Epidemic on Networks — Heterogeneous Mean-Field SIR

Heterogeneous mean-field theory (Pastor-Satorras & Vespignani 2001): on scale-free networks (P(k)~k^{-γ}), ⟨k²⟩ diverges as N→∞, causing the epidemic threshold β_c = μ·⟨k⟩/⟨k²⟩ → 0. Any nonzero β causes an endemic state! Left: SIR dynamics (S=blue, I=red, R=grey) with node size ∝ degree. Right: time series of S,I,R fractions. Hubs (high-degree) become superspreaders — they get infected early and seed many neighbors, explaining COVID superspreader events.