Convergence of Bi-Virus Epidemic Models With Non-Linear Rates on Networks—A Monotone Dynamical Systems Approach

Abstract

We study convergence properties of competing epidemic models of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Susceptible-Infected-Susceptible</i> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SIS$</tex-math> </inline-formula> ) type. The SIS epidemic model has seen widespread popularity in modelling the spreading dynamics of contagions such as viruses, infectious diseases, or even rumors/opinions over contact networks (graphs). We analyze the case of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two</i> such viruses spreading on overlaid graphs, with non-linear rates of infection spread and recovery. We call this the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-linear bi-virus model</i> and, building upon recent results, obtain precise conditions for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">global convergence</i> of the solutions to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">trichotomy</i> of possible outcomes: a virus-free state, a single-virus state, and to a coexistence state. Our techniques are based on the theory of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">monotone dynamical systems (MDS)</i> , in contrast to Lyapunov based techniques that have only seen partial success in determining convergence properties in the setting of competing epidemics. We demonstrate how the existing works have been unsuccessful in characterizing a large subset of the model parameter space for bi-virus epidemics, including all scenarios leading to coexistence of the epidemics. To the best of our knowledge, our results are the first in providing complete convergence analysis for the bi-virus system with non-linear infection and recovery rates on general graphs.