  We study the information spreading yielded by the \emph{(Parsimonious)
$1$-Flooding Protocol} in geometric Mobile Ad-Hoc Networks. We consider $n$
agents on a convex plane region of diameter $D$ performing independent random
walks with move radius $\rho$. At any time step, every active agent $v$ informs
every non-informed agent which is within distance $R$ from $v$ ($R>0$ is the
transmission radius). An agent is only active at the time step immediately
after the one in which has been informed and, after that, she is removed. At
the initial time step, a source agent is informed and we look at the
\emph{completion time} of the protocol, i.e., the first time step (if any) in
which all agents are informed. This random process is equivalent to the
well-known \emph{Susceptible-Infective-Removed ($SIR$}) infection process in
Mathematical Epidemiology. No analytical results are available for this random
process over any explicit mobility model. The presence of removed agents makes
this process much more complex than the (standard) flooding. We prove optimal
bounds on the completion time depending on the parameters $n$, $D$, $R$, and
$\rho$. The obtained bounds hold with high probability. We remark that our
method of analysis provides a clear picture of the dynamic shape of the
information spreading (or infection wave) over the time.
