ASYMPTOTIC PROFILES OF STEADY STATES FOR A DIFFUSIVE SIS EPIDEMIC MODEL WITH SPONTANEOUS INFECTION AND A LOGISTIC SOURCE

Spatial heterogeneity and movement of population play an important role in disease spread and control in reality. This paper concerns with a spatial Susceptible-Infected-Susceptible epidemic model with spontaneous infection and logistic source, aiming to investigate the asymptotic profiles of the endemic steady state (whenever it exists) for large and small diffusion rates. We firstly establish uniform upper bound of solutions. By studying the local and global stability of the unique constant endemic equilibrium when spatial environment is homogeneous, we apply the well-known Leray-Schuauder degree index formula to confirm the existence of endemic steady state. Our theoretical results suggest that spontaneous infection and varying total population strongly enhance the persistence of disease spread in the sense that disease component of the endemic steady state will not approach zero whenever the large and small diffusion rates of the susceptible or infected population is used. This gives new insights and aspects for infectious disease modeling and control.