SIR Epidemic — Heterogeneous Network

Degree-based mixing, herd immunity, and network structure effects

Susceptible

Infected

Recovered

β (infection rate): 0.30

γ (recovery rate): 0.10

Network type:

N nodes: 120

R₀ = β/γ·⟨k²⟩/⟨k⟩: —
Herd immunity threshold: —
S: — | I: — | R: —
Attack rate: 0%
⟨k⟩ = — | ⟨k²⟩/⟨k⟩ = —

On heterogeneous networks, the epidemic threshold depends on degree heterogeneity: R₀ = β/γ · ⟨k²⟩/⟨k⟩ (Pastor-Satorras & Vespignani 2001). For scale-free networks (P(k)∝k^−γ), ⟨k²⟩ diverges, making the threshold vanish — epidemics spread for any β>0. Herd immunity requires vaccinating fraction 1 − 1/R₀ of the population.