Spatial structures for an integro-differential SIS epidemic model

In this paper, we investigate an SIS epidemic model with a nonlocal infection mechanism governed by a convolution-type transmission term. Two spatial settings are considered. In a bounded domain Ω⊂Rm, we first establish well-posedness of the system and show that the associated semiflow is point dissipative and admits a global compact attractor. The basic reproduction number, R0, is characterized as the spectral radius of a next-generation operator involving a nonlocal integral term. We show that the disease-free equilibrium (DFE) is globally asymptotically stable when R0<1. If R0>1, we establish uniform persistence of the semiflow, and ensure the existence of at least one strictly positive steady state, that is the endemic equilibrium (EE). In the unbounded domain Rm (m≥1) with spatially homogeneous coefficients, we analyze the corresponding spatially homogeneous equilibria. We show that the system admits both the DFE and EE. The DFE is spectrally stable for R0<1, while for R0>1, a unique EE exists and is spectrally stable.