Abstract
This paper studies the global stability of a general discrete-time viral infection model with virus-to-cell and cell-to-cell transmissions and with humoral immune response. We consider both latently and actively infected cells. The model incorporates three types of intracellular time delays. The production and clearance rates of all compartments as well as incidence rates of infection are modeled by general nonlinear functions. We use the nonstandard finite difference method to discretize the continuous-time model. We show that the solutions of the discrete-time model are positive and ultimately bounded. We derive two threshold parameters, the basic reproduction number R0 and the humoral immune response activation number R1, which completely determine the existence and stability of the model’s equilibria. By using Lyapunov functions, we have proven that if R0≤1, then the virus-free equilibrium Q0 is globally asymptotically stable; if R1≤1< R0, then the persistent infection equilibrium without immune response Q* is globally asymptotically stable; and if R1>1, then the persistent infection equilibrium with immune response Q¯ is globally asymptotically stable. We illustrate our theoretical results by using numerical simulations. The effects of antiretroviral drug therapy and time delay on the virus dynamics are also studied. We have shown that the time delay has a similar effect as the antiretroviral drug therapy.
Highlights
During the past two decades, mathematical modeling and analysis of viral infections have become essential tools to get a better systematic and quantitative understanding of viral processes that are difficult to discern through strictly experimental approaches.1 Nowak and Bangham2 have formulated the basic viral infection model as⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ̇IUX ̇(((tt)t))===κθβXI−((ttδ))UU−((cttX))(−−t)κa,XI((tt)),U(t), (1)where U(t), I(t), and X(t) are the concentrations of susceptible cells, actively infected cells, and free viruses at time t, respectively
These general functions encompass several specific forms commonly used in the virus dynamics literature
We incorporated three types of time delays, in which the first and second delays describe the times between a virus or an infected cell contacting a susceptible cell and the cell becoming latently infected and actively infected, respectively
Summary
During the past two decades, mathematical modeling and analysis of viral infections have become essential tools to get a better systematic and quantitative understanding of viral processes that are difficult to discern through strictly experimental approaches. Nowak and Bangham have formulated the basic viral infection model as. Elaiw and Alshaikh have proposed and investigated a class of discrete-time virus infection models with humoral immune response and with only virus-to-cell transmission. Elaiw and Alshaikh have studied a discrete-time viral infection model with humoral immune response and both virus-to-cell and cell-to-cell transmissions but with specific forms of the incidence rate, production, and clearance/death rates of the viruses and cells. The aim of this paper is to investigate a discrete-time virus dynamics model which includes (i) humoral immune response, (ii) both virus-to-cell and cell-to-cell transmissions, (iii) both latently and actively infected cells, (iv) three types of time delays, (v) antiretroviral drug therapy, and (vi) general forms of the incidence rates of infection as well as the production and removal rate of all compartments. The global stability of the equilibria is established by constructing Lyapunov functions
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