Network Epidemic Threshold

Heterogeneous mean-field (HMF) theory for SIR epidemics on scale-free networks P(k)~k^{-γ}. The epidemic threshold β_c = ⟨k⟩/⟨k²⟩ vanishes for γ≤3 — any pathogen spreads on scale-free networks.

Degree exponent γ: 2.5

Infection rate β: 0.10

Recovery rate μ: 0.10

k_min: 2

k_max: 100

—

⟨k⟩ = —
⟨k²⟩ = —
β_c = ⟨k⟩/⟨k²⟩ = —

HMF equations: dS_k/dt = -βk S_k Θ | dI_k/dt = βk S_k Θ - μ I_k | dR_k/dt = μ I_k
where Θ(t) = Σ_k k P(k) I_k(t) / ⟨k⟩ is the probability that a random neighbor is infected. Threshold: β_c = μ⟨k⟩/⟨k²⟩ → 0 as k_max→∞ for γ≤3 (Pastor-Satorras & Vespignani, PRL 2001).