Convergence and Equilibria Analysis of a Networked Bivirus Epidemic Model

TL;DR

This paper analyzes the convergence and equilibrium states of a bivirus epidemic model on networks, revealing conditions for stability, coexistence, and the impact of network structure on epidemic outcomes.

Contribution

It introduces new theoretical results on the number, stability, and structure of equilibria in a networked bivirus model, using monotone systems theory.

Findings

Finite number of equilibria for generic parameters

Almost global convergence to an equilibrium

Existence of networks with multiple stable equilibria

Abstract

This paper studies a networked bivirus model, in which two competing viruses spread across a network of interconnected populations; each node represents a population with a large number of individuals. The viruses may spread through possibly different network structures, and an individual cannot be simultaneously infected with both viruses. Focusing on convergence and equilibria analysis, a number of new results are provided. First, we show that for networks with generic system parameters, there exist a finite number of equilibria. Exploiting monotone systems theory, we further prove that for bivirus networks with generic system parameters, then convergence to an equilibrium occurs for all initial conditions, except possibly for a set of measure zero. Given the network structure of one virus, a method is presented to construct an infinite family of network structures for the other virus that results in an infinite number of equilibria in which both viruses coexist. Necessary and sufficient conditions are derived for the local stability/instability of boundary equilibria, in which one virus is present and the other is extinct. A sufficient condition for a boundary equilibrium to be almost globally stable is presented. Then, we show how to use monotone systems theory to generate conclusions on the ordering of stable and unstable equilibria, and in some instances identify the number of equilibria via rapid simulation testing. Last, we provide an analytical method for computing equilibria in networks with only two nodes, and show that it is possible for a bivirus network to have an unstable coexistence equilibrium and two locally stable boundary equilibria.