The impulsive vaccination in an SVIS diffusive model

This paper examines a reaction-diffusion problem with impulses, which presents a susceptible-vaccinated-infected-susceptible (SVIS) epidemic model with impulsive vaccination. First, the existence and uniqueness of the global solution and the disease-free periodic solution have been established. Second, by considering the principal eigenvalue, the asymptotic behavior of the disease-free periodic solution and the sufficient condition for the persistence of diseases are given, in addition, the mono-tonicity of the principal eigenvalue is investigated. Third, the periodic state of the endemic disease has been explored when vaccination is not completely effective. Finally, the numerical simulations presented reveal the impacts of the infection rate and impulses, and demonstrate that effective vaccination is a reliable way to control the spread of diseases.