Concept Architecture
Concept
Theoretically, the SEIR Model is a compartmental infectious disease transmission model that describes the movement of individuals through four epidemiological states: Susceptible (S), Exposed (E), Infectious (I), and Recovered (R). It extends the classic SIR model by incorporating a latent period during which infected individuals are infected but not yet infectious. The model is founded on deterministic compartmental modelling and mathematical epidemiology and exists to represent the dynamics of diseases with an incubation period, enabling the evaluation of disease spread and intervention strategies within populations. In health economics, the SEIR model is widely used to estimate disease burden, forecast healthcare demand, and assess the cost-effectiveness of infectious disease interventions.
Mathematically, the SEIR model is represented by a system of coupled ordinary differential equations describing the rates of change between compartments. Individuals move from the susceptible to exposed compartment according to the force of infection, progress from exposed to infectious at the incubation rate, and recover at the recovery rate. The equations estimate the temporal evolution of disease transmission and are commonly solved numerically because analytical solutions are generally unavailable.
In practice, the SEIR model is implemented by estimating epidemiological parameters from surveillance, clinical, or observational data, specifying initial population conditions, and numerically solving the differential equations over time. In health economics it is applied to evaluate vaccination programmes, screening strategies, quarantine policies, pharmaceutical interventions, pandemic preparedness, healthcare capacity planning, and economic evaluations of infectious disease control programmes.
Purpose
Used to model infectious disease transmission with an explicit latent period, estimate the effects of interventions, forecast disease burden, and support economic evaluation of infectious disease policies.
Mathematical Formulae
Primary Formula
dS/dt = ??SI/N
dE/dt = ?SI/N ? �E
dI/dt = �E ? ?I
dR/dt = ?I
where:
- S = susceptible population
- E = exposed population
- I = infectious population
- R = recovered population
- N = S + E + I + R
- ? = transmission rate
- � = progression rate from exposed to infectious
- ? = recovery rate
Supporting Formulae
Basic reproduction number:
R? = ?/?
Incubation period:
T? = 1/�
Infectious period:
T? = 1/?
Related Mathematical Methods
- Compartmental modelling
- Ordinary differential equations
- Numerical integration
- Numerical optimisation
- Model calibration
- Maximum likelihood estimation
- Bayesian parameter estimation
- Sensitivity analysis
Example
A population of 100,000 individuals is modelled with:
- Initial susceptible population = 99,990
- Initial exposed population = 5
- Initial infectious population = 5
- Initial recovered population = 0
Estimated parameters:
- ? = 0.36
- � = 0.20 per day
- ? = 0.14 per day
The basic reproduction number is
R? = 0.36/0.14 = 2.57.
Numerical integration of the SEIR equations predicts the epidemic trajectory, cumulative infections, healthcare demand, and intervention impacts. These outputs are subsequently combined with costs and quality-adjusted life years to estimate the cost-effectiveness of alternative vaccination or control strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Euler update | =B2+(-$H$2*B2*D2/$H$1)*$H$6 | Updates the susceptible compartment over one time step. |
| Euler update | =C2+($H$2*B2*D2/$H$1-$H$3*C2)*$H$6 | Updates the exposed compartment. |
| Euler update | =D2+($H$3*C2-$H$4*D2)*$H$6 | Updates the infectious compartment. |
| Solver Add-in | Minimise calibration error by changing ?, � and ? | Estimates transmission parameters from observed epidemiological data. |
VBA (Optional)
A VBA procedure can automate numerical solution of the SEIR differential equations, parameter calibration, and probabilistic sensitivity analyses across multiple intervention scenarios.
Sources
- Kermack WO, McKendrick AG. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A. 1927.
- Hethcote HW. The Mathematics of Infectious Diseases. SIAM Review. 2000.
- Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on decision-analytic modelling.
Related Concepts (2)
Library
Publications
1
Conceptualizing a Model: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2 — Roberts, Russell, Paltiel, Chambers, McEwan & Krahn, Task Force Report 2 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on model conceptualisation — defining the decision problem, scoping, and choosing an appropriate model structure before implementation.
Journal ArticleView source →
Frequently Asked Questions (6)
What is an SEIR model?
An extension of the basic epidemic model adding an exposed compartment for individuals infected but not yet infectious, before they become infectious.
Source: Kermack & McKendrick 1927
For which diseases is an SEIR model suited?
An SEIR model suits infections in which a person who has caught the disease does not become infectious straight away but passes through a latent period first. Diseases with a meaningful delay between infection and infectiousness, such as measles or tuberculosis, are represented more faithfully when that waiting stage is modelled explicitly. Where the latent period is negligible, a simpler model without it suffices. Matching the model to the disease's natural timing is what determines the choice. Keeling and Rohani (2008) discuss when the extra compartment is needed.
Source: Keeling & Rohani 2008
What are the compartments in an SEIR model?
The SEIR model has four compartments: susceptible, those who can catch the disease; exposed, those who have been infected but are not yet infectious, in the latent period; infectious, those who can transmit the disease; and recovered, those who have had the disease and are immune. Individuals flow from susceptible to exposed on infection, from exposed to infectious as the latent period ends, and from infectious to recovered on recovery. The added exposed compartment distinguishes SEIR from the simpler SIR model.
Source: Kermack & McKendrick 1927
How does an SEIR model differ from an SIR model?
An SEIR model differs from an SIR model by adding an exposed compartment for individuals who are infected but not yet infectious, representing a latent period between infection and infectiousness. In an SIR model, individuals become infectious immediately upon infection, whereas in an SEIR model they first pass through the exposed stage. This makes SEIR more appropriate for diseases with a meaningful latent period, since it captures the delay before infected individuals can transmit, which affects the timing and dynamics of an epidemic.
Source: Kermack & McKendrick 1927
When is an SEIR model appropriate?
An SEIR model is appropriate for infectious diseases with a significant latent period, during which infected individuals are not yet infectious, so that the delay between infection and transmission materially affects the epidemic's dynamics. Diseases where people incubate the infection before becoming contagious are better represented by SEIR than by SIR, which assumes immediate infectiousness. Where the latent period is negligible or unimportant for the question, the simpler SIR model may suffice, so the exposed compartment is added when the latency matters.
Source: Kermack & McKendrick 1927
In the SEIR model, what does the exposed compartment represent?
The exposed compartment in an SEIR model represents individuals who have been infected but are not yet infectious, that is, in the latent period between catching the infection and being able to transmit it. These individuals will progress to the infectious compartment after a delay reflecting the latent period. Including this compartment captures the incubation delay before infectiousness, which affects how quickly an epidemic grows and the timing of its course, making the model more realistic for diseases with such a delay.
Source: Kermack & McKendrick 1927
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Oct 2025
Content version: 1.0.0
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