Pastor-Satorras & Vespignani (2001): on scale-free networks, the epidemic threshold λ_c = ⟨k⟩/⟨k²⟩ → 0 as N→∞, meaning any infection spreads.
Heterogeneous mean-field SIS: dρ_k/dt = −μρ_k + λk(1−ρ_k)Θ(ρ), where Θ = Σ_k k·P(k)·ρ_k/⟨k⟩ is the density weighted by degree. Epidemic threshold: λ_c = μ⟨k⟩/⟨k²⟩. For scale-free networks P(k)∝k^(−γ) with 2<γ<3, ⟨k²⟩ diverges → λ_c→0. Herd immunity threshold: ρ* = 1 − λ_c/λ for λ>λ_c.