SPREADING IN MEDIA WITH LONG-TIME MEMORY

We study the spreading of an agent in a medium whose susceptibility ch anges irreversibly at the first encounter with the agent. This can mod el epidemics with partial immunization or population growth with incom plete replenishment of food (in both cases the susceptibility for grow th decreases after the first attack) or epidemics in which the resista nce is weakened by the first infection (increased susceptibility). In such models one can have no growth at all, compact growth, or annular growth. We delineate the phase diagram and study the scaling behavior at the phase boundaries. Our arguments are supported by simulations in one and two dimensions. Although our model does not involve multiple absorbing states, we claim that our results explain the ''nonuniversal '' behavior seen in models with such states.