  This paper studies the endemic behavior of a multi-competitive networked
susceptible-infected-susceptible (SIS) model. Specifically, the paper deals
with three competing virus systems (i.e., tri-virus systems). First, we show
that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical
system. Using the Parametric Transversality Theorem, we show that, generically,
a tri-virus system has a finite number of equilibria and that the Jacobian
matrices associated with each equilibrium are nonsingular. The endemic
equilibria of this system can be classified as follows: a) single-virus endemic
equilibria (also referred to as the boundary equilibria), where precisely one
of the three viruses is alive; b) 2-coexistence equilibria, where exactly two
of the three viruses are alive; and c) 3-coexistence equilibria, where all
three viruses survive in the network. We provide a necessary and sufficient
condition that guarantees local exponential convergence to a boundary
equilibrium. Further, we secure conditions for the nonexistence of
3-coexistence equilibria (resp. for various forms of 2-coexistence equilibria).
We also identify sufficient conditions for the existence of a 2-coexistence
(resp. 3-coexistence) equilibrium. We identify conditions on the model
parameters that give rise to a continuum of coexistence equilibria. More
specifically, we establish i) a scenario that admits the existence and local
exponential attractivity of a line of coexistence equilibria; and ii) scenarios
that admit the existence of, and, in the case of one such scenario, global
convergence to, a plane of 3-coexistence equilibria.

Systems and Control;Systems and Control