  We consider the contact process on a dynamic graph defined as a random
$d$-regular graph with a stationary edge-switching dynamics. In this graph
dynamics, independently of the contact process state, each pair $\{e_1,e_2\}$
of edges of the graph is replaced by new edges $\{e_1',e_2'\}$ in a crossing
fashion: each of $e_1',e_2'$ contains one vertex of $e_1$ and one vertex of
$e_2$. As the number of vertices of the graph is taken to infinity, we scale
the rate of switching in a way that any fixed edge is involved in a switching
with a rate that approaches a limiting value $\mathsf{v}$, so that locally the
switching is seen in the same time scale as that of the contact process. We
prove that if the infection rate of the contact process is above a threshold
value $\bar{\lambda}$ (depending on $d$ and $\mathsf{v}$), then the infection
survives for a time that grows exponentially with the size of the graph. By
proving that $\bar{\lambda}$ is strictly smaller than the lower critical
infection rate of the contact process on the infinite $d$-regular tree, we show
that there are values of $\lambda$ for which the infection dies out in
logarithmic time in the static graph but survives exponentially long in the
dynamic graph.
