SIR Epidemic on Heterogeneous Contact Network

Scale-free networks, super-spreaders, and the heterogeneous mean-field

Network & Epidemic

Nodes N: 80 Network type: Transmission rate β: 0.30 Recovery rate γ: 0.10 Initial seeds: 3

Epidemic Stats

R₀ = β⟨k²⟩/(γ⟨k⟩)—

Susceptible—

Infected—

Recovered—

Final attack rate—

Super-spreader?—

Heterogeneous Mean-Field Theory (Pastor-Satorras & Vespignani 2001)

On homogeneous networks, the epidemic threshold is simply R₀ = β/γ > 1. But on scale-free networks with degree distribution P(k) ~ k^-γ, the second moment ⟨k²⟩ diverges as N → ∞. The heterogeneous mean-field threshold becomes R₀ = β⟨k²⟩/(γ⟨k⟩) → ∞, meaning the epidemic threshold vanishes: any nonzero β will cause an epidemic.

This result, proven by Pastor-Satorras & Vespignani using the Barabási-Albert model, explains why respiratory viruses can persist in populations with heavy-tailed contact distributions. High-degree "super-spreader" hubs (shown larger) get infected early and drive transmission. Targeted immunization of hubs is far more effective than random vaccination.

Node color: blue=Susceptible, red=Infected, green=Recovered. Edge width scales with joint degree product (weighted contacts).