# Contact Process under heavy-tailed renewals on finite graphs

## TL;DR

This paper studies a non-Markovian contact process with heavy-tailed recovery times on finite graphs, revealing conditions under which the infection persists or dies out depending on graph size and tail index.

## Contribution

It introduces a non-Markovian contact process model with heavy-tailed recovery times and characterizes survival thresholds based on graph size and tail index.

## Key findings

- Infection dies out for small graphs under certain conditions.
- Infection survives with positive probability for larger graphs.
- Survival depends on the tail index and graph size.

## 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