R0Calc

R₀ of the SIR model: formula and derivation

By Maria Esther Leyva Borges · Last updated

The basic reproduction number of the SIR model is R₀ = β / (γ + μ): the transmission rate β multiplied by the mean time a person stays infectious, 1 / (γ + μ). Without births and deaths (μ = 0) it becomes the well-known R₀ = β / γ.

R0= β γ+μ

LaTeX: R_0 = \frac{\beta}{\gamma + \mu}

What the SIR model describes

The SIR model, introduced by Kermack and McKendrick in 1927, splits a population into three compartments: Susceptible, Infectious and Recovered (or removed). It suits infections where a person becomes infectious soon after being infected and is immune after recovering. Its parameters are:

SymbolMeaningTypical unit
βTransmission rate: new infections caused per infectious person per unit time, when everyone else is susceptible1/day
γRecovery rate; 1/γ is the mean infectious period1/day
μNatural death rate (equal to the birth rate, so the population size stays constant)1/day

The SIR infection diagram

Only infected compartments are needed to compute R₀, and in the SIR model that is just I. In an infection diagram, I is an infectious node with two arrows: a new-infection arrow β that starts and ends at I, because each infectious person directly produces new infectious people, and an outflow arrow γ + μ for those who recover or die. S and R are not drawn: R₀ is defined when the whole population is susceptible, so S stays at its disease-free value and R plays no role.

SIR infection diagram: a new-infection loop β on I and an outflow μ + γ from I β μ + γ I
SIR infection diagram as drawn in R0Calc: one infectious node with a new-infection loop.

How R₀ is derived

The diagram has one infection cycle, I → I. R0Calc divides the rate entering I through new infections, β, by the rate leaving it, γ + μ, which gives R₀ = β / (γ + μ). The next-generation matrix gives the same answer: with one infected compartment, F = β and V = γ + μ, so FV⁻¹ = β / (γ + μ).

The intuition is direct. An infectious person stays infectious for 1 / (γ + μ) time units on average and causes β new infections per unit of time, so over the whole infectious period they cause β / (γ + μ) infections.

Worked example

With β = 0.3 per day, a 10-day infectious period (γ = 0.1 per day) and negligible deaths over the outbreak (μ ≈ 0), R₀ = 0.3 / 0.1 = 3. If the infectious period is cut to 5 days (γ = 0.2), R₀ drops to 1.5. These are illustrative values, not estimates for a specific disease.

Herd immunity threshold: in the SIR model an infection cannot spread once a fraction 1 − 1/R₀ of the population is immune. For R₀ = 3 that is two thirds of the population (Anderson and May, 1991).

Compute it in R0Calc

  1. Open the calculator and add an Infectious compartment; rename it I.
  2. Add a New infection arrow: select I as both start and end, and label it \beta.
  3. Add an Outflow arrow from I with the label \mu + \gamma.
  4. Choose Update YAML (→, or Ctrl+S) and R₀ appears as a formula, ready to copy as LaTeX.

Or paste this model into the YAML panel and apply it to the diagram:

non_infectious_infested: []
infectious_infested:
  - I
new_infections_arrows:
  - from: I
    to: I
    param: \beta
transfer_arrows: []
outs_arrows:
  - from: I
    param: \mu + \gamma

Adding a latent stage turns this into the SEIR model, whose R₀ has an extra factor σ / (σ + μ).

Frequently asked questions

What is R₀ for the SIR model?

R₀ = β / (γ + μ), where β is the transmission rate, γ the recovery rate and μ the natural death rate. Without births and deaths it is β / γ.

Why do S and R not appear in the R₀ calculation?

R₀ is defined at the start of an epidemic, when the whole population is susceptible. Only the infected compartments decide whether the infection grows or dies out, so S is fixed at its disease-free value and R does not enter.

Is R₀ = β / γ the same in the frequency- and density-dependent SIR model?

The formula has the same shape, but β means different things. In the frequency-dependent model β is already the infection rate per infectious person; in the density-dependent (mass-action) model it is β times the disease-free population size N, so R₀ = βN / γ.

References