Effect of ratio threshold control on epidemic dynamics: analysis of an SIR model with non-monotonic incidence

To describe non-continuous control strategy triggered by a predefined ratio of the concerned population in disease progression, an SIR epidemic model with non-monotone incidence is established, dealing with the integrated interventions of vaccination and isolation as ratio-dependent impulsive control. Based on the analysis of multiple equilibria and stability for the continuous subsystem, complex property of the impulsive controlled system is conducted theoretically and numerically. It is found that the ratio magnitude as well as the initial value plays an important role in the threshold dynamics, determining the trajectory convergence to the order-k(k \(<3\) $<3$ ) periodic solution, the endemic equilibria or the disease free equilibrium under certain conditions. Moreover, numerical examples are provided to demonstrate the theoretical results, and in particular, the case of converging to an order-2 periodic solution is presented in simulation, revealing the influence of the ratio threshold control on disease transmission.