Epidemic Wavefronts: When SIR Gets a Map

SIR, with a place attached

The classic SIR model splits a population into Susceptible, Infected and Recovered compartments and tracks how they flow into one another over time, with no notion of geography — everyone is assumed to mix with everyone else equally. Real outbreaks don't behave that way: they start at a point and spread outward, city by city, street by street. Adding a spatial dimension turns the ordinary differential equations into reaction-diffusion partial differential equations, and the result is a genuine travelling wave of infection moving across the population at a predictable speed.

The Kermack-McKendrick reaction terms

At any point in space, the local dynamics still obey the original 1927 Kermack-McKendrick equations — new infections proportional to the product of susceptible and infected density, recoveries proportional to infected density:

dS/dt = -beta * S * I  +  D * Laplacian(S)
dI/dt =  beta * S * I  -  gamma * I  +  D * Laplacian(I)
dR/dt =  gamma * I

beta  = transmission rate    gamma = recovery rate    D = spatial diffusion coefficient
R0 = beta / gamma   (basic reproduction number)

The Laplacian term is what makes this spatial: it pushes infected and susceptible density from crowded neighbouring cells into emptier ones, exactly like heat or ink diffusing outward, which is why the same mathematics — Fisher-KPP-type reaction-diffusion equations, first studied for gene-frequency spread in 1937 — describes infection fronts, forest fires and even some chemical reaction fronts.

Why the front moves at a constant speed

Once a wavefront forms, it settles into a shape that translates without changing — a travelling wave solution. The minimum speed such a wave can sustain has a clean closed form for the linearised Fisher-KPP equation, c ≈ 2·√(D·r), where r is the effective local growth rate of the infection ((β·S₀ − γ) near the leading edge) and D is the diffusion coefficient. Doubling the diffusion coefficient increases front speed by only about 41% (√2), while doubling the net growth rate has the same effect — the square-root dependence means pushing raw transmissibility down is generally a more efficient lever than trying to slow physical mixing, though real interventions usually attack both.

R0, herd immunity and the shrinking front

The basic reproduction number R₀ = β/γ sets whether an outbreak can grow at all — below 1, any local flare-up decays before it can establish a front; above 1, a wave forms and advances. As susceptibles are consumed behind the advancing front, the effective reproduction number R_eff = R₀·S/N falls, because there are fewer susceptible people left for each infected person to reach. Herd immunity is the threshold where R_eff drops to 1, at fraction of the population immune ≈ 1 − 1/R₀: at that point the wave can no longer sustain itself and burns out even though many people remain uninfected, because the immune fraction breaks up the chains of transmission the front depends on.

Why real outbreaks aren't smooth circles

A homogeneous diffusion model predicts a tidy expanding circle, but real epidemic fronts are visibly lumpy because transmission also happens instantaneously across long distances — one infected traveller can seed a new front hundreds of kilometres away, well ahead of the local diffusive wave. This long-range coupling, layered on top of local diffusion, is why models of real epidemics (like historical plague or modern flu spread) fit better with a mix of short-range diffusion and a network of long-range jumps along transportation routes, rather than diffusion alone; it also explains why travel restrictions specifically target the long-range term while local diffusion is comparatively hard to stop.

Frequently asked questions

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