  We study the contact process on a dynamic random~$d$-regular graph with an
edge-switching mechanism, as well as an interacting particle system that arises
from the local description of this process, called the herds process. Both
these processes were introduced in~\cite{da2021contact}; there it was shown
that the herds process has a phase transition with respect to the infectivity
parameter~$\lambda$, depending on the parameter~$\mathsf{v}$ that governs the
edge dynamics. Improving on a result of~\cite{da2021contact}, we prove that the
critical value of~$\lambda$ is strictly decreasing with~$\mathsf{v}$. We also
prove that in the subcritical regime, the extinction time of the herds process
started from a single individual has an exponential tail. Finally, we apply
these results to study the subcritical regime of the contact process on the
dynamic $d$-regular graph. We show that, starting from all vertices infected,
the infection goes extinct in a time that is logarithmic in the number of
vertices of the graph, with high probability.
