  We study the SIR epidemic model with infections carried by $k$ particles
making independent random walks on a random regular graph. Here we assume
$k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph,
and $\epsilon$ is some sufficiently small constant. We give an edge-weighted
graph reduction of the dynamics of the process that allows us to apply standard
results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In
particular, we show how the parameters of the model give two thresholds: In the
subcritical regime, $O(\ln k)$ particles are infected. In the supercritical
regime, for a constant $\beta\in(0,1)$ determined by the parameters of the
model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get
infected with probability $(1-\beta)$. Finally, there is a regime in which all
$k$ particles are infected. Furthermore, the edge weights give information
about when a particle becomes infected. We exploit this to give a completion
time of the process for the SI case.
