Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal

In this paper, a spatial SIR model in combination with random dispersal and non-local dispersal is proposed. We establish the existence and non-existence of the traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium for the model. The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property and the solutions lack of regularity. We use a proper iteration scheme to construct a pair of upper and lower solutions and then apply the Schauder's fixed-point theorem, polar coordinates transform to study the threshold dynamics of the model. That is, we show that if the basic reproduction number of the model R0>1, there is a positive constant critical number c* such that for each c ≥ c*, the model admits a non-trivial and positive traveling wave with wave speed c; and if R0>1 and 0 < c < c*, the model admits no non-trivial and non-negative traveling waves. In view of the numerical simulations, we see that the epidemic waves are not monotone and the non-local dispersal may cause them to oscillate more frequently.