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Stability analysis of an HIV/AIDS epidemic model with treatment. (English) Zbl 1162.92035
Summary: An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number ${\Re }_{0}$. If ${\Re }_{0}\le 1$, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if ${\Re }_{0}>1$. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.
##### MSC:
 92D30 Epidemiology 34K60 Qualitative investigation and simulation of models 34D23 Global stability of ODE 34D05 Asymptotic stability of ODE