Traveling Epidemic Wave: SIR with Diffusion

Adding spatial diffusion to the SIR model produces traveling wave solutions — epidemic fronts that propagate at speed c ≥ 2√(βD) (Fisher-KPP-like). Susceptible (S) and Infected (I) populations form a moving front, behind which S is depleted. The minimal wave speed depends on the basic reproduction number R₀ = β/γ and diffusivity D.

Transmission β 0.80 Recovery γ 0.30 Diffusion D 0.10

Susceptible S Infected I Recovered R

Wave parameters:
R₀ = β/γ = —
Min wave speed: c* = 2√(βD)
c* = —
Measured speed ≈ —

Fisher-KPP theory:
For R₀ > 1, wave speed ≥ 2√(β−γ)·√D
Front width ∝ √(D/β)
Final susceptible: S∞ = e^(−R₀(1−S∞))

Herd immunity:
1 − 1/R₀ = —