  This paper is concerned with the long-time dynamics of an epidemic model
whose diffusion and reaction terms involve nonlocal effects described by
suitable convolution operators, and the epidemic region is represented by an
evolving interval enclosed by the free boundaries in the model. In Wang and Du
\cite{WangDu-JDE}, it was shown that the model is well-posed, and its long-time
dynamical behaviour is governed by a spreading-vanishing dichotomy. The
spreading speed was investigated in a subsequent work of Wang and Du
\cite{WangDu-DCDS-A}, where a threshold condition for the diffusion kernels
$J_1$ and $J_2$ was obtained, such that the asymptotic spreading speed is
finite precisely when this condition is satisfied. In this paper, we examine
the case that this threshold condition is not satisfied, which leads to
accelerated spreading; for some typical classes of kernel functions, we
determine the precise rate of accelerated expansion of the epidemic region by
constructing delicate upper and lower solutions.
