Spatial spread for a delayed and nonlocal foot-and-mouth disease model

Abstract

Foot-and-mouth disease is a highly infectious zoonosis between human and cloven-hoofed animals and it has been endemic in almost every part of the world. In this paper, a novel mathematical model, in the form of a nonlocal delayed reaction–diffusion system on unbounded domain is studied. We concentrate on the spreading property in terms of traveling waves. By applying Schauder’s fixed point theorem on a large domain and constructing an invariant cone, we prove the existence of traveling weak-waves when R0>1 and c>c∗. Under an additional assumption that the infected individuals move more slowly than the susceptible individuals, we establish an uniform upper bound of the traveling weak-waves, which is independent of c. Based on this result, a new Lyapunov functional is then constructed to show that the traveling weak-waves converge to the endemic equilibrium at positive infinity and the existence of the critical traveling wave is proved by using a limiting argument. Finally, the nonexistence of the traveling wave is proved when R0≤1 or 0<c<c∗.