Compartmental models in epidemiology

Compartmental models in epidemiology are a family of mathematical models that divide a population into discrete compartments — typically Susceptible (S), Infected (I), and Recovered (R) — and describe the rates of transfer between these compartments using systems of ordinary differential equations. The SIR model is the canonical example, but the framework extends to include exposed (SEIR), asymptomatic, vaccinated, and spatially structured variants.

These models are not mere curve-fitting exercises. They are the macroscopic limit of individual-based stochastic dynamics, rigorously derivable under conditions of propagation of chaos where each individual interacts with a representative sample of the population. The basic reproduction number R₀ emerges naturally from the linearization of the compartmental system around the disease-free equilibrium, and its value relative to unity determines whether an epidemic will grow or die out.

Despite their parsimony, compartmental models capture the essential nonlinearities of epidemic dynamics: the depletion of susceptibles, the threshold behavior at R₀ = 1, and the herd immunity threshold. Their limitations — homogeneous mixing assumptions, lack of network structure, and deterministic approximation — are precisely what motivate extensions to network-based and agent-based models.