Shadows of the susceptible-infectious-susceptible immortality transition in small networks.

Abstract

Much of the research on the behavior of the SIS model on networks has concerned the infinite size limit; in particular the phase transition between a state where outbreaks can reach a finite fraction of the population, and a state where only a finite number would be infected. For finite networks, there is also a dynamic transition-the immortality transition-when the per-contact transmission probability λ reaches 1. If λ<1, the probability that an outbreak will survive by an observation time t tends to zero as t→∞; if λ=1, this probability is 1. We show that treating λ=1 as a critical point predicts the λ dependence of the survival probability also for more moderate λ values. The exponent, however, depends on the underlying network. This fact could, by measuring how a vertex's deletion changes the exponent, be used to evaluate the role of a vertex in the outbreak. Our work also confirms an extremely clear separation between the early die-off (from the outbreak failing to take hold in the population) and the later extinctions (corresponding to rare stochastic events of several consecutive transmission events failing to occur).