This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $N\ge 2$, in ^[18], we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work ^[19], we have obtained sharp estimates of the spreading rate when the kernel function $J(|x|)$ behaves like $|x{|}^{-\beta }$ as $|x|\to \mathrm{\infty }$ in ${\mathbb{R}}^N$ ( $N\ge 2$). In this paper, we obtain more accurate estimates for the spreading rate when $N=3$, which employs the fact that the formulas relating the involved kernel functions in the proofs of ^[19] become particularly simple in dimension $3$.

In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. ^[1] by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $R_0$ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $R_0\le 1$, we prove that the epidemic vanishes eventually. On the other hand, if $R_0>1$, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni ^[2] to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. ^[1]).