Growth R0 Gamma Generation
Use the Euler–Lotka relation for a gamma generation-interval model to infer reproduction number from exponential growth.
Description
Convert a supplied daily exponential growth rate into a reproduction number using a gamma generation-interval mean and shape.
An epidemic growth rate describes how quickly incidence changes over time. A reproduction number describes secondary infections per infection under a transmission model. Their relationship depends on the timing of those secondary infections, not only on the growth rate.1
This tool assumes a gamma distribution of generation intervals. Its mean gives the time scale, while shape controls dispersion: variance is mean squared divided by shape. Different shapes can therefore give different reproduction numbers for the same growth rate and mean.
Growth R0 Gamma Generation applies a supplied-rate conversion without estimating growth or fitting the generation distribution. Calling the result R0 requires the population and growth period to support that basic-reproduction interpretation.
When to convert growth under a gamma timing model
Use the tool when growth rate and generation-interval parameters have been estimated or selected independently. It makes the timing assumption explicit and supports sensitivity comparisons across shapes.
- Convert an exponential rate measured per day.
- Compare gamma-shape assumptions while holding mean timing fixed.
- Check the zero-growth and permitted negative-growth cases.
How to enter growth parameters and calculate reproduction
- Enter Growth rate per day as an exponential rate, such as 0.1; it is not a raw daily percent increase.
- Enter positive Mean generation days and positive Generation shape. Shape may be fractional.
- Run the conversion and read Reproduction number with its generation-distribution assumptions.
R = (1 + r×meanGenerationDays/generationShape)^generationShape The implementation requires the expression inside parentheses to be strictly positive, then raises it to the supplied shape. For negative growth, r must exceed −shape/mean.
{
"input": {
"growthRatePerDay": 0.1,
"meanGenerationDays": 5,
"generationShape": 2
},
"output": {
"reproductionNumber": 1.5625
}
}A permitted negative rate gives a number below one. A nonpositive base, nonpositive mean or shape, malformed inputs, or nonfinite values produce errors. Extreme numerical combinations can fail output validation.
Limitations and interpretation
- Growth fitting, reporting-delay correction, generation-interval estimation, and uncertainty intervals are not included.
- The gamma shape is an explicit model assumption; a mean alone does not determine this conversion.
- Observed symptom-onset intervals need not equal generation intervals and cannot be substituted without justification.
- A rate from a partially susceptible or changing population may describe effective reproduction rather than R0; the tool does not distinguish those settings.
Alternative or Complementary methods
Renewal calculations use an explicit daily lag-weight distribution and time series rather than a supplied growth rate. Paired timing summaries can help distinguish infection-to-infection intervals from onset-to-onset intervals before selecting model inputs.
References
-
How generation intervals shape the relationship between growth rates and reproductive numbers — Proceedings of the Royal Society B; J. Wallinga and M. Lipsitch (2007)
Similar or alternative tools
- Renewal Window RT
Calculate rolling incidence-to-infectiousness ratios from daily counts and supplied generation weights.
- Generation Serial Interval Pairs
Compare supplied generation and serial timing in aligned transmission pairs.