SIR Epidemic Model

The SIR model splits a population into Susceptible, Infectious, and Recovered compartments. Its dynamics are governed by non-linear ordinary differential equations: dS/dt = -βIS/N and dI/dt = βIS/N - γI. The threshold parameter R₀ = β/γ represents the basic reproduction number. If R₀ > 1, the disease spreads exponentially before reaching herd immunity (S/N < 1/R₀), which forces the infection rate to decay as the pool of susceptible individuals is depleted.

The infection curve rises, peaks, then falls — but not because the pathogen vanishes. It falls because so many people have already been infected that the virus keeps running into individuals who are already immune. That is why an epidemic burns out long before it reaches everyone: with R₀ = 1.5 only about 58% of the population is ever infected, not 100%.

The SIR model was published in 1927 by William Kermack and Anderson McKendrick. Three years earlier, a laboratory explosion had left Kermack completely blind at age 26 — yet he went on to co-derive the epidemic threshold theorem, reportedly working through much of the mathematics in his head. Their paper was largely overlooked for decades before becoming a cornerstone of modern epidemiology.