The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries

In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents u, while no dispersal is assumed in the other equation for the infective humans v. The underlying spatial region [g (t) , h(t)] (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [33] such a model was considered where the growth rate of u due to the contribution from v is given by cv for some positive constant c. Here this term is replaced by a nonlocal reaction function of v in the form c ∫(h(t))/(g(t)) K(x - y)v(t, y)dy with a suitable kernel function K, to represent g(t) the nonlocal effect of v on the growth of u. We first show that this problem has a unique solution for all t > 0, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term cv. We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [33] where the term cv was used; in particular, small nonlocal dispersal rate of u alone no longer guarantees successful spreading of the disease as in the model of [33].