Global Dynamic Analysis of a Discontinuous Infectious Disease System with Two Thresholds

In this article, we consider a SIV infectious disease control system with two-threshold guidance, in which infection rate and vaccination rate are represented by a piecewise threshold function. We analyze the global dynamics of the discontinuous system using the theory of differential equations with discontinuous right-hand sides. We find some dynamical behaviors, including the disease-free equilibrium and endemic equilibria of three subsystems, a globally asymptotically stable pseudo-equilibrium and two endemic equilibria bistable, one of the two pseudo-equilibria or pseudo-attractor is stable. Conclusions can be used to guide the selection of the most appropriate threshold and parameters to achieve the best control effect under different conditions. We hope to minimize the scale of the infection so that the system can eventually stabilize at the disease-free equilibrium, pseudo-equilibrium or pseudo-attractor, corresponding to the disease disappearing or becoming endemic to a minimum extent, respectively.