  This paper studies a distributed continuous-time bi-virus model in which two
competing viruses spread over a network consisting of multiple groups of
individuals. Limiting behaviors of the network are characterized by analyzing
the equilibria of the system and their stability. Specifically, when the two
viruses spread over possibly different directed infection graphs, the system
may have (1) a unique equilibrium, the healthy state, which is globally stable,
implying that both viruses will eventually be eradicated, (2) two equilibria
including the healthy state and a dominant virus state, which is almost
globally stable, implying that one virus will pervade the entire network
causing a single-virus epidemic while the other virus will be eradicated, or
(3) at least three equilibria including the healthy state and two dominant
virus states, depending on certain conditions on the healing and infection
rates. When the two viruses spread over the same directed infection graph, the
system may have zero or infinitely many coexisting epidemic equilibria, which
represents the pervasion of the two viruses. Sensitivity properties of some
nontrivial equilibria are investigated in the context of a decentralized
control technique, and an impossibility result is given for a certain type of
distributed feedback controller.
