Epidemic thresholds of the Susceptible-Infected-Susceptible model on networks: A comparison of numerical and theoretical results (1206.6728v2)

Abstract: Recent work has shown that different theoretical approaches to the dynamics of the Susceptible-Infected-Susceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N. We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with N with the correct exponent, but often failing to capture the correct prefactor.

• The paper presents a detailed comparison of theoretical epidemic thresholds with extensive numerical simulations on various network types.
• The research evaluates both HMF and QMF models, revealing strengths and limitations across homogeneous and heterogeneous networks.
• The findings emphasize the need to incorporate dynamical correlations to improve the precision of SIS epidemic threshold predictions.

An Analysis of Epidemic Thresholds in the SIS Model on Networks

The research paper under examination provides an intensive paper into the epidemic thresholds of the Susceptible-Infected-Susceptible (SIS) model on various network configurations, comparing numerical simulations with theoretical predictions. The authors embark on this paper due to the discrepancies found in different theoretical approaches concerning the epidemic threshold, which is critical to understanding the conditions under which an infection will persist or vanish in a networked population.

Methodological Overview

To address the differences in theoretical estimates, the authors conduct large-scale numerical simulations on diverse network types, which include both homogeneous (e.g., random regular graphs) and heterogeneous (e.g., power-law degree distributed graphs) structures. Utilizing the quasi-stationary (QS) method, they aim to retain a precise evaluation of the SIS dynamics by constraining systems to remain in active states and averting the absorbing states typical in static simulations.

Comparison of Theoretical Models

The paper provides a detailed validation against two main theoretical frameworks: the Heterogeneous Mean-Field (HMF) theory and the Quenched Mean-Field (QMF) theory.

1. HMF Theory: Traditionally applies an annealed approximation to network topology, which results in scaling predictions for the epidemic threshold based on average connectivity and degree distribution. The epidemic threshold is expected to vanish in networks with a diverging degree second moment, characteristic of scale-free networks.
2. QMF Theory: This analytic approach utilizes the actual adjacency matrix of the network, predicting the epidemic threshold based on the inverse of the largest eigenvalue. This offers more granular insights by incorporating the network's detailed structure into calculations.

Numerical and Theoretical Findings

Homogeneous Networks

• Random Regular Graphs: Theoretical predictions by both HMF and QMF fail to accurately predict the threshold in these structures due to the neglect of dynamical correlations amongst neighbors, necessitating more sophisticated models like the pair approximation theory, which better captures these dynamics.

Heterogeneous Networks

1. Star Graphs: QMF predictions scale correctly with the system size but fall short in prefactor accuracy. The divergence from theoretical models is attributed to strict correlation enforcement and indicates QMF's superiority over HMF in heterogeneous scenarios.
2. Scale-Free Networks:
   • For γ < 5/2: Both theoretical (HMF and QMF) predictions coincide closely with numerical results, indicating an asymptotic correctness in capturing the epidemic threshold.
   • For 5/2 < γ < 3: Here, QMF provides a better fit than HMF, demonstrating the threshold's scaling with network size, though discrepancies in numeric prefactors persist.
   • For γ > 3: Features a bifurcation of the epidemic transition into multiple peaks, one remaining at a finite threshold (HMF-identified) and another indicative of a vanishing threshold (QMF-identified), highlighting distinct activation mechanisms (densely connected cores versus dominant hubs).

Implications and Future Directions

This paper reinforces the practical need for models that incorporate dynamical correlations extensively, as seen by the inadequacies in HMF when faced with complex real-world networks. QMF, while a significant enhancement, still requires refinements for precise threshold determinations. Progressing forward, research should aim to develop models that intertwine dynamical correlation effects seamlessly, optimize the use of QS methods in simulations, and enhance our understanding of threshold phenomena in complex network populations.

By dissecting the SIS model across varying network types with rigorous methodology, the paper avails critical insights that bolster both theoretical epidemiology and its practical applications to networked systems, setting the stage for directed improvements in epidemic modeling and network theory at large.