Competitive epidemic networks with multiple survival-of-the-fittest outcomes

TL;DR

This paper analyzes a two-virus epidemic model on a two-layer network, establishing conditions under which either virus can dominate depending on initial states, and demonstrates the existence of such networks for any size, with practical case studies.

Contribution

It proves the existence of networks where either virus can win in a two-layer SIS model for any number of nodes, and provides a method to design such networks.

Findings

Networks satisfying the stability condition exist for any number of nodes.

A four-step procedure to design one network layer given the other.

Real-world mobility network analysis supports theoretical results.

Abstract

We use a deterministic model to study two competing viruses spreading over a two-layer network in the Susceptible--Infected--Susceptible (SIS) framework, and address a central problem of identifying the winning virus in a "survival-of-the-fittest" battle. Existing sufficient conditions ensure that the same virus always wins regardless of initial states. For networks with an arbitrary but finite number of nodes, there exists a necessary and sufficient condition that guarantees local exponential stability of the two equilibria corresponding to each virus winning the battle, meaning that either of the viruses can win, depending on the initial states. However, establishing existence and finding examples of networks with more than three nodes that satisfy such a condition has remained unaddressed. In this paper, we prove that, for any arbitrary number of nodes, such networks exist. We do this by proving that given almost any network layer of one virus, there exists a network layer for the other virus such that the resulting two-layer network satisfies the aforementioned condition. To operationalize our findings, a four-step procedure is developed to reliably and consistently design one of the network layers, when given the other layer. Conclusions from numerical case studies, including a real-world mobility network that captures the commuting patterns for people between $107$ provinces in Italy, extend on the theoretical result and its consequences.