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Global stability of HIV infection models with intracellular delays. (English) Zbl 1256.34068
We study the global stability of two mathematical models for human immunodeficiency virus (HIV) infection with intracellular delays. The first model is a 5-dimensional nonlinear differential-delay system that describes the interaction of the HIV with two classes of target cells, CD4${}^{+}$ T cells and macrophages taking into account the saturation infection rate. The second model generalizes the first one by assuming that the infection rate is given by Beddington-DeAngelis functional response. Two time delays are used to describe the time periods between viral entry the two classes of target cells and the production of new virus particles. Lyapunov functionals are constructed and a LaSalle-type theorem for delay differential equation is used to establish the global asymptotic stability of the uninfected and infected steady states of the HIV infection models. We have proven that, if the basic reproduction number ${R}_{0}$ is less than unity, then the uninfected steady state is globally asymptotically stable, and if the infected steady state exists, then it is globally asymptotically stable for all time delays.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K20 Stability theory of functional-differential equations 92C60 Medical epidemiology