Long-time dynamics of some diffusive epidemic models with free boundaries

Abstract

<p>We determine the long-time dynamics of some epidemic models with free boundaries and with local as well as nonlocal diffusions. The epidemic models originate from an ODE model proposed by Capasso and Paveri-fontana [14] to model the spread of cholera. The local diffusion version of our model reduces to the epidemic model of Capasso and Maddalena [13] when the boundary is fixed, and to the model in Ahn et al. [1] if the diffusion of infectious host population is ignored. For this local diffusion version of our free boundary model, we prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the basic reproduction number R0 obtained from the corresponding ODE model is not larger than 1, then the epidemic modeled by this system will vanish, while if <i>R0</i> > 1, then the epidemic may vanish or spread depending on its initial size, governed by certain precise criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.</p> <p>For the nonlocal diffusion version of the model, we show that it is well-posed, its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy, and we also obtain sharp criteria to determine the dichotomy. Some of the nonlocal effects in the model pose extra difficulties in the mathematical treatment, which are dealt with by introducing new approaches. Moreover, we completely determine the spreading speed of the model when spreading happens. We find a threshold condition for the kernels in the nonlocal diffusion terms such that the asymptotic spreading speed is finite precisely when this condition is satisfied. When this condition is not satisfied, and spreading is successful, we prove that the asymptotic spreading speed is infinite, namely accelerated spreading happens.</p>