# Epidemic extinction in networks: Insights from the 12,110 smallest graphs

## TL;DR

This study exactly computes extinction times for all connected graphs with 3-8 vertices in the SIS disease model, revealing how graph structure influences disease persistence and extinction under different transmission rates.

## Contribution

It provides the first exact calculations of extinction times for small graphs, uncovering structural factors affecting disease dynamics beyond stochastic simulations.

## Key findings

- Extinction time depends mainly on graph structure at high transmission rates.
- Degree of infectious vertices strongly influences extinction time rankings.
- Large transmission rates lead to extinction times independent of specific configurations.

## Abstract

We investigate the expected time to extinction in the susceptible-infectious-susceptible (SIS) model of disease spreading. Rather than using stochastic simulations, or asymptotic calculations in network models, we solve the extinction time exactly for all connected graphs with three to eight vertices. This approach enables us to discover descriptive relations that would be impossible with stochastic simulations. It also helps us discovering graphs and configurations of S and I with anomalous behaviors with respect to disease spreading. We find that for large transmission rates the extinction time is independent of the configurations, just dependent on the graph. In this limit, the number of vertices and edges determine the extinction time very accurately (deviations primarily coming from the fluctuations in degrees). We find that the rankings of configurations with respect to extinction times at low and high transmission rates are correlated at low prevalences and negatively correlated for high prevalences. The most important structural factor determining this ranking is the degrees of the infectious vertices.