Compartmental models are a mathematical framework used to simulate how populations move between different states or "compartments". While widely applied in various fields, they have become particularly fundamental to the mathematical modelling of infectious diseases. In these models, the population is divided into compartments labeled with shorthand notation – most commonly S, I, and R, representing Susceptible, Infectious, and Recovered individuals. The sequence of letters typically indicates the flow patterns between compartments; for example, an SEIS model represents progression from susceptible to exposed to infectious and then back to susceptible again.

These models originated in the early 20th century through pioneering epidemiological work by several mathematicians. Key developments include Hamer's work in 1906, Ross's contributions in 1916, collaborative work by Ross and Hudson in 1917, the seminal Kermack and McKendrick model in 1927, and Kendall's work in 1956. The historically significant Reed–Frost model, though often overlooked, also substantially influenced modern epidemiological modeling approaches.

Most implementations of compartmental models use ordinary differential equations (ODEs), providing deterministic results that are mathematically tractable. However, they can also be formulated within stochastic frameworks that incorporate randomness, offering more realistic representations of population dynamics at the cost of greater analytical complexity.

Epidemiologists and public health officials use these models for several critical purposes: analyzing disease transmission dynamics, projecting the total number of infections and recoveries over time, estimating key epidemiological parameters such as the basic reproduction number (R₀) or effective reproduction number (R_t), evaluating potential impacts of different public health interventions before implementation, and informing evidence-based policy decisions during disease outbreaks. Beyond infectious disease modeling, the approach has been adapted for applications in population ecology, pharmacokinetics, chemical kinetics, and other fields requiring the study of transitions between defined states. For such investigations and to consult decision makers, often more complex models are used.

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps, and rubella.

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious, and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t), and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control. Note that in the SIR model, ${\displaystyle R(0)}$ and ${\displaystyle R_{0}}$ are different quantities – the former describes the number of recovered at t = 0 whereas the latter describes the ratio between the frequency of contacts to the frequency of recovery.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.^{[citation needed]}