Networked Competitive Bivirus SIS Model - Analysis of the Discrete-Time Case
Abstract
The paper studies the problem of simultaneous spread of two viruses, termed virus 1 and virus 2, in a meta-population, and assumes that said viruses are in competition with each other. Against this backdrop, we provide an in-depth analysis of a discrete-time networked competitive bivirus susceptible-infected-susceptible (SIS) model. This system has three kinds of equilibria, viz. the disease-free equilibrium (DFE), the single-virus endemic equilibrium (where one virus is present and the other one is not), and coexistence equilibria (where both the viruses are present in each population node). We show that the model is strongly monotone, and, under certain assumptions, does not admit any periodic orbit. With the limiting behavior thus secured, we turn our attention to identifying conditions guaranteeing the existence of and, in some cases, convergence to different equilibria of our model. We show that, when the reproduction number for virus 1 (resp. virus 2) is greater than one, but the reproduction number of virus 2 (resp. virus 1) is less than one, the spread dynamics converge to the single-virus endemic equilibrium (boundary equilibrium) corresponding to virus 1 (resp. virus 2). When the reproduction number of virus 1 and virus 2 is larger than one, we establish a condition which ensures local exponential convergence to the single-virus equilibrium of virus 1 (resp. virus 2). Subsequently, we provide a sufficient (resp. necessary) condition for the existence of a coexistence equilibrium. It turns out that the said necessary condition for the existence of a coexistence equilibrium also ensures that the boundary equilibrium of virus 2 (resp. virus 1) is locally exponentially stable, while that of virus 1 (resp. virus 2) is unstable. Thereafter, we identify a choice of parameters that gives rise to a locally exponentially attractive line of coexistence equilibria. Finally, we provide an extensive set of simulations to illustrate our theoretical findings.