Compartments of Contagion
In 1927, William Kermack and Anderson McKendrick published a landmark paper establishing the mathematical framework for epidemic modeling. Their SIR model partitions a population into Susceptible, Infected, and Recovered compartments, with flows governed by the transmission rate β and recovery rate γ. This elegant abstraction captures the essential dynamics of outbreaks from influenza to measles.
The Threshold Theorem
The model's most powerful prediction is the epidemic threshold: an outbreak can only occur if R₀ = β/γ exceeds 1. Below this threshold, each case generates fewer than one secondary infection, and the disease fades away. Above it, exponential growth ensues until susceptibles are sufficiently depleted. This threshold concept underpins all vaccination strategies — immunize enough people to push the effective reproduction number below 1.
Anatomy of an Epidemic Curve
A typical SIR epidemic follows a characteristic trajectory: slow initial growth while infections are rare, explosive exponential rise as chains of transmission multiply, a peak when susceptibles drop to N/R₀, and gradual decline as the virus runs out of fuel. The final size of the epidemic — the total fraction infected — depends only on R₀ and can be computed from a transcendental equation. For R₀ = 3, roughly 94% of the population is eventually infected.
From SIR to Modern Epidemiology
The basic SIR model has spawned an entire family of compartment models: SEIR adds an Exposed (latent) class, SIS allows reinfection, SIRV includes vaccination, and agent-based models capture individual heterogeneity. During the COVID-19 pandemic, extensions of these models guided lockdown policies, hospital capacity planning, and vaccine rollout strategies worldwide.