R₀ of the SEIR model: formula and derivation

Q: What is R₀ for the SEIR model?

For the SEIR model with transmission rate β, rate of becoming infectious σ, recovery rate γ and natural death rate μ, the basic reproduction number is R₀ = βσ / ((σ + μ)(γ + μ)). Without births and deaths (μ = 0) it simplifies to R₀ = β / γ.

Q: Does the latent period change R₀ in the SEIR model?

Only through deaths during the latent period. The factor σ / (σ + μ) is the probability of surviving the exposed stage; when μ = 0 it equals 1 and the SEIR model has the same R₀ as the SIR model, β / γ. The latent period changes how fast an epidemic grows, not how many people each case infects.

Q: How do you derive R₀ of the SEIR model with the next-generation matrix?

With infected compartments E and I, the new-infection matrix is F = [[0, β], [0, 0]] and the transition matrix is V = [[σ + μ, 0], [−σ, γ + μ]]. The spectral radius of FV⁻¹ is βσ / ((σ + μ)(γ + μ)), which is R₀.

By Maria Esther Leyva Borges · Last updated 20 October 2026

The basic reproduction number of the SEIR model is R₀ = βσ / ((σ + μ)(γ + μ)): the number of new infections one infectious person causes in a fully susceptible population. It is the product of two factors: the infections caused during the infectious period, β / (γ + μ), and the probability of surviving the latent period to become infectious, σ / (σ + μ).

R_{0} = \frac{β σ}{(σ + μ) (γ + μ)}

LaTeX: R_0 = \frac{\beta \sigma}{\left(\gamma + \mu\right) \left(\mu + \sigma\right)}

What the SEIR model describes

The SEIR model splits a population into four compartments: Susceptible, Exposed (infected but not yet infectious), Infectious and Recovered. It fits diseases with a latent period, such as measles or influenza, where a newly infected person cannot transmit straight away. Its parameters are:

Symbol	Meaning	Typical unit
β	Transmission rate: new infections caused per infectious person per unit time, when everyone else is susceptible	1/day
σ	Rate at which exposed people become infectious; 1/σ is the mean latent period	1/day
γ	Recovery rate; 1/γ is the mean infectious period	1/day
μ	Natural death rate (equal to the birth rate, so the population size stays constant)	1/day

From the SEIR equations to the infection diagram

The infection diagram is read off the equations of the infected compartments. With births equal to deaths and frequency-dependent transmission, the SEIR model is:

\begin{aligned} \frac{d S}{d t} & = μ N - \frac{β I S}{N} - μ S \\ \frac{d E}{d t} & = \frac{β I S}{N} - (σ + μ) E \\ \frac{d I}{d t} & = σ E - (γ + μ) I \\ \frac{d R}{d t} & = γ I - μ R \end{aligned}

The infected compartments are E and I. At the disease-free equilibrium (DFE) E = I = R = 0 and S* = N, so S*/N = 1. Each term of the E and I equations becomes one arrow:

Term	Where it appears	Kind	Arrow in the diagram
β I S/N	+ in dE/dt	New infection	Dashed, I → E, label β S*/N = β
σ E	− in dE/dt, + in dI/dt	Transfer	Solid, E → I, label σ
μ E	− in dE/dt	Outflow (death)	Solid, out of E, label μ
γ I	− in dI/dt	Outflow (recovery, goes to R)	Solid, out of I, label γ
μ I	− in dI/dt	Outflow (death)	Solid, out of I, label μ

The two outflows from I are drawn as a single arrow μ + γ. Births μN and the whole equation of R are not in the diagram: they act on uninfected compartments. The term −β I S/N in dS/dt is the other side of the new-infection term, seen from the susceptibles, so it is not drawn twice. See how to build an infection diagram for the general rules.

SEIRS: waning immunity does not change the diagram

If recovered people lose immunity at rate ω (the SEIRS model; Bjørnstad et al., 2020), dS/dt gains + ωR and dR/dt gains − ωR. Neither term touches E or I, so the infection diagram and R₀ are the same as for SEIR. Waning immunity changes the long-term dynamics by refilling S, not the threshold.

The SEIR infection diagram

To compute R₀ you only need the infected compartments, E and I. An infection diagram draws them as nodes: E is non-infectious (it carries the infection but does not transmit it) and I is infectious. Three kinds of arrows connect them: a new-infection arrow β from I to E (each infectious person creates exposed people), a transfer arrow σ from E to I, and outflow arrows for the people who leave each compartment: μ from E, and μ + γ from I.

SEIR infection diagram as drawn in R0Calc. Blue: non-infectious node; red: infectious node.

How R₀ is derived

The diagram has a single infection cycle, I → E → I. R0Calc follows it and multiplies, for each node, the rate coming in by the rate going out. E receives β and loses σ + μ; I receives σ and loses γ + μ. The product is βσ / ((σ + μ)(γ + μ)). Since the cycle passes through one infectious node, that product is already R₀.

The classic next-generation matrix method (Diekmann, Heesterbeek and Metz, 1990; van den Driessche and Watmough, 2002) gives the same result. With infected compartments (E, I), the new-infection matrix is F = [[0, β], [0, 0]] and the transition matrix is V = [[σ + μ, 0], [−σ, γ + μ]]. FV⁻¹ has the eigenvalues 0 and βσ / ((σ + μ)(γ + μ)), and R₀ is the largest of them.

Worked example

Take an infection with β = 0.5 per day, a 5-day latent period (σ = 0.2 per day), a 4-day infectious period (γ = 0.25 per day) and a population whose deaths are negligible over the outbreak (μ ≈ 0). Then R₀ = (0.5 × 0.2) / (0.2 × 0.25) = 2: each case infects two others at the start of the epidemic. These are illustrative values, not estimates for a specific disease.

Why R₀ = 1 is the threshold: if R₀ > 1 the infection can spread in a fully susceptible population; if R₀ < 1 it dies out. In the SEIR model the disease-free equilibrium is locally stable exactly when R₀ < 1 (van den Driessche and Watmough, 2002).

Special case: SEIR without births and deaths

When μ = 0, the formula simplifies to R₀ = β / γ, the same as in the SIR model. Everyone who is exposed eventually becomes infectious, so the latent period does not change how many people each case infects. It only delays transmission, which slows the growth of the epidemic.

Compute it in R0Calc

Open the calculator and choose Load this example in the YAML panel. It loads exactly this SEIR model.
Or paste the YAML below into the YAML panel and apply it to the diagram.
R₀ appears under the diagram as a formula. Copy it as LaTeX, or open the cycle breakdown to see each term.
Edit the rates or add compartments (a second exposed stage, a vector, an asymptomatic class) and R₀ is recomputed.

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

Frequently asked questions

What is R₀ for the SEIR model?

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

Does the latent period change R₀ in the SEIR model?

Only through deaths during the latent period. The factor σ / (σ + μ) is the probability of surviving the exposed stage; when μ = 0 it equals 1. The latent period changes how fast an epidemic grows, not how many people each case infects.

How do you derive R₀ of the SEIR model with the next-generation matrix?

Use F = [[0, β], [0, 0]] for new infections and V = [[σ + μ, 0], [−σ, γ + μ]] for transitions between E and I. The spectral radius of FV⁻¹ is βσ / ((σ + μ)(γ + μ)).

References

Diekmann O., Heesterbeek J. A. P., Metz J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28, 365–382.
van den Driessche P., Watmough J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29–48.
Anderson R. M., May R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
Bjørnstad O. N., Shea K., Krzywinski M., Altman N. (2020). The SEIRS model for infectious disease dynamics. Nature Methods, 17, 557–558.