Viral Processes by Random Walks on Random Regular Graphs

Details Viral Processes by Random Walks on Random Regular Graphs Viral Processes by Random Walks on Random Regular Graphs

2011-04-19

arxiv.org/abs/1104.3789v3

Mathematics

Probability

Scientific paper

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. We give a edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erd\H{o}s--Renyi random graphs on the particle set. In particular, we show how the parameters of the model produce two phase transitions: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $C \in (0,1)$ determined by the parameters of the model, $Ck$ get infected with probability $C$, and $O(\ln k)$ get infected with probability $(1-C)$. 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 demonstrate how this can be exploited to determine the completion time of the process by applying a result of Janson on randomly edge weighted graphs.

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