R₀ Calculator
The basic reproduction number of an outbreak from how fast cases are doubling and how long each generation of infection takes — with the growth rate and the share of the population that would need immunity to stop it.
Cases doubling every T_d days grow at a rate r = ln 2 ÷ T_d.
How the r₀ calculator works
Cases doubling every T_d days grow at a rate r = ln 2 ÷ T_d. If each infection produces its offspring after a generation interval T_g, the number of secondary cases per case is R = e^(r × T_g) — the exponential-growth estimate of R₀ used early in an epidemic. A three-day doubling with a five-day generation interval is an R₀ near 3.2. The herd-immunity threshold follows as 1 − 1 ÷ R₀.
Formula: r = ln 2 ÷ T_d; R₀ = e^(r × T_g); HIT = 1 − 1 ÷ R₀
Worked examples
| Inputs | R₀ | Note |
|---|---|---|
| Doubling in 3 days, 5-day generations | 3.17 | R₀ ≈ 3.17 |
| Doubling in 7 days, 4-day generations | 1.49 | R₀ ≈ 1.49 |
| Measles-like: doubling in 2 days, 12-day interval | 64 | R₀ = 64 — the formula overshoots for long intervals |
FAQFrequently asked questions
Is this the true R₀?
It is the exponential-growth estimate used in the first weeks of an outbreak, and it depends heavily on the generation interval — a distribution, not one number. Later estimates from contact tracing or final size are more robust.
What is the difference between R₀ and Rₜ?
R₀ is the reproduction number in a fully susceptible population with no interventions; Rₜ is the effective number at time t, reduced by immunity and behaviour. Once Rₜ is below 1, cases fall.
Why does the generation interval matter so much?
Because the same doubling time can come from a fast-transmitting infection with a short interval or a slower one that infects more people per case. Doubling every 3 days is R₀ 2 with a 3-day interval and 5 with a 7-day one.
What are typical R₀ values?
Seasonal influenza 1.3–1.8; the original COVID-19 strain 2.5–3; chickenpox 10–12; measles 12–18 — the most contagious common disease.
Where these figures come from
- NCBI Bookshelf — Molecular Biology of the Cell — growth kinetics and molecular conventions
- CSIRO — Australia's national science agency
Last checked: September 2026. Formulas are the standard textbook forms; assumptions are stated on each page because they are where these models break.