SIR Epidemics on Evolving Erd\H{o}s-R\'enyi Graphs

TL;DR

This paper characterizes the precise conditions under which the SIR-$$ model on evolving Erd
51s-Re9nyi graphs exhibits a discontinuous phase transition in epidemic size, advancing understanding of epidemic dynamics on adaptive networks.

Contribution

It provides a necessary and sufficient condition for the emergence of discontinuous phase transitions in the SIR-$$ model on Erd
51s-Re9nyi graphs, closing previous gaps in theoretical understanding.

Findings

Identifies the exact condition for discontinuous phase transition.

Shows the phase transition behavior depends on network rewiring parameters.

Clarifies the relationship between outbreak probability and epidemic size.

Abstract

In the standard SIR model, infected vertices infect their neighbors at rate $\lambda$ independently across each edge. They also recover at rate $\gamma$. In this work we consider the SIR-$\omega$ model where the graph structure itself co-evolves with the SIR dynamics. Specifically, $S-I$ connections are broken at rate $\omega$. Then, with probability $\alpha$, $S$ rewires this edge to another uniformly chosen vertex; and with probability $1-\alpha$, this edge is simply dropped. When $\alpha=1$ the SIR-$\omega$ model becomes the evoSIR model. Jiang et al. proved in \cite{DOMath} that the probability of an outbreak in the evoSIR model converges to 0 as $\lambda$ approaches the critical infection rate $\lambda_c$. On the other hand, numerical experiments in \cite{DOMath} revealed that, as $\lambda \to \lambda_c$, (conditionally on an outbreak) the fraction of infected vertices may not converge to 0, which is referred to as a discontinuous phase transition. In \cite{BB} Ball and Britton give two (non-matching) conditions for continuous and discontinuous phase transitions for the fraction of infected vertices in the SIR-$\omega$ model. In this work, we obtain a necessary and sufficient condition for the emergence of a discontinuous phase transition of the final epidemic size of the SIR-$\omega$ model on \ER\, graphs, thus closing the gap between these two conditions.