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In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters ℜ0$$ {\mathfrak{\Re}}_0 $$ (basic reproduction number) and ℜ1$$ {\mathfrak{\Re}}_1 $$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ℜ0≤1$$ {\mathfrak{\Re}}_0\le 1 $$; the immune‐free infected steady state is globally asymptotically stable when ℜ1≤1<ℜ0$$ {\mathfrak{\Re}}_1\le 1<{\mathfrak{\Re}}_0 $$; and the immune‐present steady state is globally asymptotically stable when ℜ0>1,ℜ1>1$$ {\mathfrak{\Re}}_0>1,{\mathfrak{\Re}}_1>1 $$, and φ(I)=I$$ \varphi (I)=I $$. We perform the sensitivity and uncertainly analysis on ℜ0$$ {\mathfrak{\Re}}_0 $$ with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of stochastic human T‐cell leukemia virus type I (HTLV‐I) infection model with cytotoxic T lymphocyte (CTL) immune response is investigated. First, we show that the stochastic model exists as a unique positive global solution originating from the positive initial value. Second, we demonstrate that the stochastic model is stochastically permanent and stochastically ultimately bounded for any positive initial value. Third, we establish sufficient conditions for the existence of ergodic stationary distribution of the stochastic model. Fourth, the threshold R0∗ between extinction and persistence of the virus is obtained. Finally, numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Recent researches show that virus‐to‐cell infection and cell‐to‐cell transmission are two HIV infection modes. In this paper, we propose a delayed HIV infection model including both virus‐to‐cell infection and cell‐to‐cell transmission and cytotoxic T lymphocyte immune response. The time delay describes the phenomenon between viral entry and viral production. We show the nonnegativity and boundedness of solution, obtain the steady states, and prove local asymptotic stability of the steady states. Then the optimal control problem with antiretroviral therapy and pharmacological delay is posed. We establish and analyze two types of objective functions: one is linear control and the other is quadratic control. Numerical simulations have been performed to verify the stability of steady states and show the optimal control strategies and the effects of control on cells concentration by Matlab and Lingo. [ABSTRACT FROM AUTHOR]

In this paper, a periodic stochastic human immunodeficiency virus (HIV) model with distributed delay and cytotoxic T lymphocytes (CTL) immune response is investigated. First, by It ô's formula, we show that the solution with any positive initial value is global and positive. Then, by the stochastic comparison theorem, we obtain the sufficient conditions guaranteeing the existence and global attractivity of infection‐free periodic solution. Furthermore, we discuss the existence of the infective periodic solution by Has'minskii theory. Finally, numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Direct cell‐to‐cell transmission of HIV‐1 is a more efficient means of virus infection than virus‐to‐cell transmission. In this paper, we incorporate both these transmissions into an HIV‐1 virus model with nonlinear general incidence rate, intracellular delay, and cytotoxic T lymphocyte (CTL) immune responses. This model admits three types of equilibria: infection‐free equilibrium, CTL‐inactivated equilibrium, and CTL‐activated equilibrium. By using Lyapunov functionals and LaSalle invariance principle, it is verified that global threshold dynamics of the model can be explicitly described by the basic reproduction numbers. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for HIV-1 infection with antibody, cytotoxic T-lymphocyte immune responses and Beddington-DeAngelis functional response is investigated. The stability of the infection-free and infected steady states is investigated. The basic reproduction number R0 is identified for the proposed system. If R0 < 1, then there is an infection-free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R0 > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]