The SIRS model is an abstract representation of an infectious disease on a graph in which susceptible (S) vertices become infected (I) when adjacent to an infected vertex, infected vertices become recovered (R) and are immune to becoming infected, and recovered vertices become susceptible again. Each transition occurs at a certain (random) rate.

An important question for such models is for which topologies and rates the respective disease is epidemic, that is, it survives for a long time in expectation. The rate of infection for which a process is epidemic is known as the epidemic threshold (for a given graph). For a reduced variant of the SIRS model, which removes the recovered state from the process, known as the SIS model, epidemic thresholds on various graph classes are known. Most strikingly, stars have a very low epidemic threshold for the SIS model. Surprisingly, for the SIRS model, no such mathematical guarantees existed so far.

In this talk, I present the very first rigorous results for the epidemic threshold of the SIRS model. Our focus are the high-level ideas of our proofs. We show that, in stark contrast to the SIS model, the SIRS model has, for a large regime of the recovery rate, an expected polynomial survival time on stars, regardless of the infection rate. However, on expander graphs, the epidemic threshold is similar to that of the SIS model and thus very low. This result even holds if the expander is a subgraph, which is far from trivial in the SIRS model.