Asymptotic Behavior of the Basic Reproduction Number and Equilibria for a Partially Degenerate Nonlocal Epidemic Model

Source abstract

ABSTRACT This paper studies a partially degenerate nonlocal epidemic model in a bounded habitat and analyzes the effects of the diffusion coefficient and the diffusion range on the basic reproduction number and the positive steady state. We prove that is strictly decreasing in the diffusion coefficient and determine its limits as the coefficient tends to zero and infinity. For scaled diffusion kernels, we obtain the small‐ and large‐range limits of , including explicit formulas for both limits. We also establish the threshold dynamics, including the critical case. For the parameter‐dependent family of positive steady states, we derive the small‐diffusion limit, locally uniform convergence to the local positive equilibrium on the strictly favorable region in the small‐range regime, and convergence to a local limiting profile in the large‐range regime. Numerical simulations illustrate the analytical results.