Compartmental models (epidemiology)

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,^[1] Ross's contributions in 1916,^[2] collaborative work by Ross and Hudson in 1917,^[3][4] the seminal Kermack and McKendrick model in 1927,^[5] and Kendall's work in 1956.^[6] The historically significant Reed–Frost model, though often overlooked, also substantially influenced modern epidemiological modeling approaches.^[7]

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.