Overdispersed offspring distribution (negative binomial) and 20/80 rule
Final size: –
Extinct: –
Top 20% infected: –%
Herd immunity: –%
Negative binomial offspring distribution with mean R₀ and dispersion k.
k→∞: Poisson (homogeneous). Small k: highly overdispersed (superspreaders).
SARS-CoV-2: k ≈ 0.1 — 80% of transmission from ~10% of cases.
Extinction probability P_ext satisfies: P_ext = G(P_ext) where G is the pgf.
Herd immunity threshold: 1 - 1/R₀ for homogeneous, lower with heterogeneity.