Local-density dependent Markov processes on graphons with epidemiological applications

TL;DR

This paper studies Markov processes on large graphons with local-density dependence, deriving their limiting behavior as differential equations and analyzing epidemic thresholds for SIS models.

Contribution

It introduces a framework for analyzing local-density dependent Markov processes on graphons and derives their limiting equations and epidemic thresholds.

Findings

Convergence of process to integro-PDEs as average degree increases

Rigorous derivation of epidemic threshold for SIS on graphons

Framework applicable to large-scale network epidemiology

Abstract

We investigate local-density dependent Markov processes on a class of large graphs sampled from a graphon, where the transition rates of the vertices are influenced by the states of their neighbors. We show that as the average degree converges to infinity, the evolution of the process in the transient regime converges to the solution of a set of non-local integro-partial differential equations. We also provide rigorous derivation for the epidemic threshold in the case of the Susceptible-Infected-Susceptible (SIS) process on such graphons.