SIR Compartmental Models for Infectious Disease

Core Idea

The SIR model divides a population into Susceptible, Infected, and Recovered compartments and uses differential equations to model transitions. The force of infection (β × I/N) drives susceptible → infected transitions; the recovery rate (γ) drives infected → recovered transitions. SIR models predict epidemic dynamics, peak timing, and final size, forming the basis for control strategy evaluation.

Explainer

From your prerequisite on the basic reproduction number R₀, you understand that epidemic spread depends on the average number of secondary infections generated by a single case in a fully susceptible population, and that R₀ > 1 is necessary for an outbreak to grow. The SIR model gives R₀ a mechanistic derivation and explains its components: R₀ = β/γ, where β is the rate at which an infected individual transmits to each susceptible contact and γ is the recovery rate (the reciprocal of the average infectious period). Rather than treating R₀ as a black-box quantity estimated from case counts, the SIR model shows *why* the epidemic threshold depends on this ratio and predicts the full temporal trajectory — when the epidemic peaks, how large it gets, and what fraction of the population ultimately escapes infection.

The SIR model divides a closed population of size N into three compartments. S (susceptible) individuals can be infected; I (infectious) individuals can transmit; R (recovered) individuals are immune and no longer participate in transmission. The core dynamic is driven by the force of infection: the per-capita rate at which susceptibles become infected equals β × (I/N) — the transmission rate times the fraction of the population currently infectious. This gives dS/dt = −β(I/N)S and dI/dt = β(I/N)S − γI. The epidemic grows when dI/dt > 0, which requires (βS/N)/γ > 1 — equivalently, S/N > 1/R₀. This is the epidemic threshold: an outbreak expands when the susceptible fraction exceeds 1/R₀. The complement, 1 − 1/R₀, is the herd immunity threshold — the minimum immune fraction needed for the epidemic to self-extinguish.

The epidemic trajectory has a characteristic shape. Initially, with nearly the entire population susceptible, I grows approximately exponentially at rate β − γ = γ(R₀ − 1). As the epidemic proceeds, susceptibles are depleted, the force of infection weakens, and the I curve bends over. The peak of I occurs exactly when S/N = 1/R₀ — the moment the herd immunity threshold is first crossed. After the peak, I declines even though many people remain susceptible, because the susceptible pool has been depleted enough that new infections no longer outpace recoveries. Crucially, the epidemic ends before the entire susceptible population is infected: a fraction of susceptibles always survives uninfected, "saved" not by immunity but by the geographic depletion of infectious individuals before they could reach them. The final size equation — ln(S∞/S₀) = −R₀(1 − S∞/N) — gives the exact proportion ultimately infected as a function of R₀ alone.

Each intervention maps directly onto a model parameter. Vaccination reduces S before the epidemic starts, raising the effective immune fraction and — if vaccination coverage reaches the herd immunity threshold — preventing epidemic growth entirely. Isolation and treatment shorten the infectious period (reducing 1/γ, thus raising γ and lowering R₀). Social distancing and masking reduce β by decreasing the contact rate or per-contact transmission probability. This parameter-level clarity is why the SIR model is the standard foundation for public health modeling: interventions can be compared quantitatively, and the relative contribution of different strategies is explicit. The model's simplifying assumptions — constant β and γ, homogeneous random mixing, permanent immunity — are relaxed in extensions you will study next (the SEIR model adds an exposed/latent compartment E for diseases with incubation periods), but the core logic of thresholds, depletion dynamics, and intervention mapping originates here.