Original articlesVirus dynamic behavior of a stochastic HIV/AIDS infection model including two kinds of target cell infections and CTL immune responses
Introduction
According to the World Health Organization (WHO), by the end of 2017, approximately 36.9 million people worldwide were living with HIV, and a total of 940,000 people died of HIV-related diseases that year. The African continent was most affected, and the country of South Africa was most seriously impacted, with an estimated 7.03 million people living with HIV in 2016 [29]. Although some people can receive effective treatment, medical aid still remains a dream for millions. Moreover, the growth of HIV drug resistance poses a huge challenge to the efficacy of antibiotics [5], [13], [18]. In practice, many factors prevent the elimination of AIDS.
Computer models based on ordinary or stochastic differential equations are useful for making forward predictions, facilitating individual treatments, and analyzing disease transmission trends between communities, regions, and the world [30]. Many models can effectively perform these functions.
Numerous HIV/AIDS models have been developed in recent years. Many simply describe the interactions between the virus and CD4+ T cells [1], [12], [41], [43]. These models are insufficient compared with models that include the interactions between the virus and both CD4+ T cells and macrophages. During the first week of antiretroviral treatment, after the rapid first phase of decay, plasma virus levels decline rather slowly [9], [10], [32], [38]. This second phase of viral decay is due to the turnover of a longer-lived virus reservoir of infected cells. Thus, models with two target cells are more accurate and reasonable [6], [8], [34]. Our model includes the interactions between the virus and two target cells, CD4+ T cells and macrophages. Cytotoxic T lymphocytes (CTLs) play an important role in the antiviral mechanism and regulation of the proviral load [3], [15], [16]. The CTL immune response has been considered extensively in virus–immune response mathematical models of HIV (or HTLV-1) infections [1], [4], [12], [19], [20], [36], [41], [42]. Our mathematical model takes into account the CTL immune response with a linear incidence rate.
Epidemic systems are inevitably subject to environmental noise [7], [39], [40], which has been introduced in human populations and biological models [23], [25], [27], [31], [46]. In addition, the white noise can be used to simplify the modeling and analysis. It can be thought of as the derivative of Brownian motion that is a continuous and stationary stochastic process [26]. White noise may indirectly influence model parameters, such as the generation rates and mortality rates of different kinds of cells in vivo. Thus, it is important to construct a model that contains a random term.
The solution of stochastic differential equations is proven to be positive and global in Section 3, and the existence of an ergodic stationary distribution of the solution is proven by suitable stochastic Lyapunov functions in Section 4. Sufficient conditions are established for the extinction of the disease (the virus disappears in vivo) in Section 5, and the theoretical results are verified by numerical simulations in Section 6. Section 7 includes the conclusions, discussion of results, study’s deficiencies, and future work.
Section snippets
Mathematical model
Azoz et al. [2] have performed a deterministic HIV-1 infection model, which reveals the interactions between HIV and two co-circulating populations of target cells, CD4+ T cells and macrophages. They have considered the distributed invasion and production delays of the system, whereas this study examines the influence of environmental stochastic disturbances on the model instead of distributed invasion and production delays.
Many researchers have studied the invasion of HIV/AIDS and the CTL
Existence and uniqueness of a positive solution
The existence of a global positive solution is required before we study the dynamics of the stochastic epidemic model (2.2). We show that system (2.2) has a unique and global positive solution with any initial value. We denote the solution of system (2.2) as , with initial value .
Theorem 3.1 For any positive initial value , system (2.2) has a unique solution for , and it remains in with probability one, meaning
Existence of unique and ergodic stationary distribution
We show the existence and uniqueness of an ergodic stationary distribution. We define
Theorem 4.1 Assuming that , then the solution of system (2.2) has a unique ergodic stationary distribution.
The proof is shown in Appendix A.
Remark 2 If , then . Thus, compared to (1) in Remark 1, the result of deterministic model (2.1) is included in the result of
Extinction
We establish sufficient conditions for the extinction of a disease, which are meaningful for its treatment.
Theorem 5.1 For any initial value , if , then for the solution of system (2.2), we have where is the basic reproduction number of deterministic model (2.1), defined by (2.3), and where is the
Examples and numerical simulations
We introduce Milstein’s higher-order method [14] to illustrate our theoretical results. The discretized equations of model (2.2) are as follows:
Biological applications
We proposed a stochastic HIV/AIDS model with the CTL immune response, including interactions between the virus and two kinds of target cells, CD4+ T cells and macrophages. After proving that the solution of the stochastic model was positive and global, we obtained sufficient conditions for the existence of an ergodic stationary distribution and disease extinction.
We examined the interactions between the human immunodeficiency virus and two kinds of target cells, CD4+ T cells and macrophages,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11871473).
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