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Global stability and periodic solution of a model for HIV infection of CD4${}^{+}$ T cells. (English) Zbl 1117.92040
Summary: It is well-known that mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles. We consider the classical mathematical model with non-linear infection rate. The global dynamics of this model is rigorously established. We prove that, if the basic reproduction number ${R}_{0}⩽1$, the HIV infection is cleared from the T-cells population; if ${R}_{0}>1$, the HIV infection persists. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulations.
MSC:
 92C60 Medical epidemiology 37N25 Dynamical systems in biology 34D23 Global stability of ODE 34K20 Stability theory of functional-differential equations