In this paper, we propose a simple epidemic model with sub-optimal immunity and an arbitrary recovery rate function to understand the impact of different forms of recovery rate. It shows that a nonlinear recovery rate leads to rich dynamic behaviors, such as multiple endemic equilibria, bistability, periodicity and bifurcations. Furthermore, we are pleasantly surprised to find that between the SIS and SIR models there are very similar dynamics with a special recovery rate function. Therefore, in order to simulate the disease trends more accurately, we need to construct a reasonable recovery rate, which is as important as the reasonable incidence rate.
