Epidemic Branching Process

Galton-Watson process:
Z_{n+1} = Σᵢ Xᵢ, Xᵢ ~ offspring dist

Extinction probability q:
Smallest fixed point of q = g(q)
g(z) = probability generating function

Critical threshold R₀=1:
R₀ < 1: q = 1 (certain extinction)
R₀ = 1: q = 1 (critical)
R₀ > 1: q < 1 (major outbreak possible)

Poisson offspring g(z):
g(z) = exp(R₀(z−1))
q solves: q = exp(R₀(q−1))