Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission

We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack–McKendrick SIR models with nonlocal and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder’s fixed point theorem.