  This paper considers a deterministic Susceptible-Infected-Susceptible (SIS)
networked bivirus epidemic model (termed the bivirus model for short), in which
two competing viruses spread through a set of populations (nodes) connected by
two graphs, which may be different if the two viruses have different
transmission pathways. The networked dynamics can give rise to complex
equilibria patterns, and most current results identify conditions on the model
parameters for convergence to the healthy equilibrium (where both viruses are
extinct) or a boundary equilibrium (where one virus is endemic and the other is
extinct). However, there are only limited results on coexistence equilibria
(where both viruses are endemic). This paper establishes a set of ``counting''
results which provide lower bounds on the number of coexistence equilibria, and
perhaps more importantly, establish properties on the local
stability/instability properties of these equilibria. In order to do this, we
employ the Poincar\'e-Hopf Theorem but with significant modifications to
overcome several challenges arising from the bivirus system model, such as the
fact that the system dynamics do not evolve on a manifold in the typical sense
required to apply Poincar\'e-Hopf Theory. Subsequently, Morse inequalities are
used to tighten the counting results, under the reasonable assumption that the
bivirus system is a Morse-Smale dynamical system. Numerical examples are
provided which demonstrate the presence of multiple attractor equilibria, and
multiple coexistence equilibria.
