Pair Approximation SIR — Network Epidemics

Beyond mean-field: tracking correlated pairs [SI], [SS], [II] captures network structure. Epidemic threshold and peak height differ significantly from homogeneous mixing.

Transmission β 0.40

Recovery γ 0.15

Network degree k̄ 6

Initial infected % 0.02

S (susceptible) — MF

I (infected) — MF

R (recovered) — MF

— PA (dashed)

Pair Approximation closes at the pair level: d[SI]/dt includes triples approximated as [SI][II]/[I]. Network structure (degree k̄) matters: lower degree → more clustering → faster local burnout. R₀_PA = β(k̄−1)/γ differs from mean-field R₀_MF = βk̄/γ. Compare solid (MF) with dashed (PA) curves.