Supercriticality of the SIRS on random networks

TL;DR

This paper analyzes the persistence and extinction times of the SIRS process on various network topologies, revealing regimes of long survival and conditions for infinite re-infection.

Contribution

It provides a detailed characterization of the supercritical behavior of SIRS on finite sparse graphs, heavy-tailed networks, and infinite trees.

Findings

Existence of exponentially long survival times in finite sparse graphs.

Survival time is exponential across all parameters in heavy-tailed networks.

Conditions for infinite re-infection on infinite trees are identified.

Abstract

We study how long the SIRS process persists or how quickly it reaches extinction across various network topologies. Our results provide a three-part characterization of this process: In finite sparse graphs, we prove the existence of a regime where the process survives for an exponentially long time. In heavy-tailed networks with power-law-like exponents, we show that for all range of parameters, the survival time is exponential. Finally, for infinite trees, we find sufficient conditions for strong survival, showing the root is re-infected infinitely often even for light-tailed distributions like the Poisson distribution.