Epidemic threshold and localization of the SIS model on directed complex networks

TL;DR

This paper analyzes the SIS epidemic model on directed complex networks, revealing a phase transition, the influence of infection-rate distribution, and localization phenomena near the epidemic threshold.

Contribution

It introduces an analytic framework combining random matrix theory with fixed-point distribution analysis to characterize epidemic thresholds and localization in directed networks.

Findings

Phase transition between absorbing and endemic states at c ≥ λ^{-1}

Critical line independent of degree distribution

Infection localization near the epidemic threshold

Abstract

We study the susceptible-infected-susceptible (SIS) model on directed complex networks within the quenched mean-field approximation. Combining results from random matrix theory with an analytic approach to the distribution of fixed-point infection probabilities, we derive the phase diagram and show that the model exhibits a nonequilibrium phase transition between the absorbing and endemic phases for $c \geq \lambda^{-1}$, where $c$ is the mean degree and $\lambda$ the average infection rate. Interestingly, the critical line is independent of the degree distribution but is highly sensitive to the form of the infection-rate distribution. We further show that the inverse participation ratio of infection probabilities diverges near the epidemic threshold, indicating that the disease may become localized on a small fraction of nodes. These results provide a systematic characterization of how network heterogeneities shape epidemic spreading on directed contact networks within the quenched mean-field approximation.