We analyze a non-smooth epidemic model with nonlinear incidence rate and resource constraints, which defines a vaccination program with vaccination rate proportional to the number of susceptible individuals when this number is below the threshold level and constant otherwise. To better understand the impact of this non-smooth vaccination policy, we provide a comprehensive qualitative analysis of global dynamics for the whole parameter space. As the threshold value varies, the target model admits multistability of three regular equilibria, bistability of two regular equilibria, that of one disease-free equilibrium and one generalized endemic equilibria, and that of one disease-free equilibrium and one crossing circle. The steady-state regimes include healthy, low epidemic and high epidemic.

This suggests the key role of the threshold value, as well as the initial infection condition in disease control. Our findings demonstrate that the case number can be contained at a satisfactorily controllable level or range if eradicating it proves to be impossible.