We study the spreading of an agent in a medium whose susceptibility changes irreversibly at the first encounter with the agent. This can model epidemics with partial immunization or population growth with incomplete replenishment of food (in both cases the susceptibility for growth decreases after the first attack) or epidemics in which the resistance 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.

• Received 26 September 1996

DOI:https://doi.org/10.1103/PhysRevE.55.2488