Concept Architecture
Concept
Theoretically, the SIR Model is a compartmental infectious disease transmission model that describes the movement of individuals through three epidemiological states: Susceptible (S), Infectious (I), and Recovered (R). It is founded on deterministic compartmental modelling and mathematical epidemiology, assuming homogeneous mixing within a population and permanent immunity following recovery. The model exists to describe the dynamics of infectious disease transmission and to evaluate the impact of interventions such as vaccination, treatment and non-pharmaceutical control measures. In health economics, the SIR model is widely used to estimate disease burden, healthcare demand and the cost-effectiveness of infectious disease interventions.
Mathematically, the SIR model is represented by a system of coupled ordinary differential equations describing the rates of transition between the susceptible, infectious and recovered compartments. Disease transmission occurs according to the contact rate between susceptible and infectious individuals, while recovery occurs at a constant recovery rate. The resulting equations estimate the temporal evolution of disease spread and are generally solved numerically because closed-form analytical solutions are not available for most practical applications.
In practice, the SIR model is implemented by estimating transmission and recovery parameters from epidemiological data, specifying initial population conditions, and numerically integrating the differential equations over time. In health economics it is applied to evaluate vaccination programmes, epidemic control strategies, healthcare resource planning, infectious disease screening policies and economic evaluations of interventions designed to reduce disease transmission.
Purpose
Used to model infectious disease transmission, estimate epidemic dynamics, forecast healthcare demand and evaluate the economic consequences of infectious disease interventions.
Mathematical Formulae
Primary Formula
dS/dt = ??SI/N
dI/dt = ?SI/N ? ?I
dR/dt = ?I
where:
- S = susceptible population
- I = infectious population
- R = recovered population
- N = S + I + R
- ? = transmission rate
- ? = recovery rate
Supporting Formulae
Basic reproduction number:
R? = ?/?
Average 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 50,000 individuals is modelled with:
- Initial susceptible population = 49,990
- Initial infectious population = 10
- Initial recovered population = 0
Estimated parameters are:
- ? = 0.30
- ? = 0.10 per day
The basic reproduction number is
R? = 0.30/0.10 = 3.0.
Numerical integration of the SIR equations predicts the epidemic trajectory, cumulative infections and healthcare demand. These outputs are combined with costs and quality-adjusted life years to estimate the cost-effectiveness of vaccination and treatment strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Euler update | =B2+(-$H$2*B2*C2/$H$1)*$H$5 | Updates the susceptible compartment over one time step. |
| Euler update | =C2+($H$2*B2*C2/$H$1-$H$3*C2)*$H$5 | Updates the infectious compartment. |
| Euler update | =D2+($H$3*C2)*$H$5 | Updates the recovered 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 SIR differential equations, parameter calibration and probabilistic sensitivity analyses for 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
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What is an SIR model?
A basic epidemic model dividing a population into susceptible, infectious, and recovered compartments, representing flow between them using differential equations.
Source: Kermack & McKendrick 1927
How do people move through the compartments of an SIR model?
In an SIR model everyone begins susceptible, and contact with an infectious person can move a susceptible individual into the infectious group. After a period of infectiousness, an infected person moves on to the recovered group, where they are assumed immune and can no longer catch or spread the disease. The model tracks the size of each group over time using equations for the rates of these one-way moves. The shifting balance between the groups reproduces the rise and fall of an epidemic. Keeling and Rohani (2008) describe these flows.
Source: Keeling & Rohani 2008
What are the compartments in an SIR model?
The SIR model has three compartments: susceptible, those who can catch the disease; infectious, those currently able to transmit it; and recovered, those who have had the disease and are now immune. Individuals flow from susceptible to infectious as they become infected, at a rate depending on contact with the infectious, and from infectious to recovered as they recover. These three compartments and the flows between them capture the core dynamics of an epidemic in a homogeneously mixing population.
Source: Kermack & McKendrick 1927
How does an SIR model represent an epidemic?
An SIR model represents an epidemic through the changing numbers in its compartments over time, governed by differential equations. The rate at which susceptibles become infectious depends on the number of infectious individuals and the contact rate, so infection rises as the infectious pool grows and falls as susceptibles are depleted and individuals recover. Solving the equations traces the epidemic curve, the rise and fall of infection, capturing how an outbreak grows, peaks, and declines as immunity accumulates in the population.
Source: Kermack & McKendrick 1927
What is the basic reproduction number in an SIR model?
The basic reproduction number, often denoted R0, is the average number of secondary infections one infectious individual produces in a fully susceptible population, and it is central to the SIR model. If R0 exceeds one, an epidemic can grow, since each case produces more than one further case, while if it is below one, infection dies out. R0 depends on the contact rate, transmission probability, and infectious period. It determines whether an outbreak spreads and relates to the level of immunity needed to prevent it.
Source: Kermack & McKendrick 1927
What are the limitations of the SIR model?
The basic SIR model makes simplifying assumptions that may not hold: it typically assumes a homogeneously mixing population where everyone has equal contact, constant rates, and immediate infectiousness on infection, and it omits structure such as age, spatial spread, and latent periods. Real populations mix unevenly and diseases vary in these respects. These simplifications make the model tractable but limit its realism, so extensions add compartments and structure, such as the exposed compartment of the SEIR model, where the basic assumptions are inadequate.
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
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/sir-model
- Term code
- HE-EM-DM-092
Stable URI · Machine-readable · Resolvable · CC BY 4.0