Virus Dynamics on Starlike Graphs

TL;DR

This paper analyzes how viruses spread on starlike networks, identifying critical thresholds for virus extinction or persistence based on infection and cure rates, with implications for epidemiology and network theory.

Contribution

It introduces a mathematical model for virus dynamics on starlike graphs, establishing thresholds for virus survival or die-out and exploring long-term behavior.

Findings

Existence of a critical threshold relating cure and infection rates.

Virus dies out below the threshold; persists above it.

Results applicable to airline and social networks.

Abstract

The field of epidemiology has presented fascinating and relevant questions for mathematicians, primarily concerning the spread of viruses in a community. The importance of this research has greatly increased over time as its applications have expanded to also include studies of electronic and social networks and the spread of information and ideas. We study virus propagation on a non-linear hub and spoke graph (which models well many airline networks). We determine the long-term behavior as a function of the cure and infection rates, as well as the number of spokes n. For each n we prove the existence of a critical threshold relating the two rates. Below this threshold, the virus always dies out; above this threshold, all non-trivial initial conditions iterate to a unique non-trivial steady state. We end with some generalizations to other networks.