Global analyses of a state-dependent impulsive system for an SIRS model with a nonlinear incidence rate

This paper establishes a state-feedback control system based on a nonlinear incidence SIRS model to explore the regulatory mechanism of infectious disease control. We develop a state-dependent impulsive model with threshold-triggered vaccination and isolation interventions once the susceptible population exceeds a predefined level. After investigating the basic properties of the ODE subsystem, we define and classify the Poincaré map and analyze its domain, range and monotonicity, which provides a theoretical basis for studying positive periodic solutions of the impulsive system under different ODE dynamics. The results show that the disease-free periodic solution is globally asymptotically stable for $p>1$, where $p$ is a parameter directly related to the force of the infection. When $0<p<1$, however, the model may exhibit bistability between the disease-free periodic solution and an endemic periodic solution, where the final outcome depends sensitively on the initial infection size, which is consistent with the weak transmissibility and controllable low-level prevalence.