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We show existence of a non-trivial phase transition for the contact process,a simple model for infection without immunity, on a network which reactsdynamically to the infection trying to prevent an epidemic. This networkinitially has the distribution of an Erd\H{o}s-R\'enyi graph, but is madeadaptive via updating in only the infected neighbourhoods, at constant rate.Under these graph dynamics, the presence of infection can help to prevent thespread and so many monotonicity-based techniques fail. Instead, we approximatethe 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 thecreation of a new tree with that vertex as the root. By seeing the newlycreated trees as offspring, the CPEF generates a Galton-Watson tree and we usethat it survives if and only if its offspring number exceeds one. It turns outthat the phase transition in the adaptive model occurs at a different infectionrate when compared to the non-adaptive model.

The Contact Process on a Graph Adapting to the Infection