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Global properties of a class of HIV models. (English) Zbl 1197.34073
Summary: We study the global properties of a class of human immunodeficiency virus (HIV) models. The basic model is a 5-dimensional nonlinear ODEs that describes the interaction of the HIV with two target cells, CD4${}^{+}$ T cells and macrophages. An HIV model with exposed state and a model with nonlinear incidence rate are also analyzed. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states. We have proven that if the basic reproduction number ${R}_{0}$ is less than unity, then the uninfected steady state is globally asymptotically stable. If ${R}_{0}>1$ (or if the infected steady state exists), then the infected steady state is globally asymptotically stable. In a control system framework, we have shown that the HIV model incorporating the effect of Highly Active AntiRetroviral Therapy (HAART) is globally asymptotically controllable to the uninfected steady state.
##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 92D30 Epidemiology 92C60 Medical epidemiology 34D05 Asymptotic stability of ODE 34D20 Stability of ODE 34H05 ODE in connection with control problems