Contact Process under heavy-tailed renewals on finite graphs
Luiz Renato Fontes, Pablo Almeida Gomes, Remy Sanchis
TL;DR
This work analyzes a non-Markovian renewal analogue of the Harris contact process on finite graphs with heavy-tailed cure times having tail index $\alpha\in(\tfrac{1}{2},1)$. Using renewal theory, Dynkin-Lamperti-type results, and a Harris-style graphical construction, it derives sharp graph-size thresholds: survival with positive probability for $|V|>\frac{1}{1-\alpha}$ (for any $\lambda$) and extinction for $|V|<2+\frac{2\alpha-1}{(1-\alpha)(2-\alpha)}$ (for any $\lambda$). The proofs combine a survival mechanism based on long cure-free intervals and a domination argument yielding a negative drift in small graphs, leaving a gap of at most one vertex between thresholds. The results extend understanding of phase transitions under non-Markovian cures on finite networks and connect to prior infinite-graph results in the $\alpha<1$ regime, providing a finite-graph counterpart to those findings. Techniques include renewal-measure bounds, spanning-path propagation, and a detailed domination framework for non-Markovian epidemic processes.
Abstract
We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph G=(V,E): an individual is attached to each site x in V, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate lambda>0; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution mu such that mu(t,infty) = t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if |V| < 2 + (2 alpha -1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda; and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive probability to survive