Thresholds for epidemic spreading in networks

TL;DR

This paper investigates epidemic thresholds in quenched networks with power-law degree distributions, revealing that SIS models have vanishing thresholds due to hubs, while SIR models align with mean-field predictions.

Contribution

It demonstrates that the epidemic threshold behavior in quenched networks depends on the model type and network structure, challenging previous mean-field assumptions.

Findings

SIS threshold vanishes in large networks with diverging maximum degree.

SIR model exhibits a finite threshold consistent with mean-field theory.

The hub activity drives the epidemic spreading in SIS models.

Abstract

We study the threshold of epidemic models in quenched networks with degree distribution given by a power-law. For the susceptible-infected-susceptible (SIS) model the activity threshold lambda_c vanishes in the large size limit on any network whose maximum degree k_max diverges with the system size, at odds with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has not to do with the scale-free nature of the connectivity pattern and is instead originated by the largest hub in the system being active for any spreading rate lambda>1/sqrt{k_max} and playing the role of a self-sustained source that spreads the infection to the rest of the system. The susceptible-infected-removed (SIR) model displays instead agreement with HMF theory and a finite threshold for scale-rich networks. We conjecture that on quenched scale-rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state.