Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence and Equilibria Analysis of a Networked Bivirus Epidemic Model

Mengbin Ye, Brian D. O. Anderson, Ji Liu

TL;DR

It is shown that it is possible for a bivirus network to have an unstable coexistence equilibrium and two locally stable boundary equilibria, and how to use monotone systems theory to generate conclusions on the ordering of stable and unstableEquilibria.

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.

Convergence and Equilibria Analysis of a Networked Bivirus Epidemic Model

TL;DR

It is shown that it is possible for a bivirus network to have an unstable coexistence equilibrium and two locally stable boundary equilibria, and how to use monotone systems theory to generate conclusions on the ordering of stable and unstableEquilibria.

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.
Paper Structure (28 sections, 18 theorems, 24 equations, 2 figures, 1 table)

This paper contains 28 sections, 18 theorems, 24 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and related
  4. The bivirus equations
  5. Monotone systems theory
  6. Main results
  7. General result on equilibria and convergence
  8. Properties of the Equilibria
  9. Nongeneric networks with an infinite number of equilibria
  10. Stability of boundary equilibria
  11. Properties of trajectories and implications for equilibria
  12. Equilibria for a low order system
  13. Conclusions
  14. Proofs of main results
  15. Proof of \ref{['lem:interiorequilibria']}
  16. ...and 13 more sections

Key Result

Lemma 2.1

\newlabellem:looseinvariant0 With the above notation, suppose that the initial conditions for eq:twovirus satisfy ${\bf{0}}_n\leq x^i(0)\leq {\bf{1}}_n$ for $i=1,2$, and $x^1(0)+x^2(0)\leq {\bf{1}}_n$. Then for all $t>0$, there holds ${\bf{0}}_n\leq x^i(t)\leq {\bf{1}}_n$ for $i=1,2$ and $x^1(t)+x

Figures (2)

  • Figure 1: Illustration of Corollary \ref{['cor:equilibriumordering']}. The triangles indicate initial conditions, with the system trajectory shown converging to the circles, which indicate the equilibria. The dotted rectangle shows the intersection of the hyperrectangle $\mathcal{W}$ with the plane defined by the $i$th coordinates of $x^1$ and $x^2$.
  • Figure 2: Simulation of the bivirus system for Case 2 identified in \ref{['sssec:n2']}. Different colored lines represent the trajectories for eight different initial conditions for (a) Node 1 states $x^1_1(t)$ and $x^2_1(t)$ and (b) Node 2 states $x^1_2(t)$ and $x^2_2(t)$. The solid ball denotes the initial condition, and the cross denotes the equilibrium reached as $t\to\infty$. The black circle identifies the unstable coexistence equilibrium $(\tilde{x}^1, \tilde{x}^2)$.

Theorems & Definitions (22)

  • Lemma 2.1: liu2019bivirus
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6
  • Lemma 3.7
  • ...and 12 more