Viral processes by random walks on random regular graphs

Abstract

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\le n^{\epsilon }$, where $n$ is the number of vertices in the random graph, and $\epsilon$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdős-Rényi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $\beta \in (0,1)$ determined by the parameters of the model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get infected with probability $(1-\beta )$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.

Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Disease spreading in populations of moving agents
• Activated Random Walkers: Facts, Conjectures and Challenges
• The Opportunistic Transmission of Wireless Worms between Mobile Devices