Analysis on a Diffusive SI Epidemic Model with Logistic Source and Saturation Infection Mechanism

In this paper, we consider an SI epidemic reaction-diffusion model with logistic source and saturation infection mechanism. We first establish the uniform boundedness and the extinction and persistence of the infectious disease in terms of the basic reproductive number. We also discuss the global stability of the unique endemic equilibrium when the spatial environment is homogeneous. Then we investigate the asymptotic behavior of the endemic equilibria in the heterogeneous environment when the movement rate of the susceptible and infected populations is small. Our results, together with the other two related epidemic models , not only show that the logistic growth, the infection mechanism, and the population movement can play an important role in the transmission dynamics of disease, but also suggest that increasing the inhibitory effect of the susceptible individuals instead of reducing the mobility of the populations can control the epidemic disease modeled by the SI system under consideration.