We show that a certain model for the spread of an infection has a
phase transition in the recuperation rate. The model is as follows:
There are particles or individuals of type $A$ and type $B$, interpreted
as healthy and infected, respectively. All particles perform
independent, continuous time, simple random walks on $\mathbb{Z}^d$ with
the same jump rate $D$. The only interaction between the particles is
that at the moment when a $B$-particle jumps to a site which contains
an $A$-particle, or vice versa, the $A$-particle turns into a
$B$-particle. All $B$-particles recuperate (that is, turn back into
$A$-particles) independently of each other at a rate $\la$.
We assume that we start the system with $N_A(x,0-)$
$A$-particles at $x$, and that the $N_A(x,0-), \, x \in \mathbb{Z}^d$,
are i.i.d., mean $\mu_A$ Poisson random variables. In addition we
start with one additional $B$-particle at the origin. We show that
there is a critical recuperation rate $\la_c > 0$ such that the
$B$-particles survive (globally) with positive probability if
$\la < \la_c$ and
die out with probability 1 if $\la > \la_c$.