  We consider a system in which two viruses of the
Susceptible-Infected-Susceptible (SIS) type compete over general, overlaid
graphs. While such systems have been the focus of many recent works, they have
mostly been studied in the sense of convergence analysis, with no existing
results quantifying the non-trivial coexistence equilibria (CE) - that is, when
both competing viruses maintain long term presence over the network. In this
paper, we prove monotonicity of the CE with respect to effective infection
rates of the two viruses, and provide the first quantitative analysis of such
equilibria in the form of upper bounds involving spectral radii of the
underlying graphs, as well as positive equilibria of related single-virus
systems. Our results provide deeper insight into how the long term infection
probabilities are affected by system parameters, which we further highlight via
numerical results.
