Towards Understanding the Endemic Behavior of a Competitive Tri-virus SIS Networked Model

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) spreading over a population. 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 present in the population; (b) 2-coexistence equilibria, where exactly two of the three viruses are present in the population; and (c) 3-coexistence equilibria, where all three viruses present in the population. By leveraging the notions of basic reproduction number (i.e., the number of infections caused by an infected individual in a completely susceptible population) and invasion reproduction number (i.e., the average number of infections caused by an individual in a setting where other endemic virus(es) are at equilibrium), 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 kinds 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.