We consider long-range percolation on Z(d), where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 - exp[-lambda(r)] is an element of (0, 1) and the presence or absence of different edges are independent Here, lambda(r) is a strictly positive, nonincreasing, regularly varying function We investigate the asymptotic growth of the size of the k-ball around the origin, vertical bar B-k vertical bar, that is, the number of vertices that ale within graphdistance k of the origin, for K -> infinity, for different lambda(r) We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonmereasimi regularly varying lambda(r) exist, for which, respectively vertical bar B-k vertical bar(1/k) -> infinity almost surely, there exist 1 < a(1) < a(2) < infinity such that lim(k ->infinity)P(a(1) < vertical bar B-k vertical bar(1/k) < a(2)) = 1. vertical bar B-k vertical bar(1/k) -> 1 almost surely. This result can be applied to spatial SIR epidemics. In particular. regimes are identified for which the basic reproduction number. R-0, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.