Reproduction Number

The UK government has started publishing an interval estimate of R. This is also known as the reproduction number.

The article looks at the definition of the reproduction number. I also discuss estimation methods.

What is the reproduction number?

The reproduction number is the average number of direct infections from one case. This is over the whole time whilst people are infectious.

If the reproduction number is 2, we would expect 100 infected people to infect 200 more people. If the reproduction number is 0.5, the average group of 100 infected people infects 50 more.

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Each infected people passes it onto more people, leading to exponential growth. (Image: Stanford University)

The reproduction number can change over time. If people reduce their contact with others, a virus has fewer transmissions.

There is also the basic reproduction number. This number is often labelled as R₀ (‘R-zero’). The basic reproduction number is for when the population has no immunity. When the virus is novel, people do not have immunity.

The basic reproduction number is not a biological constant. The same virus may spread in different populations at different paces. By itself, this number does not determine how fast a virus spreads. The number of initial ‘seed’ cases is important. We also need to understand how long people are infectious for.

Suppose people can recover from the virus, conferring immunity. Over time, more people will recover, die, or get vaccinated. The susceptible population gets smaller. The effective reproduction number is for the remaining susceptible population.

Reproduction numbers are averages.

One person could pass on a virus to 100 people, and 99 others do not pass it on. In that population, the average new infections is 1. If every infected person infects one more, that would be the same reproduction number. The implications for health policy differ.

Estimating the reproduction number

Researchers need to estimate this number, through mathematical models.

Inputs for these models may include:

The MRC Biostatistics Unit (University of Cambridge) uses transmission models. Researchers stratify these models by age and region. Their work takes death statistics, mortality risks, and time from infection to death. Researchers estimate new infections over time and reproduction numbers.

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The University for Cambridge estimates are for different health regions in England. (Image: Financial Times)

These reproduction number estimates are uncertain.

There are several sources of uncertainty, including:

  • Accuracy of input data.
  • Model choice, as different approaches will give different estimates.
  • How sensitive underlying assumptions in each model are.

For the UK government estimates, modelling teams provide different estimates. Attendees to an advisory group then form a consensus over a likely range. This range reflects expert judgement, as well as uncertain data and modelling approaches.

The reproduction number can be hard to interpret.

The reproduction number refers to the average infected person. That average person can change over the epidemic.

Suppose there were two populations, each with half of all infections. Their reproduction numbers are 0.9 and 2.3. The average is 1.6.

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This is an example of Simpson’s paradox. (I adapted this example from Tom Chivers’ article.)

Imagine those reproduction numbers fell in both populations: to 0.2 and 2.2. The second population now accounts for 9 in 10 of all infections. The average reproduction number rises to 2.0.

A national reproduction number may be less useful than those of groups and areas. We need to consider reproduction number estimates with other key statistics.

The reproduction number is difficult to estimate. Uncertainty comes from several sources: data, model choice, and sensitive assumptions. Publishing a range — an interval estimate — helps reflect that uncertainty.

We should focus on plausible ranges of values, rather than a single number.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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