Dynamic behavior analysis of SIVS epidemic models with state-dependent pulse vaccination

To prevent and control the spread of infectious disease, in this paper, we formulate two susceptible-infected-vaccinated-susceptible (SIVS) epidemic models, where state-dependent pulse vaccination control strategies are introduced. The first model incorporates the density of infected individuals as the monitoring threshold value and some sufficient conditions for the existence and orbital stability of order-1 or order-2 periodic solutions are obtained by using the Poincaré map and qualitative theory of ordinary differential equations. Further, using the density of susceptible individuals as the monitoring threshold value, the second model with state-dependent pulse vaccination is proposed, and the existence and stability of the semi-trivial periodic solution and order-1 periodic solution can be obtained via utilizing the analog of the Poincaré criterion. Theoretical results imply that a state-dependent pulse vaccination strategy can eliminate the spread of infectious disease or keep the density of infected individuals at a desired low level for a long time. Finally, numerical simulations are given to verify the correctness of the theoretical results and to obtain the highest efficiency of state-dependent pulse control strategy.