Susceptible-Infected-Recovered

Susceptible-Infected-Recovered (SIR) is a compartmental model in epidemiology that divides a population into three groups: those who can contract the disease (Susceptible), those who have it and can transmit it (Infected), and those who have recovered and are now immune (Recovered). First formulated by Kermack and McKendrick in 1927, the model uses a system of ordinary differential equations to describe how individuals move between compartments over time.

The SIR model is the foundational framework of mathematical epidemiology. It produces the iconic epidemic curve: an initial exponential rise in infections, a peak, and a decline as the susceptible population is depleted. The model reveals a critical threshold — the basic reproduction number, R0 — which determines whether an outbreak will die out or become an epidemic. If R0 is less than 1, each infected person infects fewer than one other person on average, and the disease fades away. If R0 exceeds 1, the disease spreads.

The model's power lies in its simplicity. Its limitation also lies in its simplicity. Real epidemics involve asymptomatic transmission, varying immunity, behavioral change, spatial structure, and network effects that the basic SIR model cannot capture. Extensions like SEIR (adding an Exposed compartment), SIRS (allowing immunity to wane), and network-based models address some of these limitations. But the core insight remains: epidemics are dynamical systems, and their trajectories are determined by the interaction of transmission rates, recovery rates, and population structure — not by individual choices alone.