The contact process, or the SIS epidemic model, is a continuous-time Markov process used to model the spread of a recurring infection on a graph. Each vertex is either healthy or infected, and each infected vertex independently infects each of its healthy neighbors at rate $λ$ and recovers at rate $1$. We study the contact process in the presence of additional intervention measures by introducing a third possible state for vertices, which we call isolated. Vertices may enter the isolated state either because of individual decisions or due to community-influenced decisions, which leads to two distinct models that we call the isolation model and the vigilance model, respectively. In the isolation model, infected vertices self-isolate at rate $α$. In the vigilance model, each healthy vertex causes each of its infected neighbors to isolate at rate $α$. Unlike the classical contact process, these models lack the key features of attractiveness and existence of a dual, which makes analyzing them more challenging. We study the persistence times of the infection on large, finite, random graphs with heavy-tailed degree distributions. We show that the infection in the isolation model persists for at least stretched exponential time in the size of the graph for all values of $α$ and $λ$. By contrast, in the vigilance model, for every fixed $α$ the persistence time of the infection exhibits a phase transition in $λ$: for small $λ$ the infection persists for at most a linear time in the size of the graph, while for large $λ$ the infection persists exponentially long. As a corollary of our main results we show that for any $λ, α>0$ the persistence time of the SIRS model on the configuration model having $n$ vertices starting from all vertices infected is at least $\exp(n^{1-η})$ with high probability for any $η>0$.