  The dynamics of the spread of contagions such as viruses, infectious diseases
or even rumors/opinions over contact networks (graphs) have effectively been
captured by the well known \textit{Susceptible-Infected-Susceptible} ($SIS$)
epidemic model in recent years. When it comes to competition between two such
contagions spreading on overlaid graphs, their propagation is captured by
so-called \textit{bi-virus} epidemic models. Analysis of such dynamical systems
involve the identification of equilibrium points and its convergence
properties, which determine whether either of the viruses dies out, or both
survive together. We demonstrate how the existing works are unsuccessful in
characterizing a large subset of the model parameter space, including all
parameters for which the competitiveness of the bi-virus system is significant
enough to attain coexistence of the epidemics. In this paper, we fill in this
void and obtain convergence results for the entirety of the model parameter
space; giving precise conditions (necessary and sufficient) under which the
system \textit{globally converges} to a \textit{trichotomy} of possible
outcomes: a virus-free state, a single-virus state, and to a coexistence state
-- the first such result.

Systems and Control;Social and Information Networks;Systems and Control;Physics and Society;Populations and Evolution