Mean-Field Epidemic on Heterogeneous Networks

Heterogeneous mean-field (HMF) theory — degree-stratified SIR equations with scale-free network topology reveal epidemic thresholds

Susceptible

Infected

Recovered

Threshold β_c

Transmission rate β 0.30

Recovery rate μ 0.15

Network γ (power-law) 2.50

Initial seed fraction 0.010

R₀ = β⟨k²⟩/μ⟨k⟩—

Epidemic threshold β_c—

Peak infected—

Final susceptible—

Time0

HMF theory (Pastor-Satorras & Vespignani 2001): for scale-free networks P(k)~k^(-γ) with γ≤3, ⟨k²⟩/⟨k⟩ diverges — the epidemic threshold β_c = μ⟨k⟩/⟨k²⟩ → 0, meaning any nonzero transmission causes an epidemic. Degree-stratified equations track S_k, I_k, R_k per degree class.