Dynamics of the piecewise smooth epidemic model with nonlinear incidence

In this paper, a piecewise smooth SIV epidemic model with nonlinear incidence is established to describe threshold policy, in which psychological and behavior changes are triggered only when the disease infection exceeds a certain level. Firstly, the existence and stability of disease-free and endemic equilibria of two subsystems of SIV model are analyzed. Then we discuss the type of all possible equilibria, i.e., the real and virtual state of disease-free equilibrium and endemic equilibrium of the two subsystems. By using the theory of Filippov system, we examine the existence of sliding region and pseudo equilibrium state. Main results show that the complex dynamic behaviors occur, such as the bistability of the disease-free equilibrium with one of the endemic states of subsystem or the newly appeared pseudo equilibrium, or the tri-stability of disease free equilibrium, one of the endemic states of subsystem and the pseudo equilibrium. It is worth noticing that the tri-stable equilibria appear when the parameters and threshold value are properly selected. Therefore, appropriate selection of control threshold value plays an important role in the final development of the diseases.