The Contact Process on a Graph Adapting to the Infection

We show existence of a non-trivial phase transition for the contact process, a simple model for infection without immunity, on a network which reacts dynamically to the infection trying to prevent an epidemic. This network initially has the distribution of an Erd\H{o}s-R\'enyi graph, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Under these graph dynamics, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail. Instead, we approximate the infection by an idealised model, the contact process on an evolving forest (CPEF). In this model updating events of a vertex in a tree correspond to the creation of a new tree with that vertex as the root. By seeing the newly created trees as offspring, the CPEF generates a Galton-Watson tree and we use that it survives if and only if its offspring number exceeds one. It turns out that the phase transition in the adaptive model occurs at a different infection rate when compared to the non-adaptive model.