Superspreading Moment Dispersion

Estimate negative-binomial offspring dispersion k by moments; underdispersed or Poisson data do not have a finite positive k.

Description

Estimate negative-binomial offspring dispersion k from a positive mean and a larger variance of secondary-case counts.

An offspring count in an epidemic is the number of secondary cases attributed to one infected person. A mean summarizes average transmission, but it can hide large differences between people: many may transmit to no one while a few transmit to many. The spread of those individual counts is therefore relevant to describing transmission heterogeneity.^[1]

A negative-binomial offspring model uses a mean and a dispersion parameter k. At a fixed positive mean, smaller k means greater variance and more heterogeneous transmission; larger k approaches the Poisson limit. This model is often used to study superspreading, but a fitted dispersion parameter is not an identification of particular superspreading people or events.^[2]

Superspreading Moment Dispersion estimates k from supplied mean and variance using the method of moments. It also returns the variance-to-mean ratio. The variance must exceed the positive mean to produce a finite positive estimate.

When to summarize overdispersed secondary-case counts

Use the tool when mean and variance have already been calculated for counts of secondary cases per source case under a consistent observation process. Both summaries must describe the same set of individual offspring counts.

Check a negative-binomial method-of-moments calculation.
Compare heterogeneity for datasets with compatible definitions and ascertainment.
Explore how the variance above the Poisson baseline affects k.

How to supply moments and calculate dispersion

Calculate the mean and variance of secondary-case counts, including observed zero-offspring cases when they belong to the sampled population.
Enter a positive Mean secondary cases and a Variance secondary cases strictly larger than that mean.
Run the tool and read Dispersion k together with Variance mean ratio.
Keep the sample definition and variance convention with the estimate; the tool receives neither raw counts nor sample size.

variance = mean + mean²/k; dispersionK = mean²/(variance − mean); varianceMeanRatio = variance/mean

The method of moments solves the negative-binomial variance relationship for k.

The implementation checks a positive finite mean and requires variance greater than mean, then evaluates the two ratios. It does not estimate moments from records, fit a likelihood, or infer a confidence interval.

json

{
  "input": {
    "meanSecondaryCases": 2,
    "varianceSecondaryCases": 6
  },
  "output": {
    "dispersionK": 1,
    "varianceMeanRatio": 3
  }
}

json

{
  "input": {
    "meanSecondaryCases": 2,
    "varianceSecondaryCases": 22
  },
  "output": {
    "dispersionK": 0.2,
    "varianceMeanRatio": 11
  }
}

Supplied moments	Behavior
Variance greater than a positive mean	Finite positive k, subject to numeric representability.
Variance equals the mean	Input error; the Poisson limit has no finite k here.
Variance less than the mean	Input error; this model does not yield positive k.
Mean equals zero	Input error: positive mean is required.

Limitations of a moment-based dispersion estimate

A mean and variance do not verify that a negative-binomial distribution fits the data. Different offspring distributions can share the same moments.
Incomplete contact tracing, omitted zero counts, censoring, and different observation windows can distort the supplied moments.
The tool returns no sampling uncertainty, event probability, transmission forecast, or fraction of cases responsible for a given share of transmission.
The denominator is small when variance is close to mean, making k sensitive to small changes in the summaries. Sample and population variance conventions can also change the estimate.

Alternative or Complementary count-distribution summaries

Inspect the raw offspring-count distribution and sample size before interpreting k. Calculate the mean and an explicitly chosen variance from the same counts; retain zeros and relevant observation metadata. A likelihood-based count model or resampling analysis can address questions about fit and uncertainty that this two-input calculator cannot.

References

Superspreading and the effect of individual variation on disease emergence — Nature; J. O. Lloyd-Smith and colleagues (2005)
Superspreading quantified from bursty epidemic trajectories — Scientific Reports / Nature Portfolio (2021)

Similar or alternative tools

Mean Calculator
Calculate the arithmetic mean of individual secondary-case counts.
Sample Variance Calculator
Calculate variance with the n−1 denominator when treating observed counts as a sample.
Population Variance Calculator
Calculate variance with the n denominator when treating the submitted counts as the complete population of interest.