The classic SIR (Susceptible-Infected-Recovered) compartmental epidemic model maps exactly onto bond percolation on the underlying contact network. Each person is a node; each potentially infectious contact is a bond that is "occupied" with probability T = 1 - exp(-β/γ), the transmissibility. An epidemic outbreak occurs if and only if a giant percolating cluster exists in the bond-occupied contact network. The critical condition R₀ = 1 (epidemic threshold) is identical to the percolation threshold condition: T_c = 1/⟨k²/k - k⟩ for a network with degree distribution p_k, where ⟨k⟩ and ⟨k²⟩ are the first and second moments of the degree sequence. Scale-free networks (power-law degree distributions p_k ~ k^{-γ} with γ ≤ 3) have ⟨k²⟩ → ∞, driving the percolation threshold p_c → 0. This means epidemics can spread on scale-free contact networks at arbitrarily low transmissibility — explaining why sexually transmitted infections persist in heterogeneous human contact networks. Vaccination = targeted node removal to destroy the giant connected component. Random vaccination requires removing a fraction 1 - 1/κ where κ = ⟨k²⟩/⟨k⟩ (the network's epidemic spreading parameter). Targeted vaccination of high-degree hubs is far more efficient, reducing the required fraction dramatically. Herd immunity is not simply the condition R_eff < 1 in a homogeneous mixing model; it is the percolation condition that no giant connected component of susceptibles spans the network — a geometric condition that depends on network structure, not just population-averaged transmissibility.
This bridge connects Epidemiology and Network Science through shared mathematical structure. Status: Established connection.
| Domain A Term | Domain B Term | Note |
|---|---|---|
| susceptible individual | unoccupied lattice site in the percolation problem | Each susceptible person is a node that can be reached by the epidemic |
| infectious contact (transmission event) | occupied bond in bond percolation | A bond is occupied with probability T = transmissibility |
| epidemic threshold (R₀ = 1) | percolation threshold (bond occupancy p = p_c) | Both are the critical conditions for a spanning cluster / macroscopic outbreak |
| final epidemic size | size of the giant connected component S | In the infinite-size limit both are exact via the same self-consistency equation |
| basic reproduction number R₀ | network epidemic spreading parameter κ = ⟨k²⟩/⟨k⟩ | R₀ = T·κ for a network SIR model |
| herd immunity threshold | node fraction removed to destroy the giant component | Must account for network heterogeneity — not simply 1 - 1/R₀ |
| random vaccination | random node removal (site percolation) | Effective but inefficient on scale-free networks |
| targeted hub vaccination | targeted high-degree node removal (acquaintance immunisation) | Disrupts giant component with far fewer removed nodes |
| scale-free contact network (γ ≤ 3) | network with diverging ⟨k²⟩ → zero percolation threshold | Explains persistence of STIs and why scale-free networks are epidemic-prone |
Compartmental SIR models (Kermack-McKendrick) predate network science by six decades. Epidemiologists trained on ODEs often view network models as complex extensions requiring simulation, not as exact instances of a solved problem in statistical physics. The percolation connection was formalized by Newman (2002) and Pastor-Satorras & Vespignani (2001) but remains underutilized in public health modeling. The fields publish in non-overlapping journals (PLOS Pathogens, Epidemiology vs. Physical Review E, Journal of Statistical Physics).
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