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

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 equilibrium E0 and the endemic equilibrium E∗. In the end, some numerical simulations are implemented to support our analytical results.


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]. e essence of HBV infection lies in the transformation of the DNA molecule of HBV [2,4,5]. e majority of mathematical models whose research objects are classified as common three compartments has been investigated by numerous scholars [6][7][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: where H(t), I(t), D(t), and V(t) 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 viruses ese 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: In system (2) [10], the basic reproduction number is R 0 � (αβηm)/(cδμ(β + δ)). If R 0 ≤ 1, then system (2) has only the disease-free equilibrium E 0 (H 0 , 0, 0, 0) which is globally asymptotically stable, where H 0 � m/μ. If R 0 > 1, system (2) has two equilibria: E 0 (H 0 , 0, 0, 0) and E * (H * , I * , D * , V * ), and E * is globally asymptotically stable, where H * > 0, I * > 0, D * > 0, and V * > 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][12][13][14][15][16] and in the epidemiology [17][18][19][20][21][22][23][24][25][26][27], as well as in other fields [28][29][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: is a real-valued standard Brownian motion. It is defined on a complete probability space (Ω, F, P) including a filtration F t t ≥ 0 according with the general conditions, that is, it is increasing and right continuous nevertheless F 0 incorporates all P-null sets. σ i (i � 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). e initial conditions of system (3) are H(θ) � ψ 1 (θ), where C means the space in which all functions are continuous, which is expressed as is paper is organized as follows: in the Section 2, it is proved that there is a unique global positive solution of system (3) with initial value (4). e asymptotic behavior of the solution of stochastic system (3) around the equilibrium E 0 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.

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).
Proof. According to the theory of stochastic differential equations, we draw the conclusion that system (3) exists as a unique local solution (H(t), thereinto τ e 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 τ e � ∞ a.s. For this purpose, we assume k 0 ≥ 1 to be large enough. Under the circumstances, H(θ), I(θ), D(θ), and V(θ) (θ ∈ [− τ, 0]) all are contained in the interval [1/k 0 , k 0 ]. In the following, we introduce the definition of the stopping time: 2 Complexity and we let inf∅ � ∞(in general, ∅ expresses the empty set). Obviously, τ k is increasing as k ⟶ ∞. At this time, assume τ ∞ � lim k⟶∞ τ k ; hence, we can get that τ ∞ ≤ τ e 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 So, there exist an integer ≥ k 0 satisfying the following inequality: Define a C 2 − function W: R 4 + ⟶ R + as follows: where the positive constants b and f will be determined later.
Using Itô formula to W, we obtain (4). If R 0 ≤ 1 and the following conditions are satisfied,

Theorem 2. Assume that (H(t), I(t), D(t), V(t)) is the solution of system (3) with the initial value
then lim sup where λ 1 , λ 2 , A 1 , and A 2 are positive constants, and they are defined in the proof.
Proof. Since E 0 is the disease-free equilibrium of system (2), then m � μH 0 . On account of system (3), we can obtain Letting Using Itô formula, one can obtain that where the conclusion that (a + b) 2 ≤ 2a 2 + 2b 2 for any a, b ∈ R is employed. Similarly, setting then Now, we define According to (16) and (18), we can calculate that Integrating (20) from 0 to t and then taking the expectation on both sides, by virtue of eorem 1.5.8 (ii) [32], it yields us, we have lim sup Similarly, setting we have where we have used the following inequality: Define By means of (20) and (24), we obtain Let us take the integral of (27) from 0 to t and then take the expectation on both sides, by eorem 1.5.8 (ii) [32] yield erefore, we can summarize that lim sup In the following, letting then we use Itô formula and arrive at where we have applied the following inequality: (a + b) 2 ≤ 2a 2 + 2b 2 , for any a, b ∈ R.

Complexity 5
We define From (20), (24), and (31), we have Let us take the integral of (34) from 0 to t and then take the expectation on both sides, next relying on eorem 1.5.8 (ii) [32], we can attain erefore, we have lim sup Next choose that then taking advantage of Itô formula and attaining that 6 Complexity in which we have applied the following inequality: Let By means of (20), (24), (31), and (38), we can calculate that Complexity 7 In a similar way, we have lim sup e proof is completed.
□ Remark 1. For the deterministic systems (1) and (2), when R 0 ≤ 1, the equilibrium E 0 is globally asymptotically stable. is means that the disease will be extinct. However, the stochastic system (3) does not exist in the equilibrium. erefore, the significance of proving the asymptotic behavior of the solution of the stochastic system (3) around the equilibrium E 0 of system (1) is to show that diseases will be extinct.

Asymptotic Behavior Around the Endemic Equilibrium E * of Equation (2)
In the literature [10], if R 0 > 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. (4). If R 0 > 1 and the following conditions are satisfied

Theorem 3. Assume that (H(t), I(t), D(t), V(t)) is the solution of system (3) with the initial value
then lim sup where m i and B i , i � 1, 2, 3, 4, are positive constants, and they are defined in the proof.
Proof. Note that E * is the endemic equilibrium of system (2), so Letting by use of Itô formula, we arrive at 8 Complexity Taking then by means of Itô formula, we can obtain Defining we have

Complexity
Set According to (49) and (51), we can calculate that in the inequality above, we make use of the equality we can obtain Taking By virtue of (47), (53), and (56), we obtain Let us take the integral of (58) from 0 to t and then take the expectation on both sides, and we derive At this point, one obtains 10 Complexity where we have applied the following inequality to the abovementioned inequality, that is, Define By means of (58) and (62), we derive Let us take the integral of (65) from 0 to t and then take the expectation on both sides; next according to eorem 1.5.8 (ii) [32], we have erefore, we can summarize that lim sup Writing where the value of parameters m, α, μ, δ, and η 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 For the deterministic model (2), by calculating, we obtain R 0 � 2.74 > 1; therefore, the endemic equilibrium E * � (0.4866, 1.2701, 1.5001, 1.3051) is globally asymptotically stable (see Figure 2).    (2) and (3). It is shown that the solution of the stochastic system oscillates around the endemic equilibrium E * .

Conclusions
is 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). en, we obtain sufficient conditions to guarantee that the solution of the stochastic system fluctuates around the disease-free equilibrium E 0 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.