Spontaneous Infection and Periodic Evolving of Domain in a Diffusive SIS Epidemic Model

In this paper, we consider a susceptible-infective-susceptible (SIS) reaction-diffusion epidemic model with spontaneous infection and logistic source in a periodically evolving domain. Using the iterative technique, the uniform boundedness of solution is established. In addition, the spatial-temporal risk index \({{\cal R}_0}(\rho)\) depending on the domain evolution rate ρ(t) as well as its analytical properties are discussed. The monotonicity of \({{\cal R}_0}(\rho)\) with respect to the diffusion coefficients of the infected d_I, the spontaneous infection rate η(ρ(t)y) and interval length L is investigated under appropriate conditions. Further, the existence and asymptotic behavior of periodic endemic equilibria are explored by upper and lower solution method. Finally, some numerical simulations are presented to illustrate our analytical results. Our results provide valuable information for disease control and prevention.