Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries: Spreading speed

This paper is a sequel to Wang and Du [15], on the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators, and the spreading front is represented by the free boundaries in the model. In [15], it was shown that the model is well-posed, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy; however, the spreading speed was not determined. In this paper, we completely determine the spreading speed of the model when spreading happens. We find a threshold condition for the diffusion kernels $ J_1 $ and $ J_2 $ such that the asymptotic spreading speed is finite precisely when this condition is satisfied. Moreover, this speed is determined by a unique semi-wave solution which exists exactly when this threshold condition holds. When this condition is not satisfied, and spreading is successful, we prove that the asymptotic spreading speed is infinite, namely accelerated spreading happens.