State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosing p as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when p changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.