Abstract: The specific immune response plays a very important role in controlling the viral infection within host. A chronic virus infection model with bell-shaped proliferation rate of specific immune cells is proposed and investigated in this paper, where the bell-shaped expansion implies that the proliferation rate of immune cells could decrease when the virus load is sufficiently large. The impairment of virus on immune response is also incorporated in the model. For the model, the net reproduction number of the immune response with virus impairment is found, and the local dynamical behaviors are demonstrated completely. In order to determine the global dynamics, the center manifold theory is applied for some critical situations, and we also construct the suitable Dulac function to rule out the existence of perio-dic solutions. The obtained results in this paper show that the backward bifurcation may occur under certain conditions, which reflects the dependence of dynamics of the model on the initial conditions. Finally, the numerical simulation also suggests that both eventual monotonicity and sustained oscillation of viral population and immune response are possible for the model.

Key words: chronic virus infection model, immune response, net reproduction number, dynamical behaviors, backward bifurcation