Two golden times in two-step contagion models: A nonlinear map approach

Area of Science:

• Epidemiology
• Network Science
• Mathematical Modeling

Background:

• The two-step contagion model simulates real-world pandemic outbreaks.
• It identifies a critical
• golden time
• for intervention before a pandemic occurs.
• Understanding how this time scales with system size is crucial for pandemic control.

Purpose of the Study:

• To investigate the size-dependence of
• golden time
• in the two-step contagion model.
• To identify different types of
• golden time
• and their scaling behaviors.
• To explore the universality and model-dependence of these scaling exponents.

Main Methods:

• Utilized a nonlinear mapping approach to analyze the two-step contagion model.
• Examined the scaling of
• golden time
• with system size N on Erdős-Rényi networks.
• Differentiated scaling behaviors based on the initial number of infected nodes.

Main Results:

• Identified two types of
• golden time
• , scaling as O(N^{1/3}) and O(N^{ζ}) (with ζ ≈ 1/4).
• The O(N^{1/3}) scaling is universal for models with discontinuous transitions.
• The ζ exponent shows model-dependence and its asymptotic behavior remains an open question.

Conclusions:

• The study reveals two distinct
• golden time
• regimes in the two-step contagion model.
• The findings provide insights into pandemic outbreak dynamics and control strategies.
• The nonlinear mapping method is applicable to other models exhibiting hybrid percolation transitions.