We study a model that gives rise to spatially inhomogeneous population dens
ities in a system of host individuals subject to rare, randomly distributed
disease events. For stationary hosts that disperse offspring over short di
stances, evolutionary dynamics can lead to persistent populations with a va
riety of spatial structures. A mean-field analysis is shown to account for
the behavior observed in simulations of a one-dimensional system, where the
evolutionarily stable state corresponds to the solution of a straightforwa
rd optimization problem. In two dimensions, evolution drives the system to
a stable critical state that is less well understood.