The contact process on dynamic regular graphs: monotonicity and subcritical phase

TL;DR

This paper investigates the contact process on dynamic regular graphs, establishing the monotonicity of the critical infection rate with respect to edge dynamics, and demonstrating rapid extinction in the subcritical regime.

Contribution

It proves that the critical infection rate decreases as edge dynamics increase and shows exponential tail bounds for extinction times in the subcritical phase.

Findings

Critical infection rate decreases with increased edge dynamics.

Extinction time in the subcritical regime is exponentially bounded.

In the subcritical phase, infection dies out in logarithmic time with high probability.

Abstract

We study the contact process on a dynamic random~$d$-regular graph with an edge-switching mechanism, as well as an interacting particle system that arises from the local description of this process, called the herds process. Both these processes were introduced in~\cite{da2021contact}; there it was shown that the herds process has a phase transition with respect to the infectivity parameter~$\lambda$, depending on the parameter~$\mathsf{v}$ that governs the edge dynamics. Improving on a result of~\cite{da2021contact}, we prove that the critical value of~$\lambda$ is strictly decreasing with~$\mathsf{v}$. We also prove that in the subcritical regime, the extinction time of the herds process started from a single individual has an exponential tail. Finally, we apply these results to study the subcritical regime of the contact process on the dynamic $d$-regular graph. We show that, starting from all vertices infected, the infection goes extinct in a time that is logarithmic in the number of vertices of the graph, with high probability.