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In this paper, a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate is set up. We derive the basic reproductive number R0, the cytotoxic T lymphocytes immune response reproductive number R1, the humoral immune response reproductive number R2, humoral immune response competitive reproductive number R3, and cytotoxic T lymphocytes immune response competitive reproductive number R4, and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the threshold dynamics of an HIV-1 viral infection model with cell-mediated immune responses and direct cell-to-cell transmission mechanism. The model demonstrates global threshold dynamics with respect to the reproductive numbers for viral infection ℜ 0 and for CTL immune response ℜ 1 . The proofs of main results come from suitable uses of analyzing the characteristic equation and constructing Lyapunov functionals. Specifically, if ℜ 0 < 1, the infection-free equilibrium E 0 is locally and globally asymptotically stable, and the viruses are cleared. If ℜ 1 < 1 < ℜ 0 , the CTL-inactivated equilibrium E 1 is locally and globally asymptotically stable, and the infection becomes chronic but without persistent CTLs response. If ℜ 1 > 1, the CTL-activated equilibrium E 2 is locally and globally asymptotically stable, and the infection is chronic with persistent CTLs response. Numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a nonlinear model for viral infection with Cytotoxic T Lymphocyte (CTL) immune response. The model is a generalization of several models presented in the literature by considering more general functions for the: (i) intrinsic growth rate of uninfected cells; (ii) incidence rate of infection; (iii) natural death rate of infected cells; (iv) rate at which the infected cells are killed by CTL cells; (v) production and removal rates of viruses; (vi) activation and natural death rates of CTLs. We derive two threshold parameters R0 (the basic infection reproduction number) and R1 (the CTL immune response activation number) and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. By using suitable Lyapunov functions and LaSalle's invariance principle, we prove the global asymptotic stability of all equilibria of the model. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection and for CTL response such that . It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if , an infected equilibrium without immune response is globally asymptotically stable if and an infected equilibrium with immune response is globally asymptotically stable if . The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if . [Copyright &y& Elsevier]