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

Authors: Chun Lu
Journal: Journal of Function Spaces
 (2022)

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## 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 criterionR0S which relies on environmental noises. By means of Lyapunov function approach, we show that there is a stationary distribution if R0S>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 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).


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## Body

## 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:
(1)dSdt=Λ−αSI1+γI−μ0+νS,dIdt=αSI1+γI−μ0+μ1+βI,dRdt=βI+νS−μ0R,with S0>0,I0>0, and R0>0. In model (1), the birth rate is denoted by Λ. The transmission rate of hepatitis B is given by α, while μ0 and μ1, 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 E0=S0,0,R0, where the components are defined as S0=Λ/μ0+ν, and R0=Λν/μ0μ0+ν. If R0<1, E0 is globally asymptotically stable. If R0>1, E0 is unstable and there exists an endemic equilibrium E∗=S∗,I∗,R∗ which is globally asymptotically stable, where R0=αΛ/μ0+νμ0+μ1+β.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 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):
(2)dS=Λ−αξtSI1+γI−μ0+νSdt+σ1ξtSdB1t,dI=αξtSI1+γI−μ0+μ1+βIdt+σ2ξtIdB2t,dR=βI+νS−μ0Rdt+σ3ξtRdB3t,where Bit are independent standard Brownian motions and σi2 stand for the intensities of Bit,i=1,2,3. ξt,t≥0, is a right-continuous Markov chain on the complete probability space Ω,F,P with values in a finite space M=1,2,⋯,N (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 valueS0,I0,R0,ξ0∈ℝ+3×M, there exists a unique positive solution St,It,Rt,ξt∈ℝ+3×M of model (2) on t≥0 almost surely (a.s.).


## 2. Existence of a Unique and Ergodic Stationary Distribution

Theorem 2.
IfR0S>1, where
(3)R0S=∑k∈MπkαkΛμ0+ν+∑k∈Mπkσ12k/2μ0+μ1+β+∑k∈Mπkσ22k/2,then for any initial value S0,I0,R0,ξ0∈ℝ+3×M, model (2) has a unique stationary distribution which is ergodic.Proof.
In order to prove Theorem2, 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 C2-function VS,I,R,k and a compact set Dε∈ℝ+4 such that LV≤−1 for all S,I,R,k∈ℝ+3\Dε×M. Construct a C2-function
(4)VS,I,R=M−c1lnS−c2lnI+ρk+S+I+Rρ+1−lnS−lnI−lnR=MV1+V2+V3+V4+V5,where V1=−c1lnS−c2lnI+ρk,V2=S+I+Rρ+1,V3=−lnS,V4=−lnI,V5=−lnR, and 0<ρ<2μ0/maxi=1,2,3σ˘i2, where σ˘i=maxk∈Mσik, and constants M,c1,c2, compact set Dε and function ρk will be determined later. Employing Itô’s formula [28–34], we can get
(5)LV1=−c1ΛS+c1αkI1+γI+c1μ0+ν+12σ12k−c2αkS1+γI+c2μ0+μ1+β+12σ22k=+∑l∈Mζklρl−c1ΛS−c2αkS1+γI−1+γI+c1αkI1+γI+c1μ0+ν+12σ12k+c2μ0+μ1+β+12σ22k+1+γI+∑l∈Mζklρl≤−3c1c2αkΛ3+1+c1μ0+ν+12σ12k+c2μ0+μ1+β+12σ22k+γI+c1αkI1+γI+∑l∈Mζklρl.
ChooseM1k=−3c1c2αkΛ3+1+c1μ0+ν+1/2σ12k+c2μ0+μ1+β+1/2σ22k; on the basis of the irreducibility of generator matrix Γ, one can find that for Θ=Θ1,Θ2,⋯,ΘN, there exists ρ=ρ1,ρ2,⋯,ρNT satisfying the following Poisson system Γρ=∑k=1NπkΘk1⟵−Θ. Let c1 and c2 satisfy
(6)c1μ0+ν+∑k∈Mπkσ12k2=c2μ0+μ1+β+∑k∈Mπkσ22k2=∑k∈MπkαkΛμ0+ν+∑k∈Mπkσ12k/2μ0+μ1+β+∑k∈Mπkσ22k/2.
Then,(7)LV1≤−∑k∈MπkαkΛμ0+ν+∑k∈Mπkσ12k/2μ0+μ1+β+∑k∈Mπkσ22k/2+1+γI+c1αkI1+γI≤−R0S−1+γI+c1α˘I=−λ+φI,where
(8)λ=R0S−1,φI=γI+c1α˘I,and set α˘=maxk∈Mαk. Applying Itô’s formula, one can obtain
(9)LV2=ρ+1S+I+RρΛ−μ0S−μ0+μ1I−μ0R+12ρρ+1S+I+Rρ−1σ12kS2+σ22kI2+σ32kR2≤ρ+1S+I+RρΛ−μ0S+I+R+maxi=1,2,3σ˘i2ρ2ρ+1S+I+Rρ+1=Λρ+1S+I+Rρ−ρ+1μ0−ρ2maxi=1,2,3σ˘i2S+I+Rρ+1≤B−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2S+I+Rρ+1≤B−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1,where
(10)B=supS,I,R∈ℝ+3Λρ+1S+I+Rρ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2S+I+Rρ+1<∞.
Denote(11)C=supS,I,R∈ℝ+3θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1,where θ=B+μ0+ν+1/2σ˘12+μ0+μ1+β+1/2σ˘22+μ0+1/2σ˘32. By using Itô’s formula, we also have
(12)LV3=−ΛS+αkI1+γI+μ0+ν+12σ12k,LV4=−αkS1+γI+μ0+μ1+β+12σ22k,LV5=−βIR−νSR+μ0+12σ32k.
Hence, by (7), (9), and (12), we get
(13)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1,where α^=mink∈Mαk. Here, we choose that the positive constant M satisfies the following inequality:
(14)−Mλ+C≤−2.
For arbitraryε>0, define the following bounded closed set:
(15)Dε=ε≤S≤1ε,ε≤I≤1ε,ε2≤R≤1ε2,where ε satisfies the following conditions:
(16)−Λε+K≤−1,−Mλ+Mφε+α˘ε+C≤−1,−βε+K≤−1,−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+D≤−1,−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+E≤−1,−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+F≤−1,where
(17)K=supS,I,R∈ℝ+3MφI+α˘I+C,D=supMφI+α˘I+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Iρ+1+Rρ+1,E=supMφI+α˘I+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Rρ+1,F=supMφI+α˘I+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1.
Furthermore,(18)ℝ+3\Dε=D1∪D2∪D3∪D4∪D5∪D6,where
(19)D1=S,I,R∈ℝ+3,0<S<ε,D2=S,I,R∈ℝ+3,0<I<ε,D3=S,I,R∈ℝ+3,0<R<ε2,S>ε,I>ε,D4=S,I,R∈ℝ+3,S>1ε,D5=S,I,R∈ℝ+3,I>1ε,D6=S,I,R∈ℝ+3,R>1ε.
Case 1. IfS,I,R∈D1, we derive that
(20)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−Λε+K≤−1.
Case 2. IfS,I,R∈D2, we have
(21)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−Mλ+Mφε+α˘ε+C≤−1.
Case 3. IfS,I,R∈D3, we compute
(22)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−βε+K≤−1.
Case 4. IfS,I,R∈D4, we derive
(23)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+D≤−1.
Case 5. IfS,I,R∈D5, we conclude
(24)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+E≤−1.
Case 6. IfS,I,R∈D6, we have
(25)LV≤−Mλ+MφI+α˘I−ΛS−α^S1+γI−βIR−νSR+θ−12ρ+1μ0−ρ2maxi=1,2,3σ˘i2Sρ+1+Iρ+1+Rρ+1≤−12ρ+1μ0−ρ2maxi=1,2,3σ˘i21ερ+1+F≤−1.
Then, we can obtain that for a sufficiently smallε, LV<−1 for any S,I,R∈ℝ+3\Dε. Therefore, we can verify (A3) in Lemma 7 of the appendix. On the other hand, the diffusion matrix Dx,k=diagσ12kS2,σ22kI2,σ32kR2 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:
(26)dS=Λ−αSI1+γI−μ0+νSdt+σ1SdB1t,dI=αSI1+γI−μ0+μ1+βIdt+σ2IdB2t,dR=βI+νS−μ0Rdt+σ3RdB3t.Define a parameter(27)R^0=α∫0∞xπxdxμ0+μ1+β+σ22/2,where
(28)πx=Qx−2−2μ0+ν/σ12σ1−2+2μ0+ν/σ12e−2/σ12Λ/x+μ0+ν,x∈0,+∞.Similar to Theorem 3.1 in [35], it is easy to obtain the following result.Theorem 3.
LetSt,It,Rt be the solution of model (26). If R^0<1, for any initial value S0,I0,R0∈ℝ3, then the solution St,It,Rt of model (26) satisfies
(29)limt⟶+∞It=0a.s.,and the distribution of St converges weakly to the measure which has the density
(30)πx=Qx−2−2μ0+ν/σ12σ1−2+2μ0+ν/σ12e−2/σ12Λ/x+μ0+ν,x∈0,+∞,where Q is a constant such that ∫0∞πxdx=1.Remark 4.
In Theorem2, we derive R0S=R0 when αk≡α and σik≡0. 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 chainζij be
(31)Γ=−12141416−13161414−12,in which ζij is a right-continuous Markov chain taking value in M=1,2,3. By solving the linear equation πΓ=0, we obtain the unique stationary (probability) distribution π=π1,π2,π3=2/7,3/7,2/7. Choose parameters Λ=0.232,γ=0.9,μ0=0.000232,ν=0.02,μ1=0.0000547,β=0.12,α1=0.0013,α2=0.00129,α3=0.00132,σ11=0.01,σ21=0.02,σ31=0.06,σ12=0.011,σ22=0.022,σ32=0.055,σ13=0.009,σ23=0.019, and σ33=0.063. Then, R0S=1.2226>1. In view of Theorem 2, there is a stationary distribution of model (2), and it is ergodic. Phase portrait of St,It,Rt and histograms of St,It,Rt are plotted in Figure 1.Example 2.
Select parametersΛ=0.232,γ=0.9,μ0=0.000232,ν=0.02,μ1=0.0000547,β=0.12,α=0.04,σ1=0.1,σ2=0.08, and σ3=0.05. By calculation, R0=αΛ/μ0+νμ0+μ1+β=3.87>1, ∫0∞xπxdx=1.16, and R^0=0.377<1. 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 limt⟶+∞It=0a.s. and the distribution of St in model (26) converges weakly to the measure πx (see Figure 2).Figure 1
St,It, and Rt have ergodic property. The pictures in (a) are Markovian chain. The pictures in (c) are the density functions of model (2) for k∈M=1,2,3. The initial value S0=0.8,I0=0.7, and R0=1.1. Step size Δt=0.001.Figure 2
The left column reflects the simulation of number variations ofSt,Rt and It in model (26) with the initial value S0=0.8,R0=1.1, and I0=0.7 and the noise intensities given in Example 2. The right column reveals the relevant histogram of density functions of the classes St,Rt, and It. Step size Δt=0.001.


## 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),(32)dYt=F1t,ζt,Yt,∫−∞0Yt+θdμ1θdt+F2t,ζt,Yt,∫−∞0Yt+θdμ2θdBt,t≠tk,k∈N,Ytk+−Ytk=HkYtk,k∈N,where Yt+θ,−∞<θ≤0, represents Cg-value stochastic process, Cg=ψ∈C−∞,0;ℝd:∥ψ∥cg=sup−∞<s≤0eqs∣ψs∣<+∞, gs=e−qs,q>0,∣ψs∣=ψ12s+ψ22s+⋯+ψd2s, and ψ1s,ψ2s,⋯,ψds∈ℝd. Hk>−1, ζt denotes the regime switching [41, 42]. For i=1,2, μiθ is a measure on −∞,0, 0<t1<t2<⋯,limk⟶+∞tk=+∞. The initial condition Y0∈Cg and ζ0=0, where Y0=ϑ=ϑθ:−∞<θ≤0 is an F0-measurable Cg-valued random variable such that ϑ∈M2−∞,0;ℝd which is the family of all F0-measurable, ℝd-valued processes ψt,t∈−∞,0 such that E∫−∞0ψt2dt<+∞. An ℝd-value stochastic process Yt defined on ℝ is called a solution of Equation (32) with initial condition above when Yt satisfies the following criterion:
(i)
Yt is Ft-adapted and continuous on 0,t1 and tk,tk+1,k∈N; F1t,ζt,Yt,∫−∞0Yt+θdμ1θ∈L1ℝ¯+;ℝd and F2t,ζt,Yt,∫−∞0Yt+θdμ2θ∈L2ℝ¯+;ℝd×m. Here, the interpretations of L1ℝ¯+;ℝd and L2ℝ¯+;ℝd×m can be found in [43]. Bt stands for a m-dimension standard Brownian motion(ii)
For eachtk,k∈N,Ytk+=limt⟶tk+Yt and Ytk=Ytk−=limt⟶tk−Yt a.s.(iii)
Yt satisfies the equivalent integral equation of (32) for almost every t∈0,∞\tk and satisfies the impulsive criterion at each t=tk,k∈N with probability oneRemark 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.



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