Shadows of the SIS immortality transition in small networks

TL;DR

This paper investigates the SIS epidemic model on small networks, revealing how the immortality transition at transmission probability one influences outbreak survival and how network structure affects this critical behavior.

Contribution

It demonstrates that the critical point at transmission probability one predicts survival probability behavior for moderate values and introduces a method to evaluate vertex importance through exponent changes.

Findings

Survival probability follows a power-law near the critical point.

Network structure influences the exponent of the survival probability.

Early die-off is distinct from later stochastic extinction events.

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 $\lambda$ reaches one. If $\lambda < 1$, the probability that an outbreak will survive by an observation time $t$ tends to zero as $t \rightarrow \infty$; if $\lambda = 1$, this probability is one. We show that treating $\lambda = 1$ as a critical point predicts the $\lambda$-dependence of the survival probability also for more moderate $\lambda$-values. The exponent, however, depends on the underlying network. This fact could, by measuring how a vertex' 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).