A Stochastic HIV Infection Model with Latent Infection and Antiretroviral Therapy

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.


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 [1][2][3].With the progress of HIV drug treatment, some researchers have investigated the effects of drug therapies on model behaviors [4][5][6][7].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 [10][11][12][13][14][15].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 [16][17][18].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 [20][21][22][23][24][25] have been greatly developed, but the stochastic differential equation model of the virus dynamic model has just been developed.The classic threedimensional stochastic model is studied in the earlier literatures [26][27][28][29], 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.

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  [14] have analyzed the theoretical results of system (1).System (1) always has an uninfected equilibrium  0 ( 0 , 0, 0, 0), where and the basic reproduction number is When R 0 > 1, only one infected equilibrium  (, , , ) exists.Wang et al. have obtained the following theoretical results of system (1): (I) If R 0 < 1, the uninfected equilibrium  0 is globally asymptotically stable.
(II) If R 0 > 1, then the infected equilibrium  is locally asymptotically stable under certain condition, and the uninfected equilibrium  0 is unstable.
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  ∈ [0,   ), 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  0 > 0 be big enough so that (0), (0), (0) and (0) are all on the interval [1/ 0 ,  0 ].For each integer  ≥  0 , the stopping time is defined as follows: and we let inf 0 = ∞ (commonly, 0 = the empty set).It is obvious that s. for all  ≥ 0. (ii) Otherwise, we assume that there exists a pair of constants t > 0 and  ∈ (0, 1) such that P{ ∞ ≤ t} > .So, there exists an integer  1 ≥  0 that makes P{ ∞ ≤ t} ≥  for all  ≥  1 valid.
Define a  2 -function : where  is a positive constant to be determined.Employing Itô's formula [33] to  yields and : R 4 + → R + is defined by Here, we use the notation , and we choose the constant 0 <  ≤ /2, such that  − /2 ≤ 0, so  (, , , ) ≤ . ( The rest of the proof is similar to Theorem 2.2 in literature [34].At the end, we deduce that  ∞ = ∞.This completes the proof.

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 condition and denote Theorem 2. When R  0 > 1, 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  = min (,,,) where and  is a sufficiently large integer, then condition (A1) in Lemma A.1 is satisfied.
By condition (42), it is obvious to get that  ≤ −1 for any (, , , ) ∈  8 .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, is the basic reproduction number R 0 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.
Consider the following auxiliary stochastic differential equation: For the initial value (0) = (0) > 0 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 By the comparison theorem of one-dimensional stochastic differential equation [37], we obtain that Next, we denote a matrix It is easy to obtain that there must be a left eigenvector (V 1 , V 2 , V 3 ) of matrix  corresponding to the value √R 0 [38], such that where where where We have ) .

Numerical Simulations
The numerical simulation is performed to investigate the effect of the white noises on model dynamical behavior.Applying Milstein's higher order method [39], system (5) can be rewritten as the following discrete form:  Figure 2: The virus can be eradicated for the stochastic model ( 5) when the white noises  1 =  2 =  3 = 0.3 and  4 = 1.5, while the virus is prevalent for its corresponding deterministic model.