Answering FAQs: Part 1

What is the reproduction number? What about future risks?

To help the Royal Statistical Society, I wrote answers to frequently asked questions about COVID-19.

The main constraint was a limit of 400 words. These are the original version of the articles. The posts were later improved by other authors.

What is R and how is it determined?

What is the reproduction number (R)?

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

If the reproduction number is 2, we 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.

The reproduction number can change over time. If people reduce contacts, the virus has fewer transmissions.

The basic reproduction number (R₀) is for when the population has no immunity. This 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. Initial ‘seed’ cases and infectious periods are important.

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The virus passes on in each generation. (Image: Stanford University/James Holland Jones)

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

Reproduction numbers are averages.

One person could pass on the 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.

How is the reproduction number determined?

Researchers estimate this number, through mathematical models.

Model inputs could include:

  • Data on confirmed infections, hospital admissions, critical care, and deaths.
  • Social contact surveys, with self-reported data on contacts.
  • Household infection surveys, which can study current prevalence of infections.

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

Reproduction number estimates are uncertain, and can be hard to interpret.

There are several sources of uncertainty:

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

The reproduction number refers to the average infected person. That average person changes. At lower infection levels, reproduction number estimates are more volatile.

A national reproduction number may be less useful than those of groups and areas.

We should look at the reproduction number alongside other key statistics. There is a plausible range around reproduction number estimates. It could be somewhat higher or lower.

How will the risks in future compare with the risks in the peak of the epidemic?

There are a range of answers: branching forks of the future. This answer depends on the country. Countries have different experiences of this pandemic.

A confirmed case means a person with a positive test result for SARS-CoV-2.

Confirmed cases are not all cases. There was low capacity for testing in earlier stages of the pandemic. Testing is imperfect.

In some countries, confirmed cases are rising. This measure declined from an earlier peak. A common term for this rise is a second wave. Distinction between these different ‘waves’ is arbitrary and down to analytical judgement.

Daily confirmed cases could rise above that initial peak, as it has in Japan. The actual height of true infections for this second peak may be lower than the first one. This is despite daily confirmed cases being higher.

Recovering from the virus confers degrees of immunity, reducing the susceptible population. Second waves can be more deadly than the first: such as the 1918 flu pandemic.

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The 1918 flu pandemic is often called ‘Spanish flu’. Wartime censorship limited early reporting of the epidemic’. The media in neutral Spain were more free to report on this flu. That included the illness of King Alfonso XIII. (Image: Wikimedia)

Short-term future risks may be higher than in early 2020. Contacts with infected people could increase. Without mitigation, that leads to greater transmission of this new virus.

Longer term, there are three broad scenarios:

  • Endemic: The virus becomes endemic at low levels with seasonal cycles. SARS-CoV-2 circulates like other human coronaviruses and seasonal flu. The number of deaths depends on infections and how effective medical treatments are. This is what happened to the 2009 H1N1 influenza virus.
  • Containment: Testing, tracing and isolation programmes help contain the virus. Risks vary over time, with localised spread of the virus. It may take some time for containment to lead to elimination, as it did for the original SARS coronavirus.
  • Vaccination: Powered by recoveries and vaccinations, societies develop herd immunity. Countries eliminate the virus, but remain at risk of importation. There are many examples of vaccinations leading to elimination, such as smallpox worldwide.

The relative scale of those risks depends on which scenario becomes real.

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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