Original articles
Virus dynamic behavior of a stochastic HIV/AIDS infection model including two kinds of target cell infections and CTL immune responses

https://doi.org/10.1016/j.matcom.2021.05.009Get rights and content

Abstract

This study investigated the impact of white noise on the HIV/AIDS model with a cytotoxic T lymphocyte (CTL) immune response. The model introduced the interactions between the virus and two kinds of target cells, CD4+ T cells and macrophages. It was theoretically proved that the solution of the stochastic model is positive and global, as well as the existence of an ergodic stationary distribution. The sufficient conditions were established for viral eradication. By comparing these new results to those of a deterministic model, it is determined that white noise can promote the extinction of the virus. Theoretical results have been verified by numerical simulation of several examples.

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 X(t)=(x1(t),y1(t),x2(t),y2(t),v(t),z(t))T, with initial value X0=(x1(0),y1(0),x2(0),y2(0),v(0),z(0))T.

Theorem 3.1

For any positive initial value X0R+6, system (2.2) has a unique solution X(t) for t0, and it remains in R+6 with probability one, meaning

Existence of unique and ergodic stationary distribution

We show the existence and uniqueness of an ergodic stationary distribution. We define R1Sμ1α1Nrf1(0)(k1+12σ12)(r+12σ22)(d+12σ52),R2Sμ2α2Nrf2(0)(k2+12σ32)(r+12σ42)(d+12σ52), R0S=R1S+R2S.

Theorem 4.1

Assuming that R0S>1, then the solution X(t) of system (2.2) has a unique ergodic stationary distribution.

The proof is shown in Appendix A.

Remark 2

If σi=0(i=1,2,3,4,5), then R0S=R0=μ1α1Nf1(0)k1d+μ2α2Nf2(0)k2d. 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 X0R+6, if k1>12σ12,k2>12σ32, then for the solution X(t) of system (2.2), we have lim supt1tlnα1μ1k2f1(0)y1(t)+α2μ2k1f2(0)y2(t)+k1rR0v(t)ϱa.s.,where R0 is the basic reproduction number of deterministic model (2.1), defined by (2.3), and ϱ=(rd)(R01)1(R0<1)+(rd)(R01)1(R0>1)+α12μ12σ1(f1(0))2k1dR0k1(2k1σ12)+α22μ22σ3(f2(0))2k2dR0k2(2k2σ32),where 1() 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: x1(k+1)=x1(k)+[μ1k1x1(k)α1x1(k)f1(v(k))]t+σ1x1(k)ψ1kt+σ12x1(k)2t(ψ1k21),y1(k+1)=y1(k)+[α1x1(k)f1(v(k))ry1(k)βh1y1(k)z(k)]t+σ2y1(k)ψ2kt+σ22y1(k)2t(ψ2k21),x2(k+1)=x2(k)+[μ2k2x2(k)α2x2(k)f2(v(k))]t+σ3x2(k)ψ3kt+σ32x2(k)2t(ψ3k21),y2(k+1)=y2(k)+[α2x2(k)f2(v(k))ry2(k)βh2y2(k)z(k)]t+σ4y2(k)ψ4kt+σ42y2(k)2t(ψ4k21),v(k+1)=v(k)+[Nr(y1(k)+y2(

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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