  In a collection of particles performing independent random walks on $\mathbb
Z^d$ we study the spread of an infection with SIR dynamics. Susceptible
particles become infected when they meet an infected particle. Infected
particles heal and are removed at rate $\nu$. We show that when $\nu$ is small,
with positive probability the infection survives forever and grows linearly.
Furthermore, after the infection reaches a region, it quickly passes through
and leaves behind a $\textit{herd immunity}$ regime consisting of recovered
particles, a small positive density of susceptible particles, and no infected
particles. One notable feature of this model is the simultaneously existence of
supercritical and subcritical phases on either side of an infection front of
$O(1)$ width.
