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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 5175383, 14 pages
https://doi.org/10.1155/2018/5175383
Research Article

A Stochastic HIV Infection Model with Latent Infection and Antiretroviral Therapy

1College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China
2School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China
3School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Correspondence should be addressed to Yan Wang; nc.ude.cpu@naygnaw

Received 2 February 2018; Accepted 12 April 2018; Published 25 June 2018

Academic Editor: Vicenç Méndez

Copyright © 2018 Jun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Recent studies have demonstrated that the latent infection is a major obstacle to the viral elimination in HIV infection process. In this paper, we formulate a stochastic HIV infection model to include both latent infection and combination drug therapies. We derive that the model solution is unique and positive, and the solution is global. By constructing appropriate stochastic Lyapunov functions, the existence of an ergodic stationary distribution is obtained when the critical condition is greater than one. Furthermore, through rigorous analysis and deduction, the extinction of the virus is established under certain conditions. Numerical simulations are performed to show that small intensity of white noises can maintain the existence of a stationary distribution, while large intensity of white noises is beneficial to the extinction of the virus.

1. Introduction

More and more mathematical models have been developed to reflect the dynamics mechanism of HIV virus. The classical HIV virus model is based on three-dimensional ordinary differential equations, which contains the target T-cells population, infected T-cells population, and virions [13]. With the progress of HIV drug treatment, some researchers have investigated the effects of drug therapies on model behaviors [47]. The commonly used highly active antiretroviral therapy (HAART) is combined by reverse transcriptase inhibitors and protease inhibitors. Reverse transcriptase inhibitors can inhibit the activity of reverse transcriptase and prevent the formation of provirus during the virus infection; protease inhibitors can block the virus infection of new target T-cells. Unfortunately, the current drug treatment can not eradicate the virus thoroughly. With the development of cell and molecular biology, it has been discovered that the existence of latent infection is a major obstacle to clear the virus [8, 9].

In recent years, mathematical models with the inclusion of latent infected T-cells have been developed to investigate the model behaviors [1015]. However, all these models are deterministic model, and the effect of stochastic fluctuation factor is not considered. Actually, HIV transcription is an inherent random process and produces strong random fluctuations in the HIV gene products [1618]. Thus, by experimental data, it has been proved that these random factors seriously affect the evolution of HIV virus during the protease inhibitor therapy [16]. Moreover, Weinberger et al. have demonstrated that stochastic fluctuation could play an important role in delaying HIV transactivation and contributing to latency [19]. Therefore, it is necessary to consider stochastic fluctuation in HIV virus model, which will be more appropriate to reflect the virus infection process.

Recently, the stochastic differential equation models of infectious diseases [2025] have been greatly developed, but the stochastic differential equation model of the virus dynamic model has just been developed. The classic three-dimensional stochastic model is studied in the earlier literatures [2629], then the virus stochastic model with target T-cell logistic growth and CTL immune response is also developed in the last two years [30, 31]. However, all these models did not consider the latent infection mechanism. Conway and Coombs have formulated a stochastic model of latent infected T-cells [32], but they neither include the target T-cell logistic growth nor analyze the model dynamics theoretically. In this paper, we will formulate an HIV model with latent infection, healthy T-cell logistic growth, antiretroviral therapy, and random perturbations.

The organization of this paper is as follows. In Section 2, we construct our stochastic model and give the existence and uniqueness of the global positive solution. By constructing suitable stochastic Lyapunov functions, the existence of a unique ergodic stationary distribution is derived in Section 3. In Section 4, the sufficient conditions for the extinction of the virus are obtained, and numerical simulations are also employed to show how the white noises affect the model behaviors in Section 5. We summarize our results and give future work finally.

2. Model and Preliminaries

For deterministic model, Wang et al. [14] have studied the following HIV model with latent infection, T-cell logistic growth, and antiretroviral therapy,Here, , , and represent the concentrations of healthy T-cells, latent infected T-cells, and actively infected T-cells at time , respectively. denote the infectious viral particles at time . Parameter is a constant input rate, is the logistic growth rate, and is the maximum capacity of the number of healthy T-cells in the blood. is the infection rate between healthy T-cells and infectious virus, is the fraction of infections leading to latency, and is the activated rate. Parameters , , , and are the death rates of the healthy T-cells, latently infected T-cells, actively infected T-cells, and infectious virions, respectively. is the burst size of one productively infected T-cell. and denote the drug efficacy of reverse transcriptase inhibitors and protease inhibitors ( and ), respectively.

Wang et al. in literature [14] have analyzed the theoretical results of system (1). System (1) always has an uninfected equilibrium , where and the basic reproduction number isWhen , only one infected equilibrium exists. Wang et al. have obtained the following theoretical results of system (1): (I)If , the uninfected equilibrium is globally asymptotically stable.(II)If , then the infected equilibrium is locally asymptotically stable under certain condition, and the uninfected equilibrium is unstable.

In this paper, considering random fluctuation during HIV infection process [1619], we follow the assumptions of previous literatures [2631] and assume that the parameters ,  ,  , and are perturbed by the white noise type. In other words, ,  ,  , and . Thus, where are independent standard Brownian motions with and denote the intensities of the white noise, . Then, system (1) with white noises can be written as

In the following of this paper, we assume be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets), and are defined on this complete probability space. We also let .

For convenience, we use symbol to represent and use symbol to represent . We further denote the following notations:

Based on the theory of Mao [33] and the article of Mao et al. [34], we show the existence and uniqueness of the solution of system (5) in the following theorem.

Theorem 1. For any initial value in system (5), there exists a unique positive solution and the solution will be contained in with probability one, namely, for all almost surely (a.s.).

Proof. All the coefficients of system (5) are locally Lipschitz continuous, by the theory of stochastic differential equation [33], so we obtain that there exists a unique maximum local solution on , where is the explosion time.
We show the solution is global in the following; namely, we should derive that a.s. Otherwise, it is supposed that there is a finite time so that the solution can not explode to infinity. Let be big enough so that and are all on the interval . For each integer , the stopping time is defined as follows: and we let (commonly, the empty set). It is obvious that increases with . We denote , then a.s.
(i) If is satisfied a.s., then a.s., so a.s. for all .
(ii) Otherwise, we assume that there exists a pair of constants and such that . So, there exists an integer that makes valid.
Define a -function : , where is a positive constant to be determined. Employing It’s formula [33] to yields and : is defined by Here, we use the notation , and we choose the constant , such that , soThe rest of the proof is similar to Theorem 2.2 in literature [34]. At the end, we deduce that . This completes the proof.

3. Stationary Distribution

Our main concern of this paper is to investigate whether the random fluctuations could affect the model dynamic behavior or not. According to the theory of Hasminskii [35] in the Appendix, we prove the existence of a unique ergodic stationary distribution, which means the survival of the virus population in the future.

Define the critical conditionand denote .

Theorem 2. When , system (5) has a unique ergodic stationary distribution .

Proof. According to Lemma A.1 in the Appendix, we divide it into two steps to prove our conclusion. Now, we prove the validity of condition (A1) in Lemma A.1. The diffusion matrix of system (5) is Choose ; we have where and is a sufficiently large integer, then condition (A1) in Lemma A.1 is satisfied.
In the following, we illustrate the establishment of condition (A2) in Lemma A.1 in the Appendix. Denote and is a quadratic function with respect to . Suppose and are the two roots of , then Since , then , andWe define a -function : , and then, Here, we use the equality .
We construct a -function : , where , , , , and are positive constants which will be confirmed later. Applying It’s formula [33] to and using inequality (19) yield LetWe calculate that Hence, where is defined in (12) andDefine a -function : , in the following form: where . Select an appropriate constant to meet the following requirement: where It is obvious to get that where and is a continuous function. So, should have a minimum point in the interior of . Therefore, we formulate the following nonnegative -function : According to It formula,where According to inequalities (24) and (31), we obtain that Next, we define a compact subset which satisfies condition (A2) in Lemma A.1. Define the following bounded closed set: where is a small enough constant. In set , we further select sufficiently small which satisfies the following conditions:where Set is divided into the following eight domains: Obviously, .
Case 1. If , Based on condition (35), it indicates that for any .
Case 2. If , In view of (36), we get for any .
Case 3. If , According to (37), we obtain that for any .
Case 4. If , From condition (38), we deduce that for any .
Case 5. If , In view of (39), we obtain that for any .
Case 6. If , By condition (40), we conclude that for all .
Case 7. If , It derives that for any if condition (41) is satisfied.
Case 8. If , By condition (42), it is obvious to get that for any . Clearly, from (45)–(52), there exists a sufficiently small , such that By Lemma A.1 in the Appendix, we obtain that system (5) is ergodic and has a unique stationary distribution. This completes the proof.

Remark 3. For system (5), if there is no white noise, that is, , then the critical condition is the basic reproduction number of its corresponding deterministic model (1). It indicates that the existence of the stationary distribution of our stochastic model is a generalization of its corresponding deterministic model to the stability of the infected equilibrium.

4. Extinction

Theorem 4. For any initial value , one assumes be the solution of system (5). If , then for almost , the solution of system (5) is where , and the distribution of converges weakly to a measure which has the invariant density where is a constant such that . If , then , and will be eliminated with probability one; namely,

Proof. Firstly, we can obtain that the solution of system (5) is positive for any given positive initial value from Theorem 1. Consider the following auxiliary stochastic differential equation:For the initial value of system (57), denote as its solution. Following the proof process in Theorem 3.1 of article [31], by the law of large numbers theorem [36], it is easy to get that has the ergodic property with the invariant density where is a constant such that . It then follows that By the comparison theorem of one-dimensional stochastic differential equation [37], we obtain thatNext, we denote a matrix It is easy to obtain that there must be a left eigenvector of matrix corresponding to the value [38], such that where .
Define a -function : , where . By applying It’s formula, it yieldswhere We have Moreover, by using (60), we get