  The basic contact process with parameter $\mu$ altered so that infections of
sites that have not been previously infected occur at rate proportional to
$\lambda$ instead is considered. Emergence of an infinite epidemic starting out
from a single infected site is not possible for $\mu$ less than the contact
process' critical value, whereas it is possible for $\mu$ greater than that
value. In the former case the space and time infected regions are shown to
decay exponentially; in the latter case and for $\lambda$ greater than $\mu$,
the ratio of the endmost infected site's velocity to that of the contact
process is shown to be at most $\lambda / \mu$.
