In a scale-free network, the epidemic threshold is λ_c = ⟨k⟩/⟨k²⟩. With broad degree distributions (γ → 2), ⟨k²⟩ → ∞ and λ_c → 0: epidemics spread for any nonzero infection rate.
Degree-heterogeneous mean-field SIR (Pastor-Satorras & Vespignani 2001): dI_k/dt = λk(1−I_k)Θ − μI_k, where Θ = Σ_k P(k)k I_k / ⟨k⟩. The epidemic threshold λ_c = ⟨k⟩/⟨k²⟩ vanishes for power-law networks with γ ≤ 3, explaining why herd immunity requires targeting high-degree hubs.