# Asymptotic Behavior of a Stochastic Delayed Model for Chronic Hepatitis B Infection

Authors: Bo Wu; Jianwen Jia
Journal: Complexity
 (2020)

---

## Abstract

In this paper, a stochastic delayed model is constructed to describe chronic hepatitis B infection with HBV DNA-containing capsids. At first, the existence and uniqueness of the global positive solution are obtained. Secondly, the sufficient conditions are derived that the solution of the stochastic system fluctuates around the disease-free equilibriumE0 and the endemic equilibrium E∗. In the end, some numerical simulations are implemented to support our analytical results.


---

## Body

## 1. Introduction

Hepatitis B virus (HBV) infection, which is a typical liver disease, has raised great attention all over the world [1]. It is generally divided into acute and chronic. In particular, it is likely to suffer from other diseases such as cirrhosis of the liver for those patients who have been sustained infected by Hepatitis B virus [2, 3]. The essence of HBV infection lies in the transformation of the DNA molecule of HBV [2, 4, 5].The majority of mathematical models whose research objects are classified as common three compartments has been investigated by numerous scholars [6–8]. In order to better explore the mechanism of HBV infection, Manna and Chakrabarty for the first time came up with the model of chronic HBV infection including HBV DNA-containing capsids [9], and their model is given below:(1)dHtdt=m−αHtVt−μHt,dItdt=αHtVt−δIt,dDtdt=ηIt−βDt−δDt,dVtdt=βDt−cVt,where Ht,It,Dt,andVt denote the healthy hepatocytes that are not infected by the viruses, the unhealthy hepatocytes which are infected by the viruses, intracellular HBV DNA-containing capsids, and hepatitis B viruses, respectively. Furthermore, the meaning of each parameter is shown as follows:(i)
m stands for the constant recruitment rate of the uninfected hepatocytes(ii)
µ is the natural death rate of the uninfected hepatocytes(iii)
α denotes the rate that these healthy hepatocytes are infected by the viruses and infected hepatocytes come into being(iv)
δ is the rate of infected hepatocytes that are eliminated and also is the natural death rate for the capsids(v)
η represents the rate of production of intracellular HBV DNA-containing capsids(vi)
β is the rate at which the capsids are exported to the blood, producing the virion(vii)
c is the natural death rate for the virusesThese parameters all are positive constant.In fact, the process that healthy hepatocytes are infected by the viruses and then transformed into the infected hepatocyte population is not instantaneous, so the time delay cannot be ignored. Manna and Chakrabarty [10] considered the following model with delay:(2)dHtdt=m−αHtVt−μHt,dItdt=αHt−τVt−τ−δIt,dDtdt=ηIt−βDt−δDt,dVtdt=βDt−cVt.In system (2) [10], the basic reproduction number is ℜ0=αβηm/cδμβ+δ. If ℜ0≤1, then system (2) has only the disease-free equilibrium E0H0,0,0,0 which is globally asymptotically stable, where H0=m/μ. If ℜ0>1, system (2) has two equilibria: E0H0,0,0,0 and E∗H∗,I∗,D∗,V∗, and E∗ is globally asymptotically stable, where H∗>0,I∗>0,D∗>0,andV∗>0.It is worth pointing out that all biological processes are inevitably affected by numerous unpredictable environmental white noise. Hence, the deterministic models have some limitations in predicting the future dynamics of the system accurately; stochastic models produce more valuable real benefits and can predict the future dynamics of the system accurately than deterministic models, and after one studies a deterministic model, extending the results to the stochastic case becomes a hot issue. To understand the impacts due to such randomness and fluctuations, stochastic differential equation (SDE) approach is widely used in many kinds of branches of applied science; many stochastic models have been proposed and studied, such as in the population ecology [11–16] and in the epidemiology [17–27], as well as in other fields [28–30]. Many valuable and interesting results were obtained.On the basis of the abovementioned works, to make model (2) more reasonable and realistic, including the stochastic perturbation on the natural death rate with white noise, we establish a delayed stochastic model as an extension of system (2) as follows:(3)dHt=m−αHtVt−μHtdt+σ1HtdB1t,dIt=αHt−τVt−τ−δItdt+σ2ItdB2t,dDt=ηIt−βDt−δDtdt+σ3DtdB3t,dVt=βDt−cVtdt+σ4VtdB4t,where B=B1t,B2t,B3t,B4t,t≥0 is a real-valued standard Brownian motion. It is defined on a complete probability space Ω,F,ℙ including a filtration Ftt≥0 according with the general conditions, that is, it is increasing and right continuous nevertheless F0 incorporates all ℙ-null sets. σii=1,2,3,4 represents the intensities of the white noise, and they are positive. All the other parameters have the same meaning as that of system (2).The initial conditions of system (3) are(4)Hθ=ψ1θ,Iθ=ψ2θ,Dθ=ψ3θ,Vθ=ψ4θ,ψ1,ψ2,ψ3,ψ4∈C,where C means the space in which all functions are continuous, which is expressed as C−τ,0;ℝ+4, where ℝ+4=x1,x2,x3,x4∈ℝ4:xi>0,i=1,2,3,4.This paper is organized as follows: in the Section2, it is proved that there is a unique global positive solution of system (3) with initial value (4). The asymptotic behavior of the solution of stochastic system (3) around the equilibrium E0 of deterministic model (2) is discussed in Section 3. In section 4, we show that the solution of the stochastic system (3) oscillates around the infected equilibrium E∗ of deterministic model (2) under certain conditions. Numerical simulations are carried out in Section 5 to illustrate the main theoretical results. A brief discussion is given in Section 6 to conclude this work.


## 2. Existence and Uniqueness of the Global Positive Solution

In this section, we will prove that there is a unique global positive solution of system (3) with initial value (4).Theorem 1.
If we give any initial value (4), then there is a unique positive solution Ht,It,Dt,Vt for all t≥−τ for system (3). Furthermore, the solution will remain in ℝ+4 with probability one. In brief, Ht,It,Dt,Vt∈ℝ+4 for all t≥−τ almost surely (a.s).Proof.
According to the theory of stochastic differential equations, we draw the conclusion that system (3) exists as a unique local solution Ht,It,Dt,Vt on t∈−τ,τe, thereinto τe is called as the explosion time [31]. However, we want to illuminate that there is a global solution for system (3). So, it is necessary for us to prove that τe=∞ a.s. For this purpose, we assume k0≥1 to be large enough. Under the circumstances, Hθ, Iθ, Dθ, and Vθ θ∈−τ,0 all are contained in the interval 1/k0,k0. In the following, we introduce the definition of the stopping time:(5)τk=inft∈0,τe:Ht∉1k,korIt∉1k,korDt∉1k,korVt∉1k,k,and we let inf∅=∞(in general, ∅ expresses the empty set). Obviously, τk is increasing as k⟶∞. At this time, assume τ∞=limk⟶∞τk; hence, we can get that τ∞≤τe a.s. In order to finish the proof, we must prove that τ∞=∞ a.s. If this assertion is false, then there are constants T>0 and ε∈0,1 such that(6)ℙτ∞≤T>ε.
So, there exist an integer≥k0 satisfying the following inequality:(7)ℙτk≤T≥ε,for allk≥k1.
Define aC2− function W:ℝ+4⟶ℝ+ as follows:(8)WH,I,D,V=H−b−blnHb+I−1−lnI+fD+fV−1−lnV+∫t−τtαHζVζdζ,where the positive constants bandf will be determined later.
Using Itô formula toW, we obtain(9)dW=1−bHdH+1−1IdI+fdD+f1−1VdV+12bH2dH2+121I2dI2+12f1V2dV2+αHtVt−αHt−τVt−τdt=1−bHm−αHV−μH+12σ12b+1−1IαHt−τVt−τ−δI+12σ22+fηI−βD−δD+f1−1VβD−cV+12σ42f+αHV−αHt−τVt−τdt+σ1H−bdB1t+σ2I−1dB2t+fσ3DdB3t+fσ4V−1dB4t≔LWdt+σ1H−bdB1t+σ2I−1dB2t+fσ3DdB3t+fσ4V−1dB4t,where(10)LW=m−μH−mbH+αbV+μb−δI−αHt−τVt−τI+δ+fηI−fβ+δD+fβD−fcV−fβDV+fc+12σ12b+12σ22+12σ42f≤m+μb+δ+fc+αb−fcV+fη−δI+fc+12σ12b+12σ22+12σ42f.
Choose the parametersf=δ/η and b=cδ/αη such that αb−fc=0 and fη−δ=0, then(11)LW≤m+μb+δ+fc+12σ12b+12σ22+12σ42f≔K.
The following proof is similar to the method in the literature [20], so it is omitted. The proof is completed.


## 3. Asymptotic Behavior around the Disease-free EquilibriumE0 of Equation (2)

Ifℜ0≤1, then system (2) has only the disease-free equilibrium E0H0,0,0,0 which is globally asymptotically stable. However, system (3) does not exist in the equilibrium. In the following, we establish the sufficient conditions to ensure that the solution of system (3) oscillates around E0 of system (2).Theorem 2.
Assume thatHt,It,Dt,Vt is the solution of system (3) with the initial value (4). If ℜ0≤1 and the following conditions are satisfied,(12)σ12<μ,σ22<δ,σ32<β+δ,σ42<2c−32,then(13)limsupt⟶∞1tE∫0tHζ−H02dζ≤σ12H02μ−σ12,limsupt⟶∞1tE∫0tI2ζdζ≤2σ12H02δ−σ221μ−σ12μ2+δ22δ+σ12+1,limsupt⟶∞1tE∫0tD2ζdζ≤A1λ1,limsupt⟶∞1tE∫0tV2ζdζ≤A2λ2,where λ1,λ2,A1,andA2 are positive constants, and they are defined in the proof.Proof.
SinceE0 is the disease-free equilibrium of system (2), then m=μH0.
On account of system (3), we can obtain(14)dIt+τ=αHV−δIt+τdt+σ2It+τdB2t,dDt+τ=ηIt+τ−β+δDt+τdt+σ3Dt+τdB3t,dVt+τ=βDt+τ−cVt+τdt+σ4Vt+τdB4t.
Letting(15)S11=H−H022.
Using Itô formula, one can obtain that(16)LS11=−μH−H02−αH−H02V−αH0H−H0V+12σ12H2≤−μH−H02−αH0H−H0V+σ12H−H02+σ12H02=−μ−σ12H−H02−αH0H−H0V+σ12H02,where the conclusion that a+b2≤2a2+2b2 for any a,b∈ℝ is employed.
Similarly, setting(17)S12=It+δηDt+δβ+δβηVt+α∫t−τtHζVζdζ,then(18)LS12=αHV−cδβ+δβηV=αH−H0V+αH0V−cδβ+δβηV=αH−H0V+cδβ+δβηℜ0−1V≤αH−H0V.
Now, we define(19)S1=S11+H0S12.
According to (16) and (18), we can calculate that(20)LS1≤−μ−σ12H−H02+σ12H02.
Integrating (20) from 0 to t and then taking the expectation on both sides, by virtue of Theorem 1.5.8 (ii) [32], it yields(21)0≤ES1t≤S10−μ−σ12E∫0tHζ−H02dζ+σ12H02t.
Thus, we have(22)limsupt⟶∞1tE∫0tHζ−H02dζ≤σ12H02μ−σ12.
Similarly, setting(23)S13=H−H0+It+τ22,we have(24)LS13=−μH−H02−δI2t+τ−μ+δH−H0It+τ+12σ12H2+12σ22I2t+τ≤−μH−H02−δI2t+τ+δ2I2t+τ+μ+δ22δH−H02+σ12H−H02+σ12H02+12σ22I2t+τ=μ2+δ22δ+σ12H−H02−12δ−σ22I2t+τ+σ12H02,where we have used the following inequality:(25)−μ+δH−H0It+τ≤δ2I2t+τ+μ+δ22δH−H02.
Define(26)S2=S13+1μ−σ12μ2+δ22δ+σ12S1+12δ−σ22∫tt+τI2ζdζ.
By means of (20) and (24), we obtain(27)LS2≤−12δ−σ22I2t+σ12H021μ−σ12μ2+δ22δ+σ12+1.
Let us take the integral of (27) from 0 to t and then take the expectation on both sides, by Theorem 1.5.8 (ii) [32] yield(28)0≤ES2t≤S20−12δ−σ22E∫0tI2dζ+σ12H021μ−σ12μ2+δ22δ+σ12+1t.
Therefore, we can summarize that(29)limsupt⟶∞1tE∫0tI2dζ≤2σ12H02δ−σ221μ−σ12μ2+δ22δ+σ12+1.
In the following, letting(30)S14=H−H0+It+τ+δ/ηDt+τ22,then we use Itô formula and arrive at(31)LS14=−μH−H02−δβ+δ+μηH−H0Dt+τ−μH−H0It+τ−δβ+δηIt+τDt+τ−δ2β+δη2+δ22η2σ32D2t+τ+12σ12H2+12σ22I2t+τ≤−μH−H02+δ2β+δ4η2D2t+τ+β+δ+μ2β+δH−H02+μ4H−H02+μI2t+τ+δ2β+δ4η2D2t+τ+β+δI2t+τ−δ2β+δη2D2t+τ+σ12H−H02+σ12H02+12σ22I2t+τ+δ22η2σ32D2t+τ=4β+δ2+5μβ+δ+4μ24β+δ+σ12H−H02+μ+β+δ+12σ22I2t+τ−δ22η2β+δ−σ32D2t+τ+σ12H02,where we have applied the following inequality: (32)−δβ+δ+μηH−H0Dt+τ≤δ2β+δ4η2D2t+τ+β+δ+μ2β+δH−H02,−μH−H0It+τ≤μ4H−H02+μI2t+τ,−δβ+δηIt+τDt+τ≤δ2β+δ4η2D2t+τ+β+δI2t+τ,a+b2≤2a2+2b2, for any a,b∈ℝ.
We define(33)S3=S14+1μ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12S1+2δ−σ22μ+β+δ+12σ22×S13+1μ−σ12μ2+δ22δ+σ12S1+δ22η2β+δ−σ32∫tt+τD2ζdζ.
From (20), (24), and (31), we have(34)LS3≤−δ22η2β+δ−σ32D2t+σ12H02μ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12+2σ12H02δ−σ22μ+β+δ+12σ221μ−σ12μ2+δ22δ+σ12+1+σ12H02≔−λ1D2t+A1.
Let us take the integral of (34) from 0 to t and then take the expectation on both sides, next relying on Theorem 1.5.8 (ii) [32], we can attain(35)ES3t−S30≤−λ1E∫0tD2ζdζ+A1t.
Therefore, we have(36)limsupt⟶∞1tE∫0tD2ζdζ≤A1λ1.
Next choose that(37)S15=H−H0+It+τ+δ/ηDt+τ+δβ+δ/βηVt+τ22,then taking advantage of Itô formula and attaining that(38)LS15=−μH−H02−μH−H0It+τ−δμηH−H0Dt+τ−δβ+δμ+cβηH−H0Vt+τ−δcβ+δβηIt+τVt+τ−δ2cβ+δβη2Dt+τVt+τ−δ2cβ+δ2β2η2V2t+τ+12σ12H2+12σ22I2t+τ+δ22η2σ32D2t+τ+δ2β+δ22β2η2σ42V2t+τ≤−μH−H02+μ4H−H02+μI2t+τ+δ2β+δ4η2D2t+τ+μ2β+δH−H02+δ2β+δ24β2η2V2t+τ+μ+c2H−H02+δ2β+δ24β2η2V2t+τ+c2I2t+τ+δ2β+δ24β2η2V2t+τ+δ2c2η2D2t+τ−δ2cβ+δ2β2η2V2t+τ+σ12H−H02+σ12H02+12σ22I2t+τ+δ22η2σ32D2t+τ+δ2β+δ22β2η2σ42V2t+τ=4β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12H−H02+μ+c2+12σ22I2t+τ+δ24η2β+δ+4c2+2σ32D2t+τ−δ2β+δ22β2η22c−32−σ42V2t+τ+σ12H02,in which we have applied the following inequality:(39)−μH−H0It+τ≤μ4H−H02+μI2t+τ,−δμηH−H0Dt+τ≤δ2β+δ4η2D2t+τ+μ2β+δH−H02,−δβ+δμ+cβηH−H0Vt+τ≤δ2β+δ24β2η2V2t+τ+μ+c2H−H02,−δcβ+δβηIt+τVt+τ≤δ2β+δ24β2η2V2t+τ+c2I2t+τ,−δ2cβ+δβη2Dt+τVt+τ≤δ2β+δ24β2η2V2t+τ+δ2c2η2D2t+τ.
Let(40)S4=S15+1μ−σ124β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12S1+2δ−σ22μ+c2+12σ22S13+1μ−σ12μ2+δ22δ+σ12S1+β+δ+4c2+2σ322β+δ−σ32S14+1μ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12S1+2δ−σ22μ+β+δ+12σ22×S13+1μ−σ12μ2+δ22δ+σ12S1+δ2β+δ22β2η22c−32−σ42∫tt+τV2ζdζ.
By means of (20), (24), (31), and (38), we can calculate that(41)LS4≤−δ2β+δ22β2η22c−32−σ42V2t+σ12H02μ−σ124β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12+2σ12H02δ−σ22μ+c2+12σ221μ−σ12μ2+δ22δ+σ12+1+σ12H02β+δ+4c2+2σ322β+δ−2σ32+σ12H02β+δ+4c2+2σ322μ−σ12β+δ−2σ324β+δ2+5μβ+δ+4μ24β+δ+σ12+σ12H02μ+β+δ+1/2σ22β+δ+4c2+2σ32δ−σ22β+δ−σ321μ−σ12μ2+δ22δ+σ12+σ12H02≔−λ2V2t+A2.
In a similar way, we have(42)limsupt⟶∞1tE∫0tV2ζdζ≤A2λ2.
The proof is completed.Remark 1.
For the deterministic systems (1) and (2), when ℜ0≤1, the equilibrium E0 is globally asymptotically stable. This means that the disease will be extinct. However, the stochastic system (3) does not exist in the equilibrium. Therefore, the significance of proving the asymptotic behavior of the solution of the stochastic system (3) around the equilibrium E0 of system (1) is to show that diseases will be extinct.


## 4. Asymptotic Behavior Around the Endemic EquilibriumE∗ of Equation (2)

In the literature [10], if ℜ0>1, the endemic equilibrium E∗ of system (2) is globally asymptotically stable. However, system (3) does not have the endemic equilibrium E∗. In this section, we show that the solution of system (3) oscillates around E∗ of system (2) under certain conditions.Theorem 3.
Assume thatHt,It,Dt,Vt is the solution of system (3) with the initial value (4). If ℜ0>1 and the following conditions are satisfied(43)σ12<μ,σ22<δ2,σ32<β+δ2,σ42<c−34,maxB1m1,B2m2,B3m3,B4m4<dE∗,E0=H∗−H02+I∗2+D∗2+V∗2,then (44)limsupt⟶∞1tE∫0tHζ−H∗2dζ≤B1m1,limsupt⟶∞1tE∫0tIζ−I∗2dζ≤B2m2,limsupt⟶∞1tE∫0tDζ−S∗2dζ≤B3m3,limsupt⟶∞1tE∫0tVζ−x∗2dζ≤B4m4,where miandBi,i=1,2,3,4, are positive constants, and they are defined in the proof.Proof.
Note thatE∗ is the endemic equilibrium of system (2), so(45)m=αH∗V∗+μH∗,αH∗V∗=δI∗,ηI∗=β+δD∗,βD∗=cV∗.
Letting(46)S21=H−H∗−H∗lnHH∗,by use of Itô formula, we arrive at(47)LS21=1−H∗Hm−αHV−μH+12σ12H∗=m−αHV−μH−mH∗H+αH∗V+μH∗+12σ12H∗=μH∗2−HH∗−H∗H+αH∗V∗2−HH∗−H∗H+αH∗V∗VV∗+HH∗−HVH∗V∗−1+12σ12H∗=−μ+αV∗H−H∗2H−αH−H∗V−V∗+12σ12H∗.
Taking(48)S22=I−I∗−I∗lnII∗+αH∗V∗∫t−τtHζVζH∗V∗−lnHζVζH∗V∗−1dζ,then by means of Itô formula, we can obtain(49)LS22=αHt−τVt−τ−δI−αI∗Ht−τVt−τI+δI∗+12σ22I∗+αH∗V∗HVH∗V∗−Ht−τVt−τH∗V∗−lnHVH∗V∗+lnHt−τVt−τH∗V∗=αH∗V∗Ht−τVt−τH∗V∗−II∗−Ht−τI∗H∗IVt−τV∗+1+12σ22I∗+αH∗V∗HVH∗V∗−Ht−τVt−τH∗V∗−lnHVH∗V∗+lnHt−τVt−τH∗V∗≤αH∗V∗Ht−τVt−τH∗V∗−II∗−lnHt−τVt−τI∗H∗V∗I+12σ22I∗+αH∗V∗HVH∗V∗−Ht−τVt−τH∗V∗−lnHVH∗V∗+lnHt−τVt−τH∗V∗=αH∗V∗Ht−τVt−τH∗V∗−II∗−lnI∗I−lnHt−τVt−τH∗V∗+12σ22I∗+αH∗V∗HVH∗V∗−Ht−τVt−τH∗V∗−lnHVH∗V∗+lnHt−τVt−τH∗V∗=αHV−αH∗V∗II∗−αH∗V∗lnHVH∗V∗+αH∗V∗lnII∗+12σ22I∗=αHV−αH∗V∗II∗−αH∗V∗lnHH∗−αH∗V∗lnVV∗+αH∗V∗lnII∗+12σ22I∗.
Defining(50)S23=1ηI∗D−D∗−D∗lnDD∗+1βD∗V−V∗−V∗lnVV∗,we have(51)LS23=1ηI∗ηI−β+δD−ηD∗ID+β+δD∗+12ηI∗σ32D∗+1βD∗βD−cV−βV∗DV+cV∗+12βD∗σ42V∗=II∗−II∗D∗D+1−VV∗−DD∗V∗V+1+12ηI∗σ32D∗+12βD∗σ42V∗≤II∗−VV∗−lnII∗D∗D−lnDD∗V∗V+12ηI∗σ32D∗+12βD∗σ42V∗=II∗−VV∗−lnII∗+lnVV∗+12ηI∗σ32D∗+12βD∗σ42V∗.
Set(52)S1=S22+αH∗V∗S23.
According to (49) and (51), we can calculate that(53)LS1≤αHV−αH∗V∗lnHH∗−αH∗V∗VV∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗=αH∗V∗HH∗−1VV∗−1+αH∗V∗HH∗−1−lnHH∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗≤αH∗V∗HH∗−1VV∗−1+αH∗V∗HH∗+H∗H−2+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗=αH−H∗V−V∗+αV∗H−H∗2H+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗,in the inequality above, we make use of the equality(54)HH∗−1−lnHH∗=HH∗−1+lnH∗H≤HH∗+H∗H−2=H−H∗2HH∗.
Choosing(55)S24=H−H∗22,we can obtain(56)LS24=−μH−H∗2−αH−H∗2V−αH∗H−H∗V−V∗+12σ12H2≤−μ−σ12H−H∗2−αH∗H−H∗V−V∗+σ12H∗2.
Taking(57)S2=αV∗μ+αV∗S21+S1+μμ+αV∗S24.
By virtue of (47), (53), and (56), we obtain(58)LS2≤−μμ−σ12μ+αV∗Ht−H∗2+μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗≔−m1Ht−H∗2+B1.
Let us take the integral of (58) from 0 to t and then take the expectation on both sides, and we derive(59)ES2t−S20≤−m1E∫0tHζ−H∗2dζ+B1t.
Therefore, we have(60)lim supt⟶∞1tE∫0tHζ−H∗2dζ≤B1m1.
Set(61)S31=H−H∗+It+τ−I∗22.
At this point, one obtains(62)LS31=−μH−H∗2−δIt+τ−I∗2−μ+δH−H∗It+τ−I∗+12σ12H2+12σ22I2t+τ≤−μH−H∗2−δIt+τ−I∗2+δ2It+τ−I∗2+μ+δ22δH−H∗2+σ12H−H∗2+σ12H∗2+σ22It+τ−I∗2+σ22I∗2=−12δ−2σ22It+τ−I∗2+μ2+δ22δ+σ12H−H∗2+σ12H∗2+σ22I∗2,where we have applied the following inequality to the abovementioned inequality, that is,(63)−μ+δH−H∗It+τ−I∗≤δ2It+τ−I∗2+μ+δ22δH−H∗2.
Define(64)S3=S31+μ+αV∗μμ−σ12μ2+δ22δ+σ12S2+12δ−2σ22∫tt+τIζ−I∗2dζ.
By means of (58) and (62), we derive(65)LS3≤−12δ−2σ22It−I∗2+σ12H∗2+σ22I∗2+1μ−σ12μ2+δ22δ+σ12σ12H∗2+αV∗2μμ−σ12μ2+δ22δ+σ12σ12H∗+μ+αV∗2μμ−σ12μ2+δ22δ+σ12σ22I∗+αH∗V∗μ+αV∗2ημI∗μ−σ12μ2+δ22δ+σ12σ32D∗+αH∗V∗μ+αV∗2βμD∗μ−σ12μ2+δ22δ+σ12σ42V∗≔−m2It−I∗2+B2.
Let us take the integral of (65) from 0 to t and then take the expectation on both sides; next according to Theorem 1.5.8 (ii) [32], we have(66)ES3t−S30≤−m2E∫0tIζ−I∗2dζ+B2t.
Therefore, we can summarize that(67)lim supt⟶∞1tE∫0tIζ−I∗2dζ≤B2m2.
Writing(68)S32=H−H∗+It+τ−I∗+δ/ηDt+τ−D∗22,we arrive at(69)LS32=−μH−H∗2−δβ+δ+μηH−H∗Dt+τ−D∗−μH−H∗It+τ−I∗−δβ+δηIt+τ−I∗Dt+τ−D∗−δ2β+δη2Dt+τ−D∗2+12σ12H2+12σ22I2t+τ+δ22η2σ32D2t+τ≤−μH−H∗2+δ2β+δ4η2Dt+τ−D∗2+β+δ+μ2β+δH−H∗2+μ4H−H∗2+μIt+τ−I∗2+δ2β+δ4η2Dt+τ−D∗2+β+δIt+τ−I∗2−δ2β+δη2Dt+τ−D∗2+σ12H−H∗2+σ12H∗2+σ22It+τ−I∗2+σ22I∗2+δ2η2σ32Dt+τ−D∗2+δ2η2σ32D∗2=4β+δ2+5μβ+δ+4μ24β+δ+σ12H−H∗2+μ+β+δ+σ22It+τ−I∗2−δ22η2β+δ−2σ32Dt+τ−D∗2+σ12H∗2+σ22I∗2+δ2η2σ32D∗2,where we have applied the following inequality:(70)−δβ+δ+μηH−H∗Dt+τ−D∗≤δ2β+δ4η2Dt+τ−D∗2+β+δ+μ2β+δH−H∗2,−μH−H∗It+τ−I∗≤μ4H−H∗2+μIt+τ−I∗2,−δβ+δηIt+τ−I∗Dt+τ−D∗≤δ2β+δ4η2Dt+τ−D∗2+β+δIt+τ−I∗2.
Define(71)S4=S32+μ+αV∗μμ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12S2+2δ−2σ22μ+β+δ+σ22×S31+μ+αV∗μμ−σ12μ2+δ22δ+σ12S2+δ22η2β+δ−2σ32∫tt+τDζ−D∗2dζ.
According to (58), (62), and (69), we obtain(72)LS4≤−δ22η2β+δ−2σ32Dt−D∗2+σ12H∗2+σ22I∗2+δ2η2σ32D∗2+μ+αV∗μμ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗+2μ+β+δ+σ22δ−2σ22σ12H∗2+σ22I∗2+μ+αV∗μμ−σ12μ2+δ22δ+σ12μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗≔−m3Dt−D∗2+B3.
Integrating (72) from 0 to t and then taking the expectation on both sides, we have(73)ES4t−S40≤−m3E∫0tDζ−D∗2dζ+B3t.
Therefore, we have(74)limsupt⟶∞1tE∫0tDζ−D∗2dζ≤B3m3.
Define(75)S33=H−H∗+It+τ−I∗+δ/ηDt+τ−D∗+δβ+δ/βηVt+τ−V∗22,and we can derive(76)LS33=−μH−H∗2−μH−H∗It+τ−I∗−δμηH−H∗Dt+τ−D∗−δβ+δμ+cβηH−H∗Vt+τ−V∗−δcβ+δβηIt+τ−I∗Vt+τ−V∗−δ2cβ+δβη2Dt+τ−D∗Vt+τ−V∗−δ2cβ+δ2β2η2Vt+τ−V∗2+12σ12H2+12σ22I2t+τ+δ22η2σ32D2t+τ+δ2β+δ22β2η2σ42V2t+τ≤−μH−H∗2+μ4H−H∗2+μIt+τ−I∗2+δ2β+δ4η2Dt+τ−D∗2+μ2β+δH−H∗2+δ2β+δ24β2η2Vt+τ−V∗2+c2It+τ−I∗2+μ+c2H−H∗2+δ2β+δ24β2η2Vt+τ−V∗2+δ2β+δ24β2η2Vt+τ−V∗2+δ2c2η2Dt+τ−D∗2−δ2cβ+δ2β2η2Vt+τ−V∗2+σ12H−H∗2+σ12H∗2+σ22It+τ−I∗2+σ22I∗2+δ2η2σ32Dt+τ−D∗2+δ2η2σ32D∗2+δ2β+δ2β2η2σ42Vt+τ−V∗2+δ2β+δ2β2η2σ42V∗24β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12H−H∗2+μ+c2+σ22It+τ−I∗2+δ24η2β+δ+4c2+σ32Dt+τ−D∗2−δ2β+δ22β2η22c−32−2σ42Vt+τ−V∗2+σ12H∗2+σ22I∗2+δ2η2σ32D∗2+δ2β+δ2β2η2σ42V∗2,in which we have applied the following inequality:(77)−μH−H∗It+τ−I∗≤μ4H−H∗2+μIt+τ−I∗2,−δμηH−H∗Dt+τ−D∗≤δ2β+δ4η2Dt+τ−D∗2+μ2β+δH−H∗2,−δβ+δμ+cβηH−H∗Vt+τ−V∗≤δ2β+δ24β2η2Vt+τ−V∗2+μ+c2H−H∗2,−δcβ+δβηIt+τ−I∗Vt+τ−V∗≤δ2β+δ24β2η2Vt+τ−V∗2+c2It+τ−I∗2,−δ2cβ+δβη2Dt+τ−D∗Vt+τ−V∗≤δ2β+δ24β2η2Vt+τ−V∗2+δ2c2η2Dt+τ−D∗2.
Let(78)S5=S33+μ+αV∗μμ−σ124β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12S2+2δ−2σ22μ+c2+σ22S31+μ+αV∗μμ−σ12μ2+δ22δ+σ12S2+β+δ+4c2+σ322β+δ−2σ32S32+μ+αV∗μμ−σ124β+δ2+5μβ+δ+4μ24β+δ+σ12S2+2δ−2σ22μ+β+δ+σ22×S31+μ+αV∗μμ−σ12μ2+δ22δ+σ12S2+δ2β+δ22β2η22c−32−2σ42∫tt+τVζ−V∗2dζ.
By means of (58), (62), (69), and (76), we obtain(79)LS5≤−δ2β+δ22β2η22c−32−2σ42Vt−V∗2+σ12H∗2+σ22I∗2+δ2η2σ32D∗2+δ2β+δ2β2η2σ42V∗2+μ+αV∗μμ−σ124β+δμ+c2+4μ2−3μβ+δ4β+δ+σ12×μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗+2δ−2σ22μ+c2+σ22σ12H∗2+σ22I∗2+μ+αV∗μμ−σ12μ2+δ22δ+σ12×μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗+β+δ+4c2+σ322β+δ−2σ32σ12H∗2+σ22I∗2+δ2η2σ32D∗2+μ+αV∗μμ−σ12×4β+δ2+5μβ+δ+4μ24β+δ+σ12μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗β+δ+4c2+σ32μ+β+δ+σ22δ−2σ22β+δ−2σ32×σ12H∗2+σ22I∗2+μ+αV∗μμ−σ12μ2+δ22δ+σ12μμ+αV∗σ12H∗2+αV∗2μ+αV∗σ12H∗+12σ22I∗+αH∗V∗2ηI∗σ32D∗+αH∗V∗2βD∗σ42V∗≔−m4Vt−V∗2+B4.
In a similar way, we have(80)lim supt⟶∞1tE∫0tVζ−V∗2dζ≤B4m4.
The proof is completed.Remark 2.
For the deterministic systems (1) and (2), E∗ is globally asymptotically stable when ℜ0>1; this means that the disease will be persistent. However, the stochastic system (3) does not have equilibrium E∗. So, the sense of proving the asymptotic behavior of the solution of the stochastic system (3) around the equilibrium E∗ of system (2) is to illustrate that the disease will be persistent.


## 5. Numerical Simulations

In this section, we will carry out some numerical simulations to demonstrate the theoretical results obtained in this paper.Example 1.
We choose the parameters as follows:(81)m=2.6×107,α=3×10−13,μ=0.01,δ=0.053,η=150,τ=5,β=0.87,c=3.8,σ1=0.03,σ2=0.1,σ3=0.2,σ4=0.1,where the value of parameters m,α,μ,δ,η,β,andc are from the literature [9] and the rest of the parameters are assumed. In addition, we assume that the initial values of system (3) are (82)Hθ=3.412×108,Iθ=1.32×108,Dθ=2.144×1010,Vθ=4.92×109,θ∈−τ,0.
For the deterministic model (2), by calculating, we obtain ℜ0=0.5476<1; therefore, it shows that the infection-free equilibrium E0=2.6×109,0,0,0 is globally asymptotically stable (see Figure 1).
For the stochastic model (3), we have(83)σ12=0.0009<μ=0.01,σ22=0.01<δ=0.053,σ32=0.04<β+δ=0.923,σ42=0.01<2c−32=6.1.
So, the conditions of Theorem2 are satisfied, one can see the asymptotic behavior around the infection-free equilibrium E0 of system (2), that is, the infected hepatocytes I, intracellular HBV DNA-containing capsids D, and hepatitis B viruses V will become extinct almost surely. The results are supported in Figure 1.Figure 1
Numerical simulations of the solutions of systems (2) and (3). It is shown that the solution of (3) fluctuates around E0 of (2).
(a)
(b)
(c)
(d)Example 2.
We choose the parameters as follows:(84)m=0.8,α=0.8,μ=0.6,δ=0.4,η=1.5,β=0.87,c=1.0,σ1=0.03,σ2=0.03,σ3=0.03,σ4=0.03,where the value of parameters m,α,μ,δ,andη are from the literature [20] and β is from the literature [9] and the rest of the parameters are assumed. Assume that the initial values of system (3) are(85)Hθ=0.1,Iθ=1.2,Dθ=1.0,Vθ=0.8,θ∈−τ,0.
For the deterministic model (2), by calculating, we obtain ℜ0=2.74>1; therefore, the endemic equilibrium E∗=0.4866,1.2701,1.5001,1.3051 is globally asymptotically stable (see Figure 2).
For the stochastic model (3), by a simple computation, we have(86)σ12=0.0009<μ=0.8,σ22=0.0009<δ2=0.2,σ32=0.0009<β+δ2=0.635,σ42=0.0009<c−34=0.25,maxA,B,M,N=max0.0740,0.1621,1.1872,2.2392<dE∗,E0=H∗−H02+I∗2+D∗2+V∗2=2.5067.
So, the conditions of Theorem3 are satisfied. In Figure 2, one can see that the asymptotic behavior around the endemic equilibrium E∗ of system (2), that is, the infected hepatocytes I, intracellular HBV DNA-containing capsids D, and hepatitis B viruses V will become persistent almost surely (see Figures 2(b)–2(d)).Figure 2
Numerical simulations of the solutions of systems (2) and (3). It is shown that the solution of the stochastic system oscillates around the endemic equilibrium E∗.
(a)
(b)
(c)
(d)Example 3.
Based on Example2, we choose σ1=1.0,σ2=1.0,σ3=1.0,andσ4=1.0 and other parameters do not change. Here, we have ℜ0=2.74>1, σ12>μ, σ22>δ/2, σ32>β+δ/2, and σ42>c−3/4. From the numerical simulations given in Figure 3, we can see that large noise may result in infected hepatocytes, intracellular HBV DNA-containing capsids, and hepatitis B viruses of (3) become extinct almost surely, although the endemic equilibrium E∗ of system (2) is globally asymptotically stable.Figure 3
Numerical simulations of the solutions of system (3). It is shown that the larger intensity of white noise set may help to eliminate It, Dt, and Vt.
(a)
(b)
(c)
(d)


## 6. Conclusions

This paper discusses a stochastic delayed model for chronic hepatitis B infection with HBV DNA-containing capsids. At first, we illustrate that there exists a unique global positive solution for system (3) with the initial value (4). Then, we obtain sufficient conditions to guarantee that the solution of the stochastic system fluctuates around the disease-free equilibrium E0 and the endemic equilibrium E∗. At last, we carry out the numerical simulation in order to confirm the analytical results. Numerical simulations further reveal that the larger intensity of white noise may help to eliminate the infected hepatocytes, intracellular HBV DNA-containing capsids, and hepatitis B viruses (see Figure 3), and we leave these cases as our future work.



---

*Source: 1875475-2020-03-19.xml*