Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.