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

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 [68]. 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 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

These 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 . If , then system (2) has only the disease-free equilibrium which is globally asymptotically stable, where . If , system (2) has two equilibria: and , and is globally asymptotically stable, where .

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 [1116] and in the epidemiology [1727], as well as in other fields [2830]. 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:where is a real-valued standard Brownian motion. It is defined on a complete probability space including a filtration according with the general conditions, that is, it is increasing and right continuous nevertheless incorporates all -null sets. 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) arewhere C means the space in which all functions are continuous, which is expressed as , where .

This 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). The asymptotic behavior of the solution of stochastic system (3) around the equilibrium 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 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 for all for system (3). Furthermore, the solution will remain in with probability one. In brief, for all 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 on , thereinto 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 a.s. For this purpose, we assume to be large enough. Under the circumstances, , , , and all are contained in the interval . In the following, we introduce the definition of the stopping time:and we let (in general, expresses the empty set). Obviously, is increasing as . At this time, assume ; hence, we can get that a.s. In order to finish the proof, we must prove that a.s. If this assertion is false, then there are constants and such thatSo, there exist an integer satisfying the following inequality:Define a function as follows:where the positive constants will be determined later.
Using Itô formula to W, we obtainwhereChoose the parameters and such that and , thenThe 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 Equilibrium of Equation (2)

If , then system (2) has only the disease-free equilibrium 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 of system (2).

Theorem 2. Assume that is the solution of system (3) with the initial value (4). If and the following conditions are satisfied,thenwhere are positive constants, and they are defined in the proof.

Proof. Since is the disease-free equilibrium of system (2), then .
On account of system (3), we can obtainLettingUsing Itô formula, one can obtain thatwhere the conclusion that for any is employed.
Similarly, settingthenNow, we defineAccording to (16) and (18), we can calculate thatIntegrating (20) from 0 to t and then taking the expectation on both sides, by virtue of Theorem 1.5.8 () [32], it yieldsThus, we haveSimilarly, settingwe havewhere we have used the following inequality:DefineBy means of (20) and (24), we obtainLet us take the integral of (27) from 0 to t and then take the expectation on both sides, by Theorem 1.5.8 () [32] yieldTherefore, we can summarize thatIn the following, lettingthen we use Itô formula and arrive atwhere we have applied the following inequality: , for any .
We defineFrom (20), (24), and (31), we haveLet 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 () [32], we can attainTherefore, we haveNext choose thatthen taking advantage of Itô formula and attaining thatin which we have applied the following inequality:LetBy means of (20), (24), (31), and (38), we can calculate thatIn a similar way, we haveThe proof is completed.

Remark 1. For the deterministic systems (1) and (2), when , the equilibrium 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 of system (1) is to show that diseases will be extinct.

4. Asymptotic Behavior Around the Endemic Equilibrium of Equation (2)

In the literature [10], if , the endemic equilibrium of system (2) is globally asymptotically stable. However, system (3) does not have the endemic equilibrium . In this section, we show that the solution of system (3) oscillates around of system (2) under certain conditions.

Theorem 3. Assume that is the solution of system (3) with the initial value (4). If and the following conditions are satisfiedthen where , are positive constants, and they are defined in the proof.

Proof. Note that is the endemic equilibrium of system (2), soLettingby use of Itô formula, we arrive atTakingthen by means of Itô formula, we can obtainDefiningwe haveSetAccording to (49) and (51), we can calculate thatin the inequality above, we make use of the equalityChoosingwe can obtainTakingBy virtue of (47), (53), and (56), we obtainLet us take the integral of (58) from 0 to t and then take the expectation on both sides, and we deriveTherefore, we haveSetAt this point, one obtainswhere we have applied the following inequality to the abovementioned inequality, that is,DefineBy means of (58) and (62), we deriveLet 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 () [32], we haveTherefore, we can summarize thatWritingwe arrive atwhere we have applied the following inequality:DefineAccording to (58), (62), and (69), we obtainIntegrating (72) from 0 to t and then taking the expectation on both sides, we haveTherefore, we haveDefineand we can derivein which we have applied the following inequality:LetBy means of (58), (62), (69), and (76), we obtainIn a similar way, we haveThe proof is completed.

Remark 2. For the deterministic systems (1) and (2), is globally asymptotically stable when ; this means that the disease will be persistent. However, the stochastic system (3) does not have equilibrium . So, the sense of proving the asymptotic behavior of the solution of the stochastic system (3) around the equilibrium 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:where the value of parameters 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 For the deterministic model (2), by calculating, we obtain ; therefore, it shows that the infection-free equilibrium is globally asymptotically stable (see Figure 1).
For the stochastic model (3), we haveSo, the conditions of Theorem 2 are satisfied, one can see the asymptotic behavior around the infection-free equilibrium 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.

Example 2. We choose the parameters as follows:where the value of parameters 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) areFor the deterministic model (2), by calculating, we obtain ; therefore, the endemic equilibrium is globally asymptotically stable (see Figure 2).
For the stochastic model (3), by a simple computation, we haveSo, the conditions of Theorem 3 are satisfied. In Figure 2, one can see that the asymptotic behavior around the endemic equilibrium 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)).

Example 3. Based on Example 2, we choose and other parameters do not change. Here, we have , , , and . 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 of system (2) is globally asymptotically stable.

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 and the endemic equilibrium . 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

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was partially supported by the Natural Science Foundation of Shanxi Province (201801D121011).