Epidemic on Networks: Heterogeneous Mean Field

Hubs with high degree lower the epidemic threshold — scale-free networks are extremely vulnerable.

β = 0.15
γ = 0.15
α = 2.5
Scale-free
R₀ = —  |  κ = ⟨k²⟩/⟨k⟩ = —  |  Threshold β_c = —  |  Status: —
Heterogeneous mean-field (Pastor-Satorras & Vespignani 2001): dI_k/dt = β k [1−I_k] Θ(t) − γ I_k
where Θ(t) = Σ_k P(k)(k−1)/⟨k⟩ · I_k is the probability a neighbor is infected.
Epidemic threshold: β_c = γ⟨k⟩/⟨k²⟩ = γ/κ. For scale-free networks (α≤3), κ→∞ → β_c→0!
Colors: susceptible (blue), infected (red), recovered (grey).