Dynamics of a stochastic cell-to-cell HIV-1 model with distributed delay
Introduction
Human immunodeficiency virus-1 (HIV-1) continues to be a major global public health issue and priority. Many scientists have made great effort against HIV-1, and they are still going on. Mathematics and biological researchers also contribute to this by revealing its transmission and dynamics. Recently, many mathematical models have been formulated to describe the immunological response to infection with HIV-1. Most of these models focus on cell-free viral spread in a compartment such as the bloodstream, see for example, Callaway and Perelson [1], Spouge, Shrager and Dimitrov [2], Kirschner, Lenhart and Serbin [3], Kirschner and Webb [[4], [5], [6] ], McLean and Kirkwood [7], McLean and Nowak [8], Müller et al. [9], Nowak and Bangham [10], Nowak and May [[11], [12]], Perelson, Kirschner and De Boer [13], Perelson [14], Perelson and Nelson [15], Wodarz et al. [16], etc. Some most advances in areas of modeling cell-to-cell transmission and of modeling physical processes with distributed time delay [[17], [18], [19], [20] ]. And these models have been used to explain different phenomena. This is because HIV-1 mathematical models can provide insights into the dynamics of viral load in vivo and can play an important role in the development of a better understanding of HIV/AIDS and drug therapies. Especially, by assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus, Culshaw et al. [17] considered a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures where and denote the concentration of healthy cells and infected cells at time , respectively and all of the parameters are positive constants. is the effective reproductive rate of healthy cells (the term is the total reproductive rate for healthy cells minus the death rate for healthy cells ), and so denotes the number of effective reproductive cells per unit time, is the effective carrying capacity of system (1.1), denotes the infection of healthy cells by the infected cells in a well-fixed system, is the fraction of cells surviving the incubation period, is the death rate of the infected cells. It is assumed that the cells, which are productively infectious at time , were infected time units ago, where is distributed according to a probability distribution , called the delay kernel.
Taking the weak kernel function () and letting system (1.1) is equivalent to the following system There are three equilibria in system (1.2): the trivial equilibrium , the healthy equilibrium , and the infected equilibrium provided , where , , . When , the healthy cells predominate and infected cells die exponentially. In this case are unstable, is asymptotically stable. When , remains unstable and is also unstable. In this situation, healthy cells and infected cells co-exist. Furthermore, if , and , then the positive steady state is asymptotically stable, where , , [17].
On the other hand, in the real world, epidemic models are inevitably subject to the environmental noise, which is an important component in an ecosystem (see e.g. [[21], [22]]). Hence the deterministic models have some limitations in predicting the future dynamics of the system accurately. When modeling biological phenomena such as HIV dynamics, different cells and infective virus particles reacting in the same environment can often give different results. Lately, by incorporating the effects of a fluctuating environment, many authors have studied epidemic models with parameter perturbations (see e.g. [[23], [24], [25], [26], [27], [28], [29], [30], [31] ]). For example, Ji and Jiang [23] considered a stochastic HIV-1 infection model with cell-mediated immune response. They established a sufficient condition for the stochastic asymptotic stability in the large of the infection-free equilibrium and gave the conditions for the solution fluctuating around the two infection equilibria (one without CTLs being activated and the other with). Sánchez-Taltavull et al. [24] presented a stochastic model of the dynamics of the HIV-1 infection and studied the effect of the rate of latently infected cell activation on the average extinction time of the infection. Liu [30] analyzed a model of cell-to-cell HIV-1 infection to T cells perturbed by stochastic perturbations. He studied the asymptotic behavior of the solution and he also investigated the existence of ergodic stationary distribution.
There are different approaches to introduce random perturbations into the model, both from a mathematical and biological perspective. In this paper, we assume that the environmental noise is proportional to the variables and . For convenience in mathematics, we also assume that the environmental noise is proportional to (see Remark 4.1). Then the stochastic version corresponding to system (1.2) takes the following form where are mutually independent standard Brownian motions defined on a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets), denote the intensities of white noise, the other parameters in system (1.3) have the same meaning as in system (1.2).
The initial value of system (1.3) is as follows where and are continuous functions on .
Throughout this paper, we let . This paper is organized as follows. In Section 2, we show that there exists a unique global positive solution of system (1.3) with the initial value (1.4). In Section 3, we establish sufficient conditions for extinction of the disease. In Section 4, by constructing a suitable stochastic Lyapunov function and a rectangular set, we verify the existence of an ergodic stationary distribution of system (1.3). Finally, we provide some numerical examples to illustrate theoretical results and give a brief discussion.
Section snippets
Existence and uniqueness of the positive solution
To study the dynamical behavior of an epidemic model, the first concern is whether the solution is global and positive. In this section, motivated by the methods mentioned in [21] and [[32], [33], [34], [35] ], we show that there is a unique global positive solution of system (1.3).
Theorem 2.1 For any initial value (1.4), there is a unique solution
of system (1.3)on
and the solution will remain in
with probability one, that is to say,
for all
Extinction of the disease
In this section, we shall consider the extinction of the disease. For convenience and simplicity in the following analysis, define the following notations. and when is an integrable function on .
Theorem 3.1 Let
be the solution of system (1.3)with any initial value (1.4). If
and
, then
and
Stationary distribution of system (1.3)
When considering epidemic dynamical systems, we are also interested in when the disease will persist and prevail in a population. In the deterministic models, it can be solved by proving that the endemic equilibrium of the corresponding model is a global attractor or is globally asymptotically stable. But for system (1.3), there exists no endemic equilibrium. In this section, based on the theory of Has’minskii [39], we prove that there is a stationary distribution [40], which reveals that the
Discussion and conclusion
In this section, we give some examples to illustrate the obtained theoretical results and give a brief discussion. For the numerical simulation, we use Milstein’s Higher Order Method mentioned in [43] to obtain the following discretization transformation of system (1.3) where the time increment , denote the intensities
Acknowledgments
This work was supported by NSFC of China Grant Nos. 11371085, 11601043, the Fundamental Research Funds for the Central Universities (No. 15CX08011A), China Postdoctoral Science Foundation (No. 2016M590243) and sponsored by Jiangsu Province “333 High-Level Personnel Training Project” (Grant No. BRA2017468), Qing Lan Project of Jiangsu Province of 2016 and 2017.
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