7–11]. Many authors demonstrated that the white noise and colored noise have a great destabilizing influence on the epidemic transmission. Moreover, considering the effect of environment noise on the epidemic model has become a popular trend in controlling the spread of disease [12–16]. In this respect, some researches on stochastic hepatitis B virus models have been reported [17–19]. Particularly, in the epidemic model, the disease transmission rate represents an extremely important coefficient [16, 20]. In this paper, by taking into account the effect of continuous-time Markov chain on the transmission rate , we consider a stochastic analogue of the deterministic model (1): where are independent standard Brownian motions and stand for the intensities of . , is a right-continuous Markov chain on the complete probability space with values in a finite space (see [21, 22]).
It is widely known that the stability of biomathematical model has always been a hot issue in recent years [23–26]. Compared with their corresponding deterministic cases, lots of stochastic models have no traditional positive equilibrium state. Consequently, the research of ergodic stationary distribution of s stochastic biomathematical model has been a research highlight. In addition, model (2) incorporates white noise as well as colored noise possessing important practical significance [27]. The main aim of this article is to prove the existence of stationary distribution for model (2). Above all, to guarantee existence and uniqueness of globally positive solution for model (2), we establish the following conclusion. Since the proof is standard, we omit it here.
Lemma 1. For any initial value , there exists a unique positive solution of model (2) on almost surely (a.s.).
2. Existence of a Unique and Ergodic Stationary Distribution
Theorem 2. If , where then for any initial value , model (2) has a unique stationary distribution which is ergodic.
Proof. In order to prove Theorem 2, we need to validate that the feasibility of (A1), (A2), and (A3) in Lemma 7 in the appendix holds. We have assumed (A1) holds in Section 1. To verify (A3), we need to find a nonnegative -function and a compact set such that for all . Construct a -function where , and , where , and constants , compact set and function will be determined later. Employing Itô’s formula [28–34], we can get Choose ; on the basis of the irreducibility of generator matrix , one can find that for , there exists satisfying the following Poisson system . Let and satisfy Then, where and set . Applying Itô’s formula, one can obtain where Denote where . By using Itô’s formula, we also have Hence, by (7), (9), and (12), we get where . Here, we choose that the positive constant satisfies the following inequality: For arbitrary , define the following bounded closed set: where satisfies the following conditions: where Furthermore, where Case 1. If , we derive that Case 2. If , we have Case 3. If , we compute Case 4. If , we derive Case 5. If , we conclude Case 6. If , we have Then, we can obtain that for a sufficiently small , for any . Therefore, we can verify (A3) in Lemma 7 of the appendix. On the other hand, the diffusion matrix of model (2) is positive definite, which implies that condition (A2) in Lemma 7 holds. This completes the proof.☐
Now, consider the corresponding model (2) without Markov switching:
Define a parameter where
Similar to Theorem 3.1 in [35], it is easy to obtain the following result.
Theorem 3. Let be the solution of model (26). If , for any initial value , then the solution of model (26) satisfies and the distribution of converges weakly to the measure which has the density where is a constant such that .
Remark 4. In Theorem 2, we derive when and . This conclusion accords with practice.
3. Numerical Examples
In this section, we will test our theory conclusion by Milstein’s higher order method in [36].
Example 1. Let the generator of the Markov chain be in which is a right-continuous Markov chain taking value in . By solving the linear equation , we obtain the unique stationary (probability) distribution . Choose parameters , and . Then, . In view of Theorem 2, there is a stationary distribution of model (2), and it is ergodic. Phase portrait of and histograms of are plotted in Figure 1.
Example 2. Select parameters , and . By calculation, , , and . It means that there exists a unique endemic equilibrium of determined model (1), which is globally asymptotically stable. Instead, in view of Theorem 3, we have and the distribution of in model (26) converges weakly to the measure (see Figure 2).
4. Concluding Remarks
The paper successfully investigates extinction and stationary distribution of a stochastic Markov switching hepatitis B epidemic model with saturated incidence rate. Besides the effect of Markovian switching on the deterministic SIRS epidemic models [37–39], pulse vaccination strategy (PVS) has been adopted to control the outbreaks and fastly tackle the spread of disease by wide areas [40]. In order to help future research, we propose the following definition related to SIR model by taking into account Markovian switching, impulse, and infinite delay.
Definition 5. Considering the following impulsive stochastic functional differential equation with Markovian switching(ISFDM), where , represents -value stochastic process, , ,, and . , denotes the regime switching [41, 42]. For , is a measure on , . The initial condition and , where is an -measurable -valued random variable such that which is the family of all -measurable, -valued processes such that . An -value stochastic process defined on is called a solution of Equation (32) with initial condition above when satisfies the following criterion: (i) is -adapted and continuous on and ; and . Here, the interpretations of and can be found in [43]. stands for a -dimension standard Brownian motion(ii)For each and a.s.(iii) satisfies the equivalent integral equation of (32) for almost every and satisfies the impulsive criterion at each with probability one
Remark 6. Liu and Wang [44] give a new definition of a solution of an impulsive stochastic differential equation (ISDE). We propose Definition 5, which generalizes the definition of a solution of ISDE to ISFDM, because time memory and Markovian switching are very important in the fields of infectious disease, biological engineering, chemical engineering, etc.
Appendix
Let be the diffusion process described by the following equation [(31)]: where , and . For each , let be any twice continuously differentiable function; the operator can be defined by
According to theorems in [27], it follows the following lemma which provides a criterion for the ergodic stationary distribution of the solution to model (A1).
Lemma 7 ([22]). If the following conditions are satisfied:
(A1) for any .
(A2) For each is symmetric and satisfies for all , with some constant for all .
(A3) 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 (A1) is positive recurrent and ergodic. That is to say, there exists a unique stationary distribution.
Data Availability
No data were used in this study.
Conflicts of Interest
The author declares that there are no competing interests.
Acknowledgments
This work was supported by grants from the Natural Science Foundation of Shandong Province of China (No. ZR2018MA023) and a Project of Shandong Province Higher Educational Science and Technology Program of China (No. J16LI09).