  We study a generalization of the classical contact process (SIS epidemic
model) in a directed graph $G$. Our model is a continuous-time interacting
particle system in which at every time, each vertex is either healthy or
infected, and each oriented edge is either active or inactive. Infected
vertices become healthy at rate $1$, and pass the infection along each active
outgoing edge at rate $\lambda$. At rate $\alpha$, healthy individuals
deactivate each incoming edge from their infected neighbors. We study the
persistence time of this epidemic model on the lattice $\mathbb{Z}$, the
$n$-cycle $\mathbb{Z}_n$, and the $n$-star graph. We show that on $\mathbb{Z}$,
for every $\alpha>0$, there is a phase transition in $\lambda$ between almost
sure extinction and positive probability of indefinite survival; on
$\mathbb{Z}_n$ we show that there is a phase transition between
poly-logarithmic and exponential survival time as the size of the graph
increases. On the star graph, we show that the survival time is
$n^{\Delta+o(1)}$ for an explicit function $\Delta(\alpha,\lambda)$ whenever
$\alpha>0$ and $\lambda>0$. In the cases of $\mathbb{Z}$ and $\mathbb{Z}_n$,
our results qualitatively match what has been shown for the classical contact
process, while in the case of the star graph, the classical contact process
exhibits exponential survival for all $\lambda > 0$, which is qualitatively
different from our result. This model presents a challenge because, unlike the
classical contact process, it has not been shown to be monotonic in the
infection parameter $\lambda$ or the initial infected set.
