Partial immunization processes

Abstract

Partial immunization processes are generalizations of the contact process in which the susceptibility of a site to infection depends on whether or not it has been previously infected. Such processes can exhibit a phase of weak survival, in which the process survives but drifts off to infinity, even on graphs such as $\mathbb{Z}^d$, where no such phase exists for the contact process. We establish that whether or not strong survival occurs depends only on the rate at which sites are reinfected and not on the rate at which sites are infected for the first time. This confirms a prediction by Grassberger, Chaté and Rousseau. We then study the processes on homogeneous trees, where the behaviour is also related to that of the contact process whose infection rate is equal to the reinfection rate of the partial immunization process. However, the phase diagram turns out to be substantially richer than that of either the contact process on a tree or partial immunization processes on $\mathbb{Z}^d$.