It is well known that the 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. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection-free steady state. We also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.

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A differential equation model of HIV infection of CD4 + T -cells with cure rate is studied. We prove that if the basic reproduction number R 0 < 1, the HIV infection is cleared from the T -cell population and the disease dies out; if R 0 > 1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R 0 > 1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.