  In the standard SIR model, infected vertices infect their neighbors at rate
$\lambda$ independently across each edge. They also recover at rate $\gamma$.
In this work we consider the SIR-$\omega$ model where the graph structure
itself co-evolves with the SIR dynamics. Specifically, $S-I$ connections are
broken at rate $\omega$. Then, with probability $\alpha$, $S$ rewires this edge
to another uniformly chosen vertex; and with probability $1-\alpha$, this edge
is simply dropped. When $\alpha=1$ the SIR-$\omega$ model becomes the evoSIR
model. Jiang et al. proved in \cite{DOMath} that the probability of an outbreak
in the evoSIR model converges to 0 as $\lambda$ approaches the critical
infection rate $\lambda_c$. On the other hand, numerical experiments in
\cite{DOMath} revealed that, as $\lambda \to \lambda_c$, (conditionally on an
outbreak) the fraction of infected vertices may not converge to 0, which is
referred to as a discontinuous phase transition. In \cite{BB} Ball and Britton
give two (non-matching) conditions for continuous and discontinuous phase
transitions for the fraction of infected vertices in the SIR-$\omega$ model. In
this work, we obtain a necessary and sufficient condition for the emergence of
a discontinuous phase transition of the final epidemic size of the SIR-$\omega$
model on \ER\, graphs, thus closing the gap between these two conditions.
