SIR Compartmental Epidemic Model
Also known as: Kermack–McKendrick Model, Susceptible-Infectious-Recovered Model, Compartmental Epidemic Model, SIR Epidemiyoloji Modeli
The SIR model is a foundational mathematical framework for describing the spread of infectious diseases through a population. Introduced by William Ogilvy Kermack and Anderson Gray McKendrick in 1927, it partitions a closed population of size N into three mutually exclusive compartments: Susceptible (S), Infectious (I), and Recovered (R). A system of ordinary differential equations governs the flow of individuals between compartments, capturing epidemic dynamics with two key parameters — the transmission rate β and the recovery rate γ.
Key highlights
- Analytically tractable: closed-form expressions for the epidemic peak and final size exist.
- Parameter-sparse: only β and γ are needed, making fitting feasible with limited surveillance data.
- Provides the pivotal R₀ threshold that guides vaccine coverage and control policy.
- Widely validated across historical outbreaks, giving results an interpretable epidemiological meaning.
Intuition
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How it works
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When to use it
Use the SIR model when modelling diseases that confer lasting immunity after infection (e.g., measles, influenza within a season) in a well-mixed, closed population. Key assumptions include homogeneous mixing, a constant population (no births/deaths beyond the epidemic timescale), and permanent immunity post-recovery. It is unsuitable for diseases with waning immunity, strong age or spatial structure, or stochastic dynamics in small populations. For latency periods, use SEIR; for demography, use SIS or vital-dynamics extensions.
Strengths & limitations
- Analytically tractable: closed-form expressions for the epidemic peak and final size exist.
- Parameter-sparse: only β and γ are needed, making fitting feasible with limited surveillance data.
- Provides the pivotal R₀ threshold that guides vaccine coverage and control policy.
- Widely validated across historical outbreaks, giving results an interpretable epidemiological meaning.
- Assumes homogeneous mixing, ignoring age structure, spatial heterogeneity, and social networks.
- Permanent immunity assumption fails for diseases like COVID-19 with waning or partial immunity.
- Deterministic formulation cannot capture stochastic fade-out in small or isolated populations.
- Ignores the latent (exposed but not yet infectious) period, overestimating early epidemic speed.
Common pitfalls
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Applications
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Frequently asked
What is the difference between β and R₀?
β is the per-capita transmission rate (contacts per unit time × probability of transmission per contact), while R₀ = β/γ normalises β by the recovery rate to give the average number of secondary infections per primary case in a fully susceptible population. R₀ is dimensionless and comparable across settings; β is not.
Can the SIR model predict the exact size of an epidemic?
Yes, the deterministic SIR model has a transcendental equation for the final epidemic size (fraction ultimately infected) as a function of R₀ and the initial susceptible fraction. However, this prediction assumes the model's simplifying assumptions hold; deviations — heterogeneity, behavioural change, interventions — can cause substantial discrepancies from observed outcomes.
When should I use SEIR instead of SIR?
Use SEIR when the disease has a meaningful latent period during which infected individuals are not yet infectious (e.g., COVID-19 incubation ≈ 5 days, measles ≈ 8–12 days). The exposed compartment E delays the epidemic curve and lowers peak incidence compared with SIR. For diseases with very short latency (e.g., some influenza strains), SIR is often a reasonable approximation.
Sources
- 1.Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A, 115(772), 700–721.
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ScholarGate. (2026, June 2). SIR Model. ScholarGate. https://scholargate.app/epidemiology/sir-model