Quenched mean-field (QMF): Each node i has individual infection probability ρᵢ(t). dρᵢ/dt = −μρᵢ + λ(1−ρᵢ)Σⱼ Aᵢⱼρⱼ. Linearizing near ρ=0: epidemic survives iff λ/μ > 1/ρ(A), where ρ(A) is the largest eigenvalue (spectral radius) of the adjacency matrix A. Heterogeneous mean-field (HMF): Groups nodes by degree k. Threshold: λ_c = ⟨k⟩/⟨k²⟩. For scale-free networks with γ≤3, ⟨k²⟩→∞ so λ_c→0 — no epidemic threshold (Barabási-Albert). Erdős–Rényi: λ_c ≈ 1/⟨k⟩ (homogeneous). Compare the thresholds across network types.