DUAL-THRESHOLD SWITCHING STRATEGIES IN EPIDEMIC CONTROL: A NON-SMOOTH SIR FRAMEWORK AND ITS DYNAMICS

We propose a non-smooth dynamical model based on the classical SIR framework, by incorporating a dual-threshold switching strategy to describe the activation and deactivation of control measures in response to the disease state. Breaking the traditional single-threshold modeling paradigm, this framework establishes a dynamic mechanism for initiating and terminating control actions according to epidemic severity. A theoretical analysis of the non-smooth system is conducted, including the classification and stability of equilibria for each subsystem and for the overall switched system. We obtain the existence of an order-1 periodic solution for the proposed system, which extends the pseudo-equilibrium of the Filippov system with a single threshold. In particular, we show that, when no equilibrium lies between the two thresholds, solutions either converge to the endemic equilibria of the subsystems or approach a unique order-1 periodic solution. In contrast, the presence of an equilibrium between the thresholds gives rise to more complex dynamical behaviors, including the coexistence of multiple stable attractors, namely bistability between two endemic equilibria, bistability between an endemic equilibrium and a stable periodic solution, as well as tristability involving two endemic equilibria and one stable periodic solution.