Summary: We consider long-range percolation on , where the probability that two vertices at distance are connected by an edge is given by and the presence or absence of different edges are independent. Here, is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the -ball around the origin, , that is, the number of vertices that are within graph-distance of the origin, for , for different . We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying exist, for which, respectively:
there exist such that
This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, , which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.