Split a population into Susceptible, Infected, and Recovered, let infection flow S→I→R, and one number governs everything: R₀, the average people each case infects. Above 1, the infected curve rises to a peak and an epidemic sweeps through; below 1, it fizzles. It’s the model behind ‘flatten the curve’. Slide R₀ across the threshold.
The SIR model: dS/dt = −βSI, dI/dt = βSI − γI, dR/dt = γI, with basic reproduction number R₀ = β/γ (times initial S). The infected fraction grows only while S > γ/β, i.e. while R₀·S > 1: above threshold I rises to a peak then falls (an epidemic wave); below 1 it decays monotonically. The final attack size solves 1 − R∞ = e^(−R₀·R∞). A fail-loud self-check throws unless R₀>1 produces a peak and R₀<1 dies out. ◆ real epidemiology, node-verified.
Homogeneous mixing and fixed rates are simplifications (real spread has structure, waning immunity, behaviour change); the R₀=1 threshold and the peak-then-decline shape are the robust, exact content of the model.