Dynamical Behavior of Stochastic Markov Switching Hepatitis B Epidemic Model with Saturated Incidence Rate

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 signi ﬁ cant criterion R S 0 which relies on environmental noises. By means of Lyapunov function approach, we show that there is a stationary distribution if R S 0 > 1 . 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 su ﬃ cient criteria for extinction of the disease. Finally, we propose a de ﬁ nition of the solution to an impulsive stochastic functional di ﬀ erential equation with Markovian switching (ISFDM).


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][3][4][5]. Recently, Khan et al. [6] investigated a hepatitis B epidemic model with saturated incidence rate: with Sð0Þ > 0, Ið0Þ > 0, and Rð0Þ > 0. In model (1), the birth rate is denoted by Λ. The transmission rate of hepatitis B is given by α, while μ 0 and μ 1 , respectively, demonstrated the natural and disease-induced death rates. Recovery rate is denoted by β, while the vaccination and saturation rates are ν and γ, respectively. According to the theory in [6], model (1) always has the disease-free equilibrium E 0 = ðS 0 , 0, R 0 Þ, where the components are defined as S 0 = Λ/ðμ 0 + νÞ, and R 0 = Λν/ðμ 0 ðμ 0 + νÞÞ. If R 0 < 1, E 0 is globally asymptotically stable. If R 0 > 1, E 0 is unstable and there exists an endemic equilibrium E * = ðS * , I * , R * Þ which is globally asymptotically stable, where R 0 = αΛ/ððμ 0 + νÞðμ 0 + μ 1 + βÞÞ.
In fact, epidemic models are inherently subject to a continuous spectrum of disturbances [7][8][9][10][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][13][14][15][16]. In this respect, some researches on stochastic hepatitis B virus models have been reported [17][18][19]. Particularly, in the epidemic model, the disease transmission rate α 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 α, we consider a stochastic analogue of the deterministic model (1): where B i ðtÞ are independent standard Brownian motions and σ 2 i stand for the intensities of B i ðtÞ, i = 1, 2, 3. ξðtÞ, t ≥ 0, is a right-continuous Markov chain on the complete probability space ðΩ, F, P Þ with values in a finite space M = f1, 2,⋯,Ng (see [21,22]).
It is widely known that the stability of biomathematical model has always been a hot issue in recent years [23][24][25][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.

Existence of a Unique and Ergodic Stationary Distribution
then for any initial value ðSð0Þ, Ið0Þ, Rð0Þ, ξð0ÞÞ ∈ ℝ 3 + × M, model (2) has a unique stationary distribution which is ergodic.
where Q is a constant such that Remark 4. In Theorem 2, we derive R S 0 = R 0 when αðkÞ ≡ α and σ i ðkÞ ≡ 0. This conclusion accords with practice.

Numerical Examples
In this section, we will test our theory conclusion by Milstein's higher order method in [36].

Concluding Remarks
The paper successfully investigates extinction and stationary distribution of a stochastic Markov switching hepatitis B epi-demic model with saturated incidence rate. Besides the effect of Markovian switching on the deterministic SIRS epidemic models [37][38][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.  [41,42]. For i = 1, 2, μ i ðθÞ is a measure on ð−∞, 0, 0 < t 1 < t 2 <⋯, lim k⟶+∞ t k = +∞. The initial condition Y 0 ∈ C g and ζð0Þ = 0, where Y 0 = ϑ = fϑðθÞ: −∞<θ ≤ 0g Step size Δt = 0:001.