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

Abstract

<p>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 <i>u</i>, while no dispersal is assumed in the other equation for the infective humans <i>v</i>. The underlying spatial region [<i>g</i> (<i>t</i>) , <i>h</i>(<i>t</i>)] (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 <i>u</i> due to the contribution from <i>v</i> is given by <i>cv</i> for some positive constant <i>c</i>. Here this term is replaced by a nonlocal reaction function of <i>v</i> in the form <i>c</i> ∫(<i>h</i>(<i>t</i>))/(<i>g</i>(<i>t</i>)) <i>K</i>(<i>x - y</i>)<i>v</i>(<i>t, y</i>)<i>dy</i> with a suitable kernel function <i>K</i>, to represent g(t) the nonlocal effect of <i>v</i> on the growth of <i>u</i>. We first show that this problem has a unique solution for all <i>t</i> > 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 <i>cv</i>. 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 <i>cv</i> was used; in particular, small nonlocal dispersal rate of <i>u</i> alone no longer guarantees successful spreading of the disease as in the model of [33].</p>