# Stochastic Hybrid Hepatitis B Epidemic Model with Markovian Switching

Authors: A. El Koufi; N. Seshagiri Rao
Journal: Complexity
 (2022)

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## Abstract

In this paper, we consider a stochastic epidemic model under regime-switching. We show that our model is well-posed by proving the existence of solution, positivity, and uniqueness. Furthermore, we investigate sufficient conditions for the existence of ergodic stationary distribution to the model, which shows that the hepatitis B epidemic will persist. In addition, we establish sufficient conditions for the extinction of the hepatitis B epidemic. We perform numerical simulations to confirm and illustrate the theoretical results.


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## Body

## 1. Introduction

The research of new approaches and techniques in epidemiology has taken the attention of many scholars [1–4] in order to give a theoretical base that can be used to support the government to put programs to control the propagation of epidemics in the population. Likewise, the research in areas of mathematic epidemiology has perceived an immense development and proved their efficiency in solving some real medical problems. Generally, there are two approaches to formulating mathematical epidemic models: (i) deterministic based on ordinary differential equations, fractional, or partial [5–7]; (ii) stochastic founded on stochastic differential equations and their generalization [8–11]. The deterministic epidemic model has some limitations in predicting the future dynamics perfectly, and the stochastic one can show it [12]. In the literature review, many authors are involved in the formulation of the mathematical model of transmission of hepatitis B epidemic (see [13–17]) and provided some significant analysis results in order to understand the mechanism of the translation of this type of epidemic and proposing appropriate solutions to controlling their propagation.For a long time, the problem of infectious diseases presents a tall threat to the people in the world. In addition, a significant number of people in the world die due to infectious diseases. Among these dangerous diseases, we find hepatitis B (which causes liver inflammation from the hepatitis B virus infection [18]). Additionally, the propagation of hepatitis B disease is compatible with the stochastic propriety since the contact between the individuals is not predictable. Generally, the studied populations can be divided into four classes (see [19]), namely, Susceptible St, acute Infected I1t, chronically Infected I2t, and Recovered Rt. In this paper, we are interested in the following mathematic hepatitis B epidemic model given by the stochastic system:(1)dSt=A−βStI1t−ν+μ0Stdt+σ1StdW1t,dI1t=βStI1t−γ1+μ0+αI1tdt+σ2I1tdW2t,dI2t=αI1t−γ2+μ0+μ1I2tdt+σ3I2tdW3t,dRt=νSt+γ2I2t−μ0Rt+γ1I1tdt+σ4RtdW4t,where all parameters of the system are supposed to be nonnegative. A is the birth recruitment rate to the susceptible population, ν is the vaccination rate of hepatitis B virus, μ0 is the natural death rate, and μ1 is the death rate due to hepatitis B virus. The parameter β is the disease transmission rate between the infected and susceptible individuals. Therefore, the acutely infected individuals are recovered with the rate γ1 and chronically infected individuals are recovered with γ2. However, α represents the rate of the acute class going to the chronic class. Witi=1,2,3,4 are mutually independent standard Brownian motions defined on a complete probability space Ω,ℱ,ℱttt≥0,ℙ be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all ℙ-null sets), and σii=1,2,3,4 is the intensity of noise. The above model describes the fact that the epidemic model is often subject to random noise [9, 20], which conduit to represent the effect of environmental fluctuation on the parameters of the system, which are not absolute constants but fluctuate randomly around some average values due to continuous fluctuation in the environment. Furthermore, many approaches to formulate the stochastic epidemic model existed [21–24]. Therefore, in model (1), we have assumed that the white noise is directly proportional to the variable. Many works examined the effect of white noise in the epidemic model. Liu et al. in [25] proposed a stochastic model for the heroin epidemic and showed the existence of a unique global positive solution to the system. Also, they showed a sufficient condition for the extinction of the drug users. In addition, they prove the existence of ergodic stationary distribution that implied the persistence of the drug users in the population. In [26], Liu et al. used the parameters perturbation method to express the randomness in an SIRI epidemic model and employed the stochastic Lyapunov technics to show conditions for the extinction and persistence of disease in the system. Wang et al. [27] supposed that the stochastic perturbation is around the positive equilibrium states. They showed that the endemic equilibrium is stochastically asymptotically stable in the large.Frequently, in the natural population, the system switches between two or more environmental regimes that change in terms of factors [28]. Thus, the hepatitis B epidemic models need perturbing by colored noise to allow the system to switch between two or more environmental regimes [23, 29]. The switching between the environmental regimes is memoryless, and the waiting time for the next switch follows the exponential distribution [30, 31]. Hence, the regime-switching can be modeled by a continuous-time Markov chain rtt≥0 with values in a finite-state space L=1,2,…,m. Many researchers have added the Markov chain process in their models to express the change of regimes. For example, Liu and Wang (see Refs. [32]), have studied a stochastic single-species model with Markovian switching. Other interesting papers on the epidemic of Markovian switches models are listed in references [23, 29, 33–37]. Then, in this paper, we consider the hepatitis B epidemic model with both white and color noises disturbances presented by the following hybrid stochastic differential equation system:(2)dSt=Art−βrtSI1−νrt+μ0,rtSdt+σ1,rtSdW1t,dI1t=βrtSI1−γ1,rt+μ0,rt+αrtI1dt+σ2,rtI1dW2t,dI2t=αtI1−γ2,rt+μ0,rt+μ1,rtI2dt+σ3,rtI2dW3t,dRt=νrtS+γ2,rtI2−μ0,rtR+γ1,rtI1dt+σ4,rtRdW4t,where rtt>0 is a right-continuous Markov chain defined in a complete probability space Ω,ℱ,ℱtt≥0,ℙ and supposed independent of Brownian motion W.. The Markov chain takes values in a finite-state space L=1,2,…,m with the generator Φ=ϕuv1≤u,v≤m given, for Δt>0, by(3)ℙrt+Δt=v|rt=u=ϕuvΔt+oΔt,ifu≠v,1+ϕuuΔt+oΔtifu=v,with ϕuv is the transition rate from u to v and ϕuv≥0 if u≠v, while(4)ϕuu=−∑u≠vϕuv.Assume that the Markov chainr. is irreducible, independent of the Brownian motion W.. Therefore, there exists a unique stationary distribution π=π1,…,πm of rt such that πΦ=0 and ∑i=1mπi=1,πi>0,i∈L. For any sequence U=U1,…,UmT, let U^=mini∈LUi and Uˇ=maxi∈LUi. Now, we are interested in the following stochastic equation:(5)dξt=Ht,ξt,rtdt+Gt,ξt,rtdWt,(i)
Wt is a d-dimensional standard Wiener process defined on a complete probability space Ω,ℱ,ℱtt≥0,ℙ(ii)
C2ℝd×L;ℝ+ represents the family of all nonnegative functions Y defined on ℝd×L such that they are continuously twice differentiable in ξWe define the operatorLY (see [38]) from ℝd×L to ℝ by(6)LYξ,k=Ytt,ξ,k+Yξt,ξ,kHt,ξ,k+12Gt,ξ,kYξ,ξt,ξ,kGt,ξ,k,+∑i∈LϕkiYt,ξ,i,where(i)
Yξ=∂Y/∂ξ1,…,∂Y/∂ξd and Yξ,ξ=∂2Y/∂ξi∂ξjd×d(ii)
GT is the transposed matrix of GEmploying Itô’s formula [38], we obtain for Y defined on ℝd×L:(7)dYξt,k=LYξt,kdt+Yξξt,kGξt,kdWt.We put the following lemma existing in [28] which we used to prove the existence of ergodic stationary distribution of the solution to equation (17).Lemma 1 (see [28]).
If the following conditions are satisfied,(i)
ϕij>0 for any i≠j,(ii)
For eachk∈L, Dx,k=dijx,k is symmetric and satisfies(8)λζ2≤Dx,kζ,ζ≤λ−1ζ2for allζ∈ℝn,with some constantλ∈0,1 for all x∈ℝn,(iii)
There exists a nonempty open setD with compact closure, satisfying that, for each k∈L, there is a nonnegative function V.,k: Dc⟶ℝ such that V.,k is twice continuously differential and that for some α>0,(9)LV≤−αfor allx,k∈Dc×L,then xt,rt of system (4) is positive recurrent and ergodic. That is to say, there exists a unique stationary distribution μ.,. such that for any Borel measurable function f:ℝn×L⟶ℝ satisfying(10)∑k=1N∫ℝnfx,kμdx,k<∞,and we have(11)ℙlimt⟶∞1t∫0tfXs,rsds=∑k=1N∫ℝnfx,kμdx,k=1.Lemma 2 (see [39]).
Define anm×m matrix(12)B=diagχ1,…,χm−Φ,then B is a nonsingular M-matrix [40]. Therefore, for any vector z∈ℝ+n, the linear equation Bx=z has a unique solution x∈ℝ+n.Now, we consider the following system:(13)dXt=Art−νrt+μ0,rtXdt+σ1,rtXdW1t.Then, we have the following lemma used in the extinction section.Lemma 3 (see [39]).
The solutionXt,rt of system (13) satisfies the following equality:(14)limt⟶∞1t∫0tβrsXsds=∑k∈LπkAkuk,where uk verifies βk=ukνk+μ0,k−∑l=1mϕklul.


## 2. Existence of Unique Positive Solution

The investigation in this work is to study a stochastic hepatitis B epidemic model with Markovian commutation. Our models represent a general case of many works existing in the literature and give an additional degree of realism in modeling the spread of the hepatitis B epidemic. We establish the existence and uniqueness of the positive solution to system (2). We give a threshold value that permits us to determine the extinction and the persistence of hepatitis B disease in (2). We present numerical simulation by using the Milstein Higher Order Method to confirm the theoretical results.Our system (2) describes the dynamics of a biological population, thus investigating the problem of global existence, uniqueness, and positivity of the solution is so necessary. Hence, the following theorem guaranteed this statement for initial value S0,I10,I20,R0,r0 in ℝ+4×L.Theorem 1.
For any initial valueS0,I10,I20,R0,r0∈ℝ+4×L, there is a unique solution St,I1t,I2t,Rt,rt to equation (2) on t≥0 and the solution will remain in ℝ+4 with probability one, namely, St,I1t,I2t,Rt,rt∈ℝ+4×L for all t≥0 almost surely (briefly a.s.).Proof.
Since the coefficients of the stochastic system (2) are locally Lipschitz continuous, for any given initial value S0,I10,I20,R0,r0∈ℝ+4×L, there is a unique local solution St,I1t,I2t,Rt,rt for all t∈0,τe, where τe denotes the explosion time [38]. If we prove that τe=∞ a.s., then this local solution is global. Let n0>0 be sufficiently large such that all component of S0,I10,I20,R0 remains in interval 1/n0,n0. For each n≥n0, define the stopping time as(15)τn=inft∈0,τe:minSt,I1t,I2t,Rt≤1normaxSt,I1t,I2t,Rt≥n.
Obviously,τn is increasing as n tends to ∞. Set limn,∞τn=τ∞, then τ∞≤τe a.s. To prove that the solution is global, we need to confirm that τ∞=∞ a.s. If τ∞<∞ a.s., then there is a pair of constants θ≥0 and ε∈0,1 such that(16)ℙτ∞≤θ≥ε.
Therefore, there is an integern1≥n0 such that(17)ℙτn≤θ≥ε,for alln≥n1.
We consider aC2-function J:ℝ+4⟶ℝ+ defined by(18)JS,I1,I2,R=S−c−clnSc+I1−1−lnI1+I2−1−lnI2+R−1−lnR,where c:=μ^0+α^/βˇ. Applying Itô’s formula to equation (2), we obtain(19)dJS,I1,I2,R=ℒJS,I1,I2,Rdt+σ1,rtS−1dW1t+σ2,rtI1−1dW2t+σ3,rtI2−1dW3t+σ4,rtR−1dW4t,where(20)ℒJS,I1,I2,R=1−cSAk−βkSI1−νk+μ0,kS,+1−1I1βkSI1−γ1,k+μ0,k+αkI1,+1−1I2αkI1−γ2,k+μ0,k+μ1,kI2,+1−1RνkSt+γ2,kI2t−μ0,kRt+γ1,kI1t,+σ1,k+σ2,k+σ3,k+σ4,k2,≤Ak+cβk−μ0,k+αkI1+cνk+μ0,k+γ1,k+μ0,k,+αk+γ2,k+μ0,k+μ1,k+μ0,k+σ1,k+σ2,k+σ3,k+σ4,k2,≤Aˇ+cνˇ+μˇ0+γˇ1+μˇ0+αˇ+γˇ2+μˇ0+μˇ1+μˇ0+σˇ1+σˇ2+σˇ3+σˇ42,≜C˜.
The rest of the proof is similar to [31] therefore is omitted.


## 3. The Existence of Stationary Distribution

This section is sacred to provide sufficient conditions for the existence of a unique ergodic stationary distribution. To establish this statement, we define the following threshold value:(21)ℛsw=∑k=1mπkAkuk∑k=1mπkγ1,k+μ0,k+αk+σ2,k/2,where uk is the solution of the following equation:(22)βk=ukνk+μ0,k−∑l=1mϕklul.To prove that system (2) has a unique ergodic stationary distribution, we must show the conditions (i), (ii), and (iii) in Lemma 1.(i)
Using the assumptionϕij>0 for i≠j in Section 2, we obtain that condition (i) in Lemma 3.1 is satisfied.(ii)
It is easy to show that the diffusion matrix(23)DS,I1,I2,R=diagσ12kS2,σ22kI12,σ32kI22,σ42kR2,of system (2) is positive definitely, and then condition (ii) is verified.It remains to verify the condition (iii) in Lemma 1. For this, we propose the following theorem.Theorem 2.
Letℛsw>1, then for any initial value S0,I10,I20,R0,r0∈ℝ+4×L, the stochastic switched system (2) has a unique ergodic stationary distribution.Proof.
Define aC2-function F^ by(24)F^S,I1,I2,R,k=−MlnI1+ukS+I1−ηk+1κ+1S+I1+I2+Rκ+1+S+I1+I2+R−lnR−lnI2−lnS=−MU1+U2+U3+U4+U5+U6,where(25)U1=−lnI1−ukS+I1+ηk,U2=1κ+1S+I1+I2+Rκ+1,U3=S+I1+I2+R,U4=−lnR,U5=−lnI2,M satisfying the following condition:(26)K2−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1≤−2,and κ∈0,1, K2, uk, ηk will be determined later.
Clearly,(27)liminfk⟶∞,S,I1,I2,R∈ℝ+4/UkF^S,I1,I2,R,k=+∞,where Uk=1,1/k×1,1/k×1,1/k×1,1/k. Furthermore, F^S,I1,I2,R,k is a continuous function on U¯k, and then F^S,I1,I2,R,k has a minimum value point S^,I^1,I^2,R^,k^ in the interior of ℝ+4.
We define aC2-function U:ℝ+4×L⟶ℝ as follows:(28)US,I1,I2,R,k=F^S,I1,I2,R,k−F^S^,I^1,I^2,R^,k^.
By using Itô’s formula and system (2), we obtain(29)ℒU1=−βkS+γ1,k+μ0,k+αk+σ2,k2−ukAk+ukνk+μ0,kS+ukγ1,k+μ0,k+αkI1,−∑l=1mϕklulS−∑l=1mϕklulI1−∑l=1mϕklηl,=γ1,k+μ0,k+αk+σ2,k2−ukAk−∑l=1mϕklηl−βk−ukνk+μ0,k+∑l=1mϕklulS,+ukγ1,k+μ0,k+αk−∑l=1mϕklulI1.
Letuk be the solution of the following equation:(30)βk=ukνk+μ0,k−∑l=1mϕklul,then(31)ℒU1=γ1,k+μ0,k+αk+σ2,k2−ukAk−∑l=1mϕklηl+βk+ukγ1,k+μ0,k−νkI1≜−R¯k−∑l=1mϕklηl+βk+ukγ1,k+μ0,k−νkI1.
Since the generator matrixΦ is irreducible, then for R¯=R¯1,…,R¯mT, there exists η=η1,…,ηmT solution of the following Poisson system (see [41]):(32)Φη=−R¯+∑k=1mπkR¯k11⋮1.
Then, we have(33)ℒU1≤−∑k=1mπkR¯k+βˇ+uˇγˇ1+μˇ0I1=−∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+βˇ+uˇγˇ1+μˇ0I1.
In addition, we can obtain(34)ℒU2=S+I1+I2+RκAk−μ0,kS+I1+I2+R−μ1,kI2+κ2S+I1+I2+Rκ−1σ1,k2S2+σ2,k2I12+σ3,k2I22+σ4,k2R2≤S+I1+I2+RκAˇ−μ^0S+I1+I2+R+κ2S+I1+I2+Rκ+1σˇ12∨σˇ22∨σˇ32∨σˇ42=AˇS+I1+I2+Rκ−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42S+I1+I2+Rκ+1≤K1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42S+I1+I2+Rκ+1≤K1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42Sκ+1+I1κ+1+I2κ+1+Rκ+1,with(35)K1≜supS+I1+I2+R∈ℝ+4AˇS+I1+I2+Rκ−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42S+I1+I2+Rκ+1.
We have(36)ℒU3=Ak−μ0,kS+I1+I2+R−μ1,kI2≤Aˇ,(37)ℒU4=−νkSR−γ2,kI2R−γ1,kI1R+μ0,k+σ4,k22≤−ν^SR−γ^2I2R−γ^1I1R+μˇ0+σˇ422,(38)ℒU5=−αkI1I2+γ2,k+μ0,k+μ1,k+σ3,k22≤−α^I1I2+γˇ2+μˇ0+μˇ1+σˇ322,(39)ℒU6=−AkS+βkI1+νk+μ0,k+σ1,k22≤−A^S+βˇI1+νˇ+μˇ0+σˇ122.
Combining (33) and (34) with (36)–(39) yields(40)ℒU≤−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42Sκ+1+I1κ+1+I2κ+1+Rκ+1−A^S−ν^SR−γ^2I2R−γ^1I1R−α^I1I2+K2,where(41)K2=Aˇ+γˇ2+3μˇ0+μˇ1+νˇ+K1+σˇ122+σˇ322+σˇ422.
Next, we define a compact set(42)D=ε≤S≤1ε,ε≤I1≤1ε,ε′≤I2≤1ε′,ε′≤R≤1ε′,where ε and ε′ are a sufficiently small positive constant satisfying the following conditions:(43)K2+h∗−A^ε<−1,(44)M2βˇ+uˇγˇ1+μˇ0ε≤1,(45)K2+h∗−α^ε≤−1,(46)K2+h∗−γ1^ε≤−1,(47)K2+h∗−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421εκ+1≤−1,(48)K2−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421εκ+1+C≤−1,(49)−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421εκ+1+E≤−1,(50)−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421ε′κ+1+S≤−1,where K2, C, E, and S are positive constants to be determined. We divide ℝ+4,D into the following eight domains:(51)D1=S,I1,I2,R∈ℝ+4,0<S<ε,D2=S,I1,I2,R∈ℝ+4,0<I1<ε,D3=S,I1,I2,R∈ℝ+4,S≥ε,I1≥ε,0<I2<ε′,D4=S,I1,I2,R∈ℝ+4,I1≥ε,0<R<ε′,D5=S,I1,I2,R∈ℝ+4,S>1ε,D6=S,I1,I2,R∈ℝ+4,I1>1ε,D7=S,I1,I2,R∈ℝ+4,I2>1ε′,D8=S,I1,I2,R∈ℝ+4,R>1ε′,with(52)Dc=D1∪D2∪D3∪D4∪D5∪D6∪D7∪D8.
In the next step, we must prove thatℒU<−1 for any S,I1,I2,R,k∈Dc×L. For this, we discuss the following eight cases:Case 1. If S,I1,I2,R,k∈D1×L, for (40) and (43), we can obtain that(53)ℒU≤K2+hI1−A^S≤K2+h∗−A^ε≤−1,where(54)hI1=−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I1κ+1,andh∗=supI1∈0,∞hI1.Case 2. If S,I1,I2,R,k∈D2×L, from (7), (40), and (44), we obtain(55)ℒU≤K2−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1≤K2−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0ε≤−1.Case 3. If S,I1,I2,R,k∈D3×L, choosing ε′=ε2, we have(56)ℒU≤K2+hI1−α^I1I2≤K2+h∗−α^εε′≤K2+h∗−α^ε.According to (46), we obtain(57)ℒU≤K2+h∗−α^ε≤−1.Case 4. If S,I1,I2,R,k∈D4×L, from (40) and (46), we have(58)ℒU≤K2+hI1−γ1^I1R≤K2+h∗−γ1^εε′≤K2+h∗−γ1^ε≤−1.Case 5. If S,I1,I2,R,k∈D5×L, from (40) and (47), we have(59)ℒU≤K2+hI1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42Sκ+1≤K2+h∗−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421εκ+1≤−1.Case 6. If S,I1,I2,R,k∈D5×L, from (40) and (48), we get(60)ℒU≤K2−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I1κ+1=K2−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I1κ+1−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I1κ+1≤K2−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421εκ+1+C≤−1,where(61)C=supI1∈0,∞−M∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1+M2βˇ+uˇγˇ1+μˇ0I1−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I1κ+1.Case 7. If S,I1,I2,R,k∈D7×L, from (40) and (49), we have(62)ℒU≤K2+hI1−12μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I2κ+1≤−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I2κ+1+E≤−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421ε′κ+1+E≤−1,where(63)E=supI2∈0,∞K2+h∗−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42I2κ+1.Case 8. If S,I1,I2,R,k∈D7×L, from (40) and (50) and using the same method in Case 7, we obtain(64)ℒU≤−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ421ε′κ+1+S≤−1,where(65)S=supR∈0,∞K2+h∗−14μ^0−κ2σˇ12∨σˇ22∨σˇ32∨σˇ42Rκ+1.
From the above eight cases, one can conclude that(66)ℒU≤−1,for allS,I1,I2,R,k∈Dc×L.
Thus, condition (iii) in Lemma 1 holds. Therefore, we conclude that system (2) is ergodic and has a unique stationary distribution.Remark 1.
Theorem2 confirms the existence of a unique ergodic stationary distribution to system (2) that represents significant propriety across which we obtain from that the hepatitis B disease will persist in the population.


## 4. Extinction

The principal question of investigating the dynamics of the epidemic model is to give a process to reduce and control the spread of the epidemic in a population. Then, in this section, we investigate the extinction ofI1 of the stochastic system (2).Theorem 3.
LetSt,I1t,I2t,Rt,rt be the solution of system (2) with any initial value S0,I10,I20,R0,r0∈ℝ+4×L. If ℛsw≤1, then the acute infected I1 will die out exponentially with probability one, i.e.,(67)limsupt⟶∞lnI1tt≤∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1a.s.Proof.
Making use of Itô’s formula to the second equation of system (2) leads to(68)dlnI1t=βrtSt−γ1,rt+μ0,rt+αrt+σ2,rt2dt+σ2,rtdW2t.
Integrating both sides of (68) from 0 to t and dividing by t, we get(69)lnI1tt=1t∫0tβrsSs−γ1,rs+μ0,rs+αrs+σ2,rs2ds+1t∫0tσ2,rsdW2s−lnI10t≤1t∫0tβrsXs−γ1,rs+μ0,rs+αrs+σ2,rs2ds+Mtt−lnI10t,where(70)Mt=∫0tσ2,rsdW2s,is a real-valued continuous local martingale, and its quadratic variation is presented by(71)Mt,Mtt=∫0tσ2,rs2ds≤σˇ2t.
Using the large number theorem for local martingales (see Ref. [38]), we obtain(72)limsupt⟶∞Mtt=0a.s.
Taking the superior limit on the both sides of (69), combining with (72) and the ergodic property of Markov chain [38], we obtain(73)limsupt⟶∞lnI1tt≤∑k=1mπkγ1,k+μ0,k+αk+σ2,k2ℛsw−1a.s.
Sinceℛsw≤1, we obtain(74)limt⟶∞I1t=0a.s.Remark 2.
According to Theorem3, we conclude that the large noises can show the extinction of disease in the system.Remark 3.
By Theorems2 and 3, the value of ℛsw can represent the stochastic threshold and determine the persistence and extinction of disease, namely, if ℛsw<1 , then the infected extinct. On the other hand, the disease persist when ℛsw>1.


## 5. Numerical Example

In this section, we simulate our model (2) using Milstein Higher Order Method explained in [42] to validate the analysis results concerning the dynamical of the stochastic hepatitis B model in a switching environment. Firstly, we can write the model as the following discretization systems:(75)Si+1=Si+AriΔt−βriΔtSiI1,i−νriΔt+μ0,riΔtSiΔt+σ1,riΔtSiΔtζ1,i,I1,i+1=I1,i+βriΔtSiI1,i−γ1,riΔt+μ0,riΔt+αriΔtI1,iΔt+σ2,riΔtI1,iΔtζ2,i,I2,i+1=I2,i+αriΔtI1,i−γ2,riΔt+μ0,riΔt+μ1,riΔtI2,iΔt+σ3,riΔtI2,iΔtζ3,i,Ri+1=Ri+νriΔtSi+γ2,riΔtI2,i−μ0,riΔtRi+γ1,riΔtI1,iΔt+σ4,riΔtRiΔtζ4,i,where riΔt,i=0,1,2… is a discrete-time Markov chain. ζj,i,i≥0, j=1,2,3,4, are random numbers with standard normal distribution: N0,1 and Δt are arbitrary time step.Then, we consider the Markov chainrt taking value in state space L=1,2 with the generator as follows:(76)Φ=−112−2.The Markov chainrt has a unique stationary distribution as follows:(77)π=π1,π2=23,13.The path of the Markov chainrt is plotted in Figure 1.Figure 1
Trajectories of Markov chainrt.The parameter values of system (2) are chosen as follows:(78)A1=0.78;β1=0.8;ν1=0.01;μ01=0.6,γ11=0.2,γ21=0.4,α1=0.1,μ11=0.2,σ11=0.15,σ21=0.12,σ31=0.15,σ41=0.2,A2=0.75;β2=0.7,ν2=0.01,μ02=0.6,γ12=0.2,γ22=0.4,α2=0.1,μ12=0.2,σ12=0.2,σ22=0.15,σ32=0.2,σ42=0.14.By computation, we obtain(79)ℛsw=∑k=1mπkAkuk∑k=1mπkγ1,k+μ0,k+αk+σ2,k/2≈1.1130>1.Hence, according to Theorem3, the solution St,I1t,I2t,Rt,rt of system (2) with an initial value 0.5,0.4,0.3,0.2 has a unique stationary distribution with the ergodic property, which means that the disease will be prevailing in the population. Figure 2 confirms this result. In the following, we explore the extinction of the diseases for the stochastic system (2). To proceed, we take the parameters as follows:(80)A1=0.5,β1=0.1,ν1=0.01,μ01=0.5,γ11=0.2,γ21=0.4,α1=0.1,μ11=0.2,σ11=0.4,σ21=0.6,σ31=0.45,σ41=0.4;A2=0.5,β2=0.1,ν2=0.01,μ02=0.5,γ12=0.2,γ22=0.4,α2=0.1,μ12=0.2,σ12=0.4,σ22=0.5,σ32=0.44,σ42=0.4.Figure 2
Numerical simulations of the pathSt, I1t, I2t, and Rt for the stochastic systems (2).We obtain(81)ℛsw=∑k=1mπkAkuk∑k=1mπkγ1,k+μ0,k+αk+σ2,k/2≈0.0530<1,then, according to Theorem 3, the infected I1t will tend to zero exponentially with probability one. Therefore, the stochastic hepatitis B infection process (2) switches between states 1 and 2 and also goes extinct (see Figure 3).Figure 3
Numerical simulations of the pathSt, I1t, I2t, and Rt for the stochastic systems (2).


## 6. Conclusion

In this work, we addressed a stochastic epidemic model characterizing the dynamics of hepatitis B by using hybrid environmental perturbations modeled with two types of noise, namely, the white and colored noises. The objective of this investigation is to study the random noise effect in a population divided into four groups, namely,S Susceptible, I1 acute Infected, I2 chronically Infected, and R recovered. Firstly, the existence of a unique global positive solution with any positive initial value is established. Then, we showed a sufficient condition for the existence of ergodic stationary distribution of the solution for system (2) and conditions for the extinction of the infection. Precisely,(i)
Ifℛsw<1, then the infected I1t will die out exponentially with probability one.(ii)
Ifℛsw>1, then for any initial value S0,I10,I20,R0,r0∈ℝ+4×L, the solution St,I1t,I2t,Rt,rt of system (2) has an ergodic unique stationary distribution.



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