Concept Architecture
Concept
Theoretically, a Compartmental Model is a mathematical modelling framework that partitions a population into mutually exclusive compartments representing health states, disease stages or epidemiological conditions, with individuals transitioning between compartments over time. Originating in infectious disease epidemiology and systems modelling, compartmental models describe the aggregate dynamics of populations rather than individual patients. In health economics, they are used to estimate disease burden, intervention impact, healthcare resource utilisation and the cost-effectiveness of public health programmes.
Mathematically, compartmental models are represented by systems of ordinary differential equations or discrete-time difference equations describing the rates of movement between compartments. The transition rates determine the evolution of the population over time while conserving the total population unless births, deaths or migration are explicitly incorporated. Common examples include the Susceptible?Infectious?Recovered (SIR) and Susceptible?Exposed?Infectious?Recovered (SEIR) models.
In practice, compartmental models are parameterised using epidemiological, demographic and clinical data. Parameters are estimated from surveillance systems, clinical studies, transmission studies and published literature. The models are implemented to evaluate vaccination programmes, infectious disease control strategies, screening policies and other population-level interventions, with outputs linked to healthcare costs, quality-adjusted life years and budget impact analyses.
Purpose
Used to model disease transmission and population health dynamics, estimate the health and economic consequences of public health interventions, and support cost-effectiveness analyses of population-based healthcare programmes.
Mathematical Formulae
Primary Formula
For the classical SIR model:
dS/dt = ??SI
dI/dt = ?SI ? ?I
dR/dt = ?I
where:
- S = susceptible population
- I = infectious population
- R = recovered population
- ? = transmission rate
- ? = recovery rate
Supporting Formulae
Population conservation:
N = S + I + R
where:
- N = total population
Basic reproduction number:
R? = ?/?
Related Mathematical Methods
- Ordinary differential equations
- Difference equation modelling
- SIR modelling
- SEIR modelling
- Dynamic transmission modelling
- Numerical integration
- Model calibration
Example
A population of 100,000 individuals begins with:
- Susceptible = 99,900
- Infectious = 100
- Recovered = 0
The estimated transmission rate is 0.30 and the recovery rate is 0.10, giving:
R? = 0.30/0.10 = 3.0
The model predicts disease transmission over successive time periods. The resulting numbers of infections, recoveries and healthcare resource use are then combined with treatment costs and QALYs to evaluate the cost-effectiveness of alternative vaccination strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:D2) | Calculates the total population across compartments. |
| EXP | =EXP(-B2*C2) | Supports exponential transition calculations in discrete-time approximations. |
| Solver | Solver Optimisation | Calibrates transmission and recovery parameters to observed epidemiological data. |
| Data Table | What-If Analysis | Evaluates alternative transmission rates or vaccination scenarios. |
VBA (Optional)
Automate numerical simulation of compartmental disease models across multiple time steps and generate health and economic outcomes for alternative intervention strategies.
Sources
- Kermack WO, McKendrick AG. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A. 1927.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Keeling MJ, Rohani P. Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
- ISPOR-SMDM Modeling Good Research Practices Task Force reports.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
Related Concepts (2)
Library
Publications
5
Dynamic Transmission Modeling: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5 — Pitman, Fisman, Zaric, Postma, Kretzschmar, Edmunds & Brisson, Task Force Report 5 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on dynamic transmission models for infectious disease, capturing indirect (herd) effects that static models cannot, and their use in cost-effectiveness analysis of vaccination and control programmes.
Journal ArticleView source →A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →Applying Dynamic Simulation Modeling Methods in Health Care Delivery Research — The SIMULATE Checklist: Report of the ISPOR Simulation Modeling Emerging Good Practices Task Force — Marshall, Burgos-Liz, IJzerman, Osgood, Padula, Higashi, Wong, Pasupathy & Crown, Vol. 18, No. 1 ed., 2015 (Value in Health)
The first ISPOR dynamic-simulation good-practice report, introducing system dynamics, discrete event simulation and agent-based modelling for health care delivery problems and providing the SIMULATE checklist for their application.
Journal ArticleView source →Selecting a Dynamic Simulation Modeling Method for Health Care Delivery Research — Part 2: Report of the ISPOR Dynamic Simulation Modeling Emerging Good Practices Task Force — Marshall, Burgos-Liz, IJzerman, Crown, Padula, Wong, Pasupathy, Higashi & Osgood, Vol. 18, No. 2 ed., 2015 (Value in Health)
The second ISPOR dynamic-simulation report, giving decision guidance on choosing between system dynamics, discrete event simulation and agent-based modelling based on the structure and complexity of the health care delivery problem.
Journal ArticleView source →Methods for Health Economic Evaluation of Vaccines and Immunization Decision Frameworks: A Consensus Framework from a European Vaccine Economics Community — Ultsch, Damm, Beutels, Bilcke, et al., Vol. 34, No. 3 ed., 2016 (PharmacoEconomics)
A consensus framework on the immunisation-specific methodological issues in economic evaluation of vaccines — herd/indirect effects, discounting, dynamic transmission modelling — developed to support national vaccine HTA guidelines in Europe.
Journal ArticleView source →
Media
1
Infectious Disease Modelling Specialization — Imperial College London, 3-course specialization ed., 2023 (Coursera)
An Imperial College London specialization introducing mathematical modelling of infectious disease in R — compartmental and dynamic transmission models — foundational for the economic evaluation of vaccines and control programmes.
Online CourseView source →
Frequently Asked Questions (6)
What is a compartmental model?
A modelling approach dividing a population into distinct compartments, such as susceptible, infected, and recovered, and tracking flow between them over time.
Source: Kermack & McKendrick 1927
Where do compartmental models come from?
The approach originated in early mathematical epidemiology, most influentially in the work of Kermack and McKendrick in the 1920s, who divided a population into groups by infection status and wrote equations for the rates at which people moved between them. Their formulation of susceptible, infected, and recovered groups became the template for later models of disease spread. The same compartmental idea is now used well beyond infectious disease. Kermack and McKendrick (1927) set out the original model.
Source: Kermack & McKendrick 1927
How does a compartmental model work?
A compartmental model works by defining compartments for the relevant states and specifying the rates at which individuals move between them, then following how the numbers in each compartment change over time, usually through differential equations. In the susceptible-infected-recovered model, susceptible individuals become infected at a rate depending on contact with the infected, and infected individuals recover at another rate. Solving the equations traces the course of an epidemic, showing how infection rises and falls in the population.
Source: Kermack & McKendrick 1927
What are the compartments in an SIR model?
In the classic susceptible-infected-recovered, or SIR, model, the population is divided into three compartments: susceptible, those who can catch the disease; infected, those currently infectious; and recovered, those who have had the disease and are now immune. Individuals flow from susceptible to infected as they become infected, and from infected to recovered as they recover. Variants add compartments, such as exposed or vaccinated, but the SIR structure is the foundation of compartmental epidemic modelling.
Source: Kermack & McKendrick 1927
What are the assumptions of a compartmental model?
A compartmental model assumes that individuals within a compartment are alike and mix in a defined way, often that the population mixes homogeneously, so that everyone has the same chance of contact. It represents the population by aggregate flows between compartments rather than tracking individuals, and it uses rates that may be assumed constant. These simplifications make the model tractable but may not capture heterogeneity or structured contact, so their appropriateness for a given disease and population is considered.
Source: Kermack & McKendrick 1927
How are compartmental models used in health?
Compartmental models are used in health to study the spread of infectious disease and to evaluate interventions such as vaccination, treatment, and control measures. By representing how infection moves through susceptible, infected, and recovered compartments, they project the course of an epidemic and how it would change under different measures, capturing effects such as herd immunity. They underlie much infectious-disease modelling and the economic evaluation of interventions whose benefits depend on transmission dynamics.
Source: Kermack & McKendrick 1927
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 29 Sep 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-EM-DM-011
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