Thresholds for epidemic spreading in networks (1010.1646v2)

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.

• The paper shows the vanishing epidemic threshold in SIS models due to the largest hub sustaining infection, contradicting heterogeneous mean-field theory predictions.
• The paper demonstrates that SIR models yield finite thresholds on scale-rich networks (γ > 3), aligning with HMF estimates.
• The paper posits that the existence of a steady state is key in determining epidemic thresholds, highlighting the need for refined theoretical models in complex networks.

Thresholds for Epidemic Spreading in Networks: An Analysis

The paper "Thresholds for epidemic spreading in networks," authored by Claudio Castellano and Romualdo Pastor-Satorras, explores the epidemic threshold dynamics of the Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Removed (SIR) models on complex networks, focusing particularly on quenched networks with power-law degree distributions.

Key Findings

1. SIS Model on Quenched Networks:
   • The analysis reveals that in the SIS model, the epidemic threshold $\lambda_c$ vanishes in the large network size limit. This finding contradicts the predictions of heterogeneous mean-field (HMF) theory, which anticipates a finite threshold based on the divergence of the second moment of the degree distribution for scale-free networks with $\gamma \leq 3$.
   • The vanishing threshold is attributed to the largest hub in the network acting as a self-sustained infection source when the spreading rate $\lambda$ exceeds $1/\sqrt{k_{max}}$, where $k_{max}$ is the degree of the most connected node.
2. SIR Model on Quenched Networks:
   • For the SIR model, the research confirms agreement with HMF predictions, revealing a finite threshold on scale-rich networks ($\gamma > 3$). The absence of a steady state in SIR dynamics results in less impact from high-degree nodes compared to SIS dynamics.
3. Generic Epidemic Models:
   • The paper conjectures that the threshold behavior of epidemic models in quenched scale-rich networks—whether vanishing or finite—depends on the presence or absence of a steady state. Models allowing a steady state like SIS tend to have a vanishing threshold, while those without such a state, similar to SIR, adhere more closely to HMF predictions.

Implications of Findings

These results challenge existing theories on network dynamics by suggesting that the scale-free structure does not inherently define epidemic threshold behaviors as previously believed. The role of large hubs in quenched networks alters the critical dynamics significantly, indicating that traditional mean-field approaches may not adequately capture the dynamics in finite or rapidly-changing network structures.

Future Directions

The paper opens several avenues for further research:

• Dynamics Over Annealed Networks: Evaluating epidemic thresholds in dynamically evolving networks, representative of real-world interactions, may yield thresholds consistent with HMF predictions for $\gamma > 3$.
• Extensions to Other Dynamical Models: Investigating other models on complex networks may help generalize the understanding of epidemic spread and its dependency on network topology.
• Refinement of Theoretical Models: There is a need to refine theoretical models to consider both static and dynamic role distributions realistically, adapting HMF techniques for broader application across heterogeneous networks.

In summary, this paper contributes to the theoretical understanding of epidemic spread in networks, highlighting critical discrepancies in threshold behavior predictions and urging a reconsideration of network topology's role in dynamic processes.