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 $\mathbb{Z}^d$.