Social contact processes and the partner model

TL;DR

This paper analyzes a stochastic infection model with dynamic, monogamous partnerships on a complete graph, identifying conditions for infection extinction or persistence and characterizing the metastable state.

Contribution

It introduces a novel model combining contact processes with edge dynamics and provides rigorous analysis of its extinction and survival behavior.

Findings

Infection dies out if R0<1 within logarithmic time.

Infection persists exponentially long if R0>1.

Existence of a metastable proportion of infectious individuals.

Abstract

We consider a stochastic model of infection spread on the complete graph on $N$ vertices incorporating dynamic partnerships, which we assume to be monogamous. This can be seen as a variation on the contact process in which some form of edge dynamics determines the set of contacts at each moment in time. We identify a basic reproduction number $R_0$ with the property that if $R_0<1$ the infection dies out by time $O(\log N)$, while if $R_0>1$ the infection survives for an amount of time $e^{\gamma N}$ for some $\gamma>0$ and hovers around a uniquely determined metastable proportion of infectious individuals. The proof in both cases relies on comparison to a set of mean-field equations when the infection is widespread, and to a branching process when the infection is sparse.