Mathematical Analysis on the SIS Reaction‐Diffusion Model With Logistic Source and Mass Action Incidence

ABSTRACT This article investigates the dynamics of an SIS epidemic reaction‐diffusion model with logistic source and mass action infection mechanism. The uniform boundedness of the solution is obtained. The stability of the trivial and disease‐free equilibria is discussed. For homogeneous environments, global attractivity of equilibria is rigorously demonstrated through Lyapunov function methods. In heterogeneous environments, the existence of endemic equilibrium is established by the uniform persistence theory. Moreover, the asymptotic behaviors of the endemic equilibrium for large or small diffusion rate are characterized. Our results reveal that restricting the movement of infected individuals while reducing the growth rate of susceptible populations can effectively suppress disease prevalence. These findings provide some theoretical foundations for formulating effective disease control strategies.