Existence of traveling wave solutions for an SIS epidemic model with spatio-temporal delays

This paper proposes an SIS epidemic model with nonlocal effects and time delays, and explores the existence of traveling wave solutions via the geometric singular perturbation theory. Through selecting a specific strong kernel function and applying a time-scale transformation, the model is reformulated as a singular perturbation system with a small parameter. Taking the limit of the small parameter to zero in the corresponding traveling wave system, we obtain a normal hyperbolic critical manifold and subsequently derive a locally invariant manifold thanks to the geometric singular perturbation theory. Utilizing the Fredholm alternative theorem and addressing challenges arising from the non-monotonicity of the system, we prove that there exists a traveling wave solution when the basic reproduction number R0>1 and the wave speed c≥dI(β−γ). Numerically, some simulations are presented to verify the theoretical results obtained and discuss the impact of the movement of individuals on disease transmission. Our findings provide valuable guidance and assistance for the disease control.