the SIR model
/ ess-eye-ar /
When a disease sweeps through a town, you do not track every person individually — you sort the population into three bins by status: S for those still Susceptible (can catch it), I for those currently Infectious (have it and can pass it on), and R for those Removed (recovered and now immune, or sadly deceased). The SIR model watches the number in each bin change over time as people flow S to I to R. It is the founding model of mathematical epidemiology, and the source of the now-familiar 'curve' everyone wants to flatten.
The three equations are S' = -beta S I, I' = beta S I - gamma I, and R' = gamma I. The term beta S I is the engine: new infections happen when a susceptible meets an infectious person, and the number of such meetings is proportional to the product S times I, with beta the transmission rate. The term gamma I says infectious people recover (or are removed) at rate gamma, so 1/gamma is the average time a person stays infectious. Notice S only ever falls, R only ever rises, and the total S + I + R stays constant — nobody is born or migrates in this basic version.
The model's central insight lives in the I equation: I grows only while beta S - gamma > 0, that is, only while S exceeds gamma/beta. At the start almost everyone is susceptible, so the disease takes off; but every infection moves someone from S to R, and once S drops below the threshold gamma/beta the epidemic turns the corner and fades — even though susceptibles still remain. That is why an outbreak ends BEFORE everyone is infected: the disease runs out of fuel. This threshold is the seed of the basic reproduction number and the epidemic threshold.
SIR is deliberately spare: it assumes everyone mixes uniformly, immunity is permanent, and the rates are constant — no age structure, no behaviour change, no spatial spread. Yet from these three lines come the genuinely useful conclusions that herd immunity exists, that the peak height and final size are predictable, and that lowering beta (distancing) or shortening 1/gamma (treatment) bends the curve.
In a town of 1000 with beta = 0.001 and gamma = 0.1, the threshold S level is gamma/beta = 100. Start with one infected and 999 susceptible: since 999 > 100 the disease spreads, I climbs to a peak, then falls as S is depleted past 100, and the outbreak dies out leaving some people never infected.
The infection peaks and then recedes once susceptibles fall below gamma/beta — not when they hit zero.
SIR assumes recovered people stay immune forever. For diseases where immunity wanes or never forms (so people return to S), the right model is SIS or SIRS, which can sustain the disease indefinitely instead of letting it burn out.