Analysis of the survival time of the SIRS process via expansion

Abstract

We study two continuous-time Markov chains modeling the spread of infections on graphs, namely the SIS and the SIRS model. In the SIS model, vertices are either susceptible or infected; each infected vertex becomes susceptible at rate 1 and infects each of its neighbors independently at rate 𝜆. In the SIRS model, vertices are either susceptible, infected, or recovered; each infected vertex becomes recovered at rate 1 and infects each of its susceptible neighbors independently at rate 𝜆; each recovered vertex becomes susceptible at a rate 𝜚 , which we assume to be independent of the size of the graph. The survival time of the SIS process, i.e., the time until no vertex of the host graph is infected, is fairly well understood for a variety of graph classes. Stars are an important graph class for the SIS model, as the survival time of SIS on stars has been used to show that the process survives on real-world graphs for a long time. For the SIRS model, however, to the best of our knowledge, there are no rigorous results, even for simple graphs such as stars.We analyze the survival time of the SIS and the SIRS process on stars and cliques. We determine three threshold values for 𝜆 such that when 𝜆 < 𝜆 ℓ , the expected survival time of the process is at most logarithmic, when 𝜆 < 𝜆 𝑝 , it is at most polynomial, and when 𝜆 > 𝜆 𝑠 , it is at least super-polynomial in the number of vertices. Our results show that the survival time of the two processes behaves fundamentally di erent on stars, while it behaves fairly similar on cliques. Our analyses bound the drift of potential functions with globally stable equilibrium points. On the SIRS process, our two-state potential functions are inspired by Lyapunov functions used in mean-eld theory.

The Irrelevance of Influencers: Information Diffusion with Re-Activation and Immunity Lasts Exponentially Long on Social Network Models

“…Once again, due to space limitations, the technical details of the proof of Theorem 5 and all related lemmas are in the full version of the paper (Friedrich et al 2022). What follows is a high-level explanation of our proof.…”

“…Due to space limitations, for the complete proofs and all of the technical details, we refer the reader to the full version of the paper (Friedrich et al 2022).…”

“…The detailed proof of Theorem 1 can be found in the full version of the paper (Friedrich et al 2022), however, we…”

“…Due to lack of space, the detailed proofs of Corollaries 3 and 4 are included in the full version of the paper (Friedrich et al 2022).…”

“…Theorem 12 ( (Friedrich and Krohmer 2015)). Let G be a hyperbolic random graph with n vertices that follows a powerlaw degree distribution with exponent γ ∈ (2, 3).…”