Long-time dynamics of a parabolic-ODE SIS epidemic model with saturated incidence mechanism

In this paper, we investigate a parabolic-ODE SIS epidemic model with no-flux boundary conditions in a heterogeneous environment. The model incorporates a saturated infection mechanism $SI/(m\left(x\right)+S+I)$ with $m\ge ,⁄\equiv 0$. This study is motivated by disease control strategies, such as quarantine and lockdown, that limit population movement. We examine two scenarios: one where the movement of the susceptible population is restricted, and the other where the movement of the infected population is neglected. We establish the long-term dynamics of the solutions in each scenario. Compared to previous studies that assume the absence of a saturated incidence function (i.e., $m\equiv 0$), our findings highlight the novel and significant interplay between total population size, transmission risk level, and the saturated incidence function in influencing disease persistence, extinction, and spatial distribution. Numerical simulations are performed to validate the theoretical results, and the implications of the results are discussed in the context of disease control and eradication strategies.