Spatial Epidemics with Large Finite Range

By Mathew D. Penrose.

In the epidemic with removal with range $r$, each site $z$ in the 2-dimensional integer lattice, once infected, remains so for a period of time $T_z$, the variables $T_z$ being independent identically distributed with mean $\mu$. While infected, a site infects its healthy $r$-neighbours independently at total rate $\alpha$. After infection, sites become immune. We show that the critical value of $\alpha$, above which an epidemic starting from a single site may continue forever, converges to $1/\mu$ as $r$ goes to infinity.

Journal of Applied Probabiility 33, 933-939 (1996).