Epidemic Threshold: Heterogeneous Network SIR

Epidemic threshold on heterogeneous networks: for a network with degree distribution P(k), the SIR threshold is β_c = μ·⟨k⟩/⟨k²⟩. For scale-free networks (P(k)∝k^{−γ}, γ≤3), ⟨k²⟩→∞ as N→∞, so β_c→0 — no epidemic threshold! Even infinitesimal transmission probability can cause an epidemic. For Erdős-Rényi networks (Poisson degree dist), ⟨k²⟩=⟨k⟩²+⟨k⟩, giving β_c = μ/(⟨k⟩+1). The effective reproduction number R₀=β⟨k²⟩/(μ⟨k⟩) determines outbreak size. Shown: degree distribution P(k), SIR dynamics, and final infected fraction vs β/β_c.