Competing Epidemics on Graphs -- Global Convergence and Coexistence

TL;DR

This paper provides a comprehensive analysis of competing epidemics on networks, establishing conditions for global convergence to various outcomes including coexistence, which was previously uncharacterized.

Contribution

It offers the first complete characterization of the entire parameter space for bi-virus epidemic models, including coexistence scenarios.

Findings

Established necessary and sufficient conditions for global convergence.

Identified all possible equilibrium outcomes: virus-free, single-virus, and coexistence.

Filled gaps in existing models by analyzing a large subset of parameters.

Abstract

The dynamics of the spread of contagions such as viruses, infectious diseases or even rumors/opinions over contact networks (graphs) have effectively been captured by the well known \textit{Susceptible-Infected-Susceptible} ($SIS$) epidemic model in recent years. When it comes to competition between two such contagions spreading on overlaid graphs, their propagation is captured by so-called \textit{bi-virus} epidemic models. Analysis of such dynamical systems involve the identification of equilibrium points and its convergence properties, which determine whether either of the viruses dies out, or both survive together. We demonstrate how the existing works are unsuccessful in characterizing a large subset of the model parameter space, including all parameters for which the competitiveness of the bi-virus system is significant enough to attain coexistence of the epidemics. In this paper, we fill in this void and obtain convergence results for the entirety of the model parameter space; giving precise conditions (necessary and sufficient) under which the system \textit{globally converges} to a \textit{trichotomy} of possible outcomes: a virus-free state, a single-virus state, and to a coexistence state -- the first such result.