The contact process over a dynamical d-regular graph

We consider the contact process on a dynamic graph defined as a random d-regular graph with a stationary edge-switching dynamics. In this graph dynamics, independently of the contact process state, each pair {e₁,e₂} of edges of the graph is replaced by new edges {e1́,e2́} in a crossing fashion: each of e_1́, e_2́ contains one vertex of e₁ and one vertex of e₂. As the number of vertices of the graph is taken to infinity, we scale the rate of switching in a way that any fixed edge is involved in a switching with a rate that approaches a limiting value v, so that locally the switching is seen in the same time scale as that of the contact process. We prove that if the infection rate of the contact process is above a threshold value λ̄ (depending on d and v), then the infection survives for a time that grows exponentially with the size of the graph. By proving that λ̄ is strictly smaller than the lower critical infection rate of the contact process on the infinite d-regular tree, we show that there are values of λ for which the infection dies out in logarithmic time in the static graph but survives exponentially long in the dynamic graph.