A virus dynamics model with logistic function, general incidence function, and cure rate is considered. By carrying out mathematical analysis, we show that the infection-free equilibrium is globally asymptotically stable if the basic reproduction number ℛ0≤1. If ℛ0>1, then the infection equilibrium is globally asymptotically stable under some assumptions. Furthermore, we also obtain the conditions for which the model exists an orbitally asymptotically stable periodic solution. Examples are provided to support our analytical conclusions.