Network Epidemics (SIR)

SIR model on a heterogeneous network. High-degree hubs get infected first and drive spreading. Threshold: β/μ > ⟨k⟩/⟨k²⟩. Compare hub vs random immunization.

Network (size ∝ degree)

SIR curves

Susceptible

Infected

Recovered

Immunized

R₀ eff.

threshold β/μ·⟨k²⟩/⟨k⟩

peak infected

final recovered

β infection: 0.3

μ recovery: 0.1

Immunize %: 0

On scale-free networks (hubs), the epidemic threshold β_c = μ⟨k⟩/⟨k²⟩ can approach zero as the network grows, making them very vulnerable. But this also means immunizing just 5-10% of hubs can eliminate epidemics entirely, far more efficiently than random immunization (Pastor-Satorras & Vespignani 2001).