A Piecewise-Smooth Dynamic System with Two Thresholds and Its Application to Epidemic Control

A general type of piecewise-smooth dynamic system with two thresholds is analyzed in this work. We define some fundamental notions for the proposed system such as oscillating space and real/virtual equilibrium, which generalizes the counterparts for the planar Filippov system. Moreover, we show that under certain conditions the planar switched system with two thresholds generates a novel limit cycle, and analyze the properties of this periodic solution such as existence, stability, amplitude and period. Interestingly, the existence and stability of this periodic solution in the oscillating space are consistent with the pseudo-equilibrium of the corresponding planar Filippov system. Hence, we establish the connection between the planar switched system with two thresholds and the planar Filippov system. Finally, we apply the modeling and analytical approaches to a piecewise-smooth epidemic model with density-dependent interventions, describing the control measure that is triggered when the number of infected individuals increases and reaches a critical level while being suspended when it decreases down to another level. We prove that the epidemic model stabilizes at either the endemic equilibrium of the free system (the one not under control) or the new periodic solution induced by the two thresholds, depending on the threshold levels. The two-threshold measure is able to suppress the number of infected individuals during the evolution of an infectious disease.