Brownian snails with removal: epidemics in diffusing populations

Two stochastic models of susceptible/infected/removed (SIR) type are introduced for the spread infection through a spatially-distributed population. Individuals initially distributed at random in space, and they move continuously according to independent diffusion processes. The disease may pass from an infected individual uninfected when sufficiently close. Infected individuals permanently removed some given rate α. Such processes reminiscent so-called frog models, but differ action removal, as well fact that frogs jump whereas snails slither. studied here, termed ‘delayed diffusion’ ‘diffusion’ models. In first, stationary until infected, which time begin move; second, all start initial 0. Using perturbative argument, conditions established under infects a.s. only finitely many individuals. It is proved delayed model there exists critical value αc∈(0,∞) survival epidemic.