SIR on locally converging dynamic random graphs

TL;DR

This paper analyzes the spread of an SIR epidemic on dynamic random graphs that evolve locally, demonstrating that the epidemic's trajectory can be approximated by its behavior on local limit graphs, with a focus on dynamic local convergence.

Contribution

It introduces a framework for understanding SIR epidemics on dynamically evolving random graphs using dynamic local convergence theory, a novel approach in this context.

Findings

Epidemic spread can be approximated by local neighborhood dynamics.

Dynamic local convergence is essential for the main results.

Provides a detailed theory of dynamic local convergence in random graphs.

Abstract

In this paper, we study the trajectory of a classic SIR epidemic on a family of dynamic random graphs of fixed size, whose set of edges continuously evolves over time. We set general infection and recovery times, and start the epidemic from a positive, yet small, proportion of vertices. We show that in such a case, the spread of an infectious disease around a typical individual can be approximated by the spread of the disease in a local neighbourhood of a uniformly chosen vertex. We formalize this by studying general dynamic random graphs that converge dynamically locally in probability and demonstrate that the epidemic on these graphs converges to the epidemic on their dynamic local limit graphs. We provide a detailed treatment of the theory of dynamic local convergence, which remains a relatively new topic in the study of random graphs. One main conclusion of our paper is that a specific form of dynamic local convergence is required for our results to hold.