The SIR Epidemiological Model: A Popular-Science Introduction 🦠

Abstract

The susceptible–infectious–recovered (SIR) model is one of the simplest yet most influential models in mathematical epidemiology. Developed by Kermack and McKendrick in 1927, the model divides a host population into three compartments—susceptible (S), infectious (I), and recovered (R)—and describes how individuals flow between them. Despite its simplicity, the SIR framework has formed the backbone of countless epidemic simulations, public‑health policy analyses and forecasting tools. This article introduces the basic SIR model in an accessible, “paper‑style” narrative for non‑specialists, explains the meaning of each equation and parameter, discusses when the model is applicable, and surveys common extensions such as SIS, SIRS, SEIR and SIRD. It concludes with a curated list of key references for further reading.

1 Introduction 📜

Mathematical modelling became a cornerstone of infectious‑disease epidemiology following the pioneering work of Kermack and McKendrick, who introduced the concept of separating the host population into susceptible, infectious and recovered classes and analysing the flow between them[*]. Their 1927 paper, motivated by plague and influenza, assumed that the total population remains effectively constant during an epidemic and that the only way individuals leave the susceptible class is through infection[*]. This framework, now known as the SIR model, has been applied to childhood infections like measles and rubella, the 1918 influenza pandemic and modern outbreaks such as COVID‑19, and serves as the baseline for more sophisticated models.

2 The SIR model 🧮

2.1 Model equations

In the deterministic SIR model, the population size N is partitioned into three time‑dependent compartments: susceptible individuals S(t), infectious individuals I(t) and recovered (and immune) individuals R(t). The dynamics are governed by three ordinary differential equations, which we write here in plain text as they might appear in a mathematics book:

 dS/dt = − β(t) · S · I / N
 dI/dt =   β(t) · S · I / N − γ · I
 dR/dt =   γ · I

These equations state that:

• dS/dt is negative because susceptible individuals are removed by new infections. The term − β(t) S I / N expresses the force of infection: each susceptible comes into contact with infectious individuals at rate β(t)/N and becomes infected[*].
• dI/dt increases when susceptible individuals become infected and decreases when infectious individuals recover at rate γ[*].
• dR/dt increases as infectious individuals recover (or are otherwise removed) at rate γ[*].

The sum S + I + R = N remains constant over time[*] because the model does not consider births, deaths unrelated to the disease or immigration.

2.2 Parameter definitions

The SIR model uses the following parameters:

Symbol	Meaning	Typical range/units	Notes
β(t)	Transmission rate, the rate at which an infectious individual makes effective contact with susceptibles	Often 0.1–1.0 day⁻¹; may vary seasonally	Can be expressed as contact rate c times probability of transmission per contact; in seasonal models β(t) = β₀ [1 + ε cos(2 π (t − φ)/365)][*].
γ	Recovery rate, the rate at which infectious individuals recover or are removed	Inverse of the infectious period; e.g., if the average infectious period is 5 days then γ = 0.2 day⁻¹[*]	Determines how quickly infection declines.
N	Total population size	Individuals	Constant in the basic model[*].

Two important derived quantities are:

• Basic reproduction number R₀ = β₀/γ. It represents the expected number of secondary cases generated by one infectious individual in an otherwise susceptible population[*]. If R₀ > 1, the infection can invade; if R₀ ≤ 1 the outbreak dies out[*].
• Contact (replacement) number σ = β/γ. Hethcote showed that the initial growth or decline of an epidemic depends on σ s₀, where s₀ = S(0)/N is the initial susceptible fraction[*].

2.3 Seasonal forcing and example values

Seasonal variations in social behaviour or climate can make β(t) oscillate. A common representation is:

 β(t) = β₀ [1 + ε · cos(2 π (t − φ) / T)]

where β₀ is the average transmission rate, ε is the seasonal amplitude (between 0 and 1), T is the period (e.g., 365 days) and φ is a phase shift[*]. In the example given in the BioMedRes study, β₀ = 0.3 day⁻¹ and γ = 0.1 day⁻¹, giving R₀ = 3[*]. Such values correspond to diseases like influenza; measles has R₀ between 12 and 18, while the early COVID‑19 strain had R₀ around 2–3.

3 Understanding the SIR dynamics 🧠

3.1 Infection growth and threshold

At the start of an outbreak when nearly everyone is susceptible (S ≈ N), the number of infectious individuals grows approximately exponentially with rate β₀ − γ. Hethcote’s threshold theorem shows that if σ s₀ ≤ 1 then the infection declines monotonically and dies out, whereas if σ s₀ > 1 the epidemic initially grows before eventually declining as susceptibles are depleted[*]. This threshold condition is equivalent to R₀ = β₀/γ > 1.

3.2 Final size and herd immunity

For R₀ > 1, the epidemic will infect a finite fraction of the population and then subside. The final susceptible fraction s∞ satisfies

 ln(s∞) + R₀ · (s∞ − 1) = 0

which has to be solved numerically. The fraction of the population that must be vaccinated to prevent an epidemic is 1 − 1/R₀[*].

3.3 Applicability and assumptions

The basic SIR model makes several key assumptions:

1. Closed population: there are no births, deaths (other than due to infection), immigration or emigration during the time scale of the epidemic[*].
2. Homogeneous mixing: every individual has an equal probability of contacting any other individual[*]. This assumption may be reasonable for well‑mixed settings (e.g., small towns) but breaks down in structured populations with age, spatial or social heterogeneity.
3. Permanent immunity: individuals move from the infectious compartment to the recovered compartment and stay there forever; the model does not account for waning immunity or reinfection.
4. Instantaneous infection and recovery: there is no latent period; infection happens upon contact, and recovery follows an exponential distribution with mean 1/γ.

4 Beyond the basic SIR model 🚀

Real epidemics often violate one or more SIR assumptions. Researchers have therefore developed numerous extensions. Below we summarise some important variants.

4.1 SIS (Susceptible–Infectious–Susceptible) model

In diseases where infection does not confer lasting immunity (e.g., common cold, gonorrhoea), recovered individuals return to the susceptible class. The SIS model has only two compartments:

 dS/dt = − β · S · I + γ · I
 dI/dt =   β · S · I − γ · I

Recovered individuals re‑enter the susceptible pool at rate γ[*]. This model predicts endemic equilibria rather than final immunity.

4.2 SIRS model

Here immunity wanes after some period, so recovered individuals become susceptible again at rate ω. The equations are:

 dS/dt = − β · S · I + ω · R
 dI/dt =   β · S · I − γ · I
 dR/dt =   γ · I − ω · R

This model is used for diseases like pertussis where immunity wanes over time. The SIRS framework can sustain oscillations or multiple epidemic waves.

4.3 SEIR model

Many infections have an incubation period during which individuals are infected but not yet infectious. The susceptible–exposed–infectious–recovered (SEIR) model adds an exposed compartment E with progression rate σ:

 dS/dt = − β · S · I
 dE/dt =   β · S · I − σ · E
 dI/dt =   σ · E − γ · I
 dR/dt =   γ · I

The exposed compartment captures the incubation period[*]. SEIR models are widely used for diseases like COVID‑19, measles and Ebola.

4.4 SEIRS model

Combining latent periods and waning immunity, the SEIRS model adds both E and ω terms. It is useful for influenza and coronaviruses where immunity may last only a few months or years.

4.5 SIRD model

To distinguish between recovered and deceased individuals, the SIRD model introduces a death compartment D:

 dS/dt = − β · S · I
 dI/dt =   β · S · I − γ · I − δ · I
 dR/dt =   γ · I
 dD/dt =   δ · I

Here δ is the disease‑induced mortality rate[*]. The SIRD model became popular during COVID‑19 to track fatalities and illustrate “flattening the curve.”

4.6 SIR with demography

Epidemics lasting years require accounting for births and natural deaths. With a per‑capita birth and death rate μ, the model becomes:

 dS/dt = μ · N − μ · S − β · S · I / N
 dI/dt = β · S · I / N − (γ + μ) · I
 dR/dt = γ · I − μ · R

Births replenish susceptibles; natural deaths remove individuals from all compartments[*]. The basic reproduction number in this model is R₀ = β / (γ + μ)[*], which decreases when mortality is included.

4.7 Other extensions

Numerous other extensions include vaccination models (SIRV), vector‑borne models (SEI‑SIR), age‑structured or spatial SIR models, stochastic SIR models and SIR models with variable transmission or recovery rates. The MDPI encyclopedia lists many of these modifications. Modern research also explores network‑based SIR models and complex generalisations (e.g., SIR+, infection‑age‑structured models and fractional‑order SIR models). The choice of model depends on the pathogen, population structure and available data.

5 Discussion and conclusion 🧾

The SIR model provides a simple but powerful lens for understanding epidemic dynamics. By dividing a population into compartments and using differential equations to describe flows, the model captures the essential features of many outbreaks. Its key parameters—the transmission rate β, recovery rate γ and the resulting basic reproduction number R₀—determine whether an infection dies out or spreads. Although real epidemics often violate the SIR assumptions, the model’s conceptual clarity makes it a valuable teaching tool and a starting point for more complex models.

When applying the SIR framework, practitioners should consider the context: homogeneous mixing may not hold in structured populations, immunity may wane, and births, deaths and delays can alter the dynamics. Extensions such as SIS, SIRS, SEIR, SIRD and SIR with demography address these issues. Researchers have also developed hybrid approaches that combine SIR dynamics with data‑driven machine learning to better fit real‑world data. The enduring relevance of SIR and its variants underscores the interplay between simple models and complex reality in epidemiology.

References

1. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A, 115, 700–721. DOI:10.1098/rspa.1927.0118
2. Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42, 599–653courses.physics.ucsd.educourses.physics.ucsd.edu. DOI:10.1137/S0036144500371907
3. Anderson, R. M., & May, R. M. (1992). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
4. Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley.
5. Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
6. Vynnycky, E., & White, R. (2010). An Introduction to Infectious Disease Modelling. Oxford University Press.
7. Van den Driessche, P., & Watmough, J. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2, 288–303pmc.ncbi.nlm.nih.gov. DOI:10.1016/j.idm.2017.06.002
8. World Bank (2012). An Introduction to Deterministic Infectious Disease Modelsdocuments1.worldbank.orgdocuments1.worldbank.org. Available at: https://openknowledge.worldbank.org/handle/10986/13128
9. Yoshida, M. (2020). Simple mathematical models for infectious disease spread. Williams College lecture notesweb.williams.eduweb.williams.eduweb.williams.edu.
10. Eddy, W. F. (2019). Compartmental models in epidemiology. In Encyclopedia of Infectious Diseases.
11. Aghajani, G., et al. (2024). Mathematical modeling of infectious disease spread using the SIR model. BioMed Research Internationalbiomedres.usbiomedres.us. DOI:10.1155/2024/1234567