Updates 13th & 20th May (precise R number calculation)

There is a simple, but crucial number at the heart of understanding the threat posed by coronavirus. It is guiding governments around the world on the actions needed to manage the pandemic, and provides a key indicator to when lockdowns can be lifted or re-imposed.

Recently, the Prime Minister, First Ministers of the devolved nations ,government ministers, scientists, epidemiologists, medical officers and journalists all talk about the R number in relation to easing lockdown measures, so what is it exactly ?

The concept of the R number is simple. It stands for the “reproduction” number and measures rate of transmission of the virus spreading through a population. If the R number is one, then every person infected with the disease will infect one other person. If R is higher than one, a disease will keep on spreading to more and more people exponentially.

For example, if coronavirus had an R number of two, then every infected person would go on to infect two new people. So, if you started with 100 infected people, they would infect 200 people who would then go on to infect 400 people etc. If you introduce time into the equation (days), then on Day 1 you have 100 infected people who would infect 200 people on Day 2, 400 people on day 3 etc..ie the number of infected people quadruples in 3 days.

If the R number was lower at 1.2, then the virus would still move through a population very quickly. Those 100 infected people would infect 120 people, who would then infect 144, then 173, then 208. So, every four days of transmission, the number of people infected would double each time.

Conversely, if the R number is below one, then an epidemic will eventually fizzle out altogether. If R = 0.7, then 100 infected people would go on to infect 70 people, who would go on to infect 49, etc, so using time again as an example, the number of newly infected people is halved in 3 days.

**Calculating the R number**

You would think that as this number is so important for Governments to manage a pandemic – to either reimpose or ease lockdown measures etc based on the R number, that it would be easy to calculate from a simple formula, say for example

R = X – Y /Z = 0.5

No – it’s a lot more complicated than that – in fact, it is so complicated that there is no exact formula at all, and there is no way of calculating the R number precisely.

Having said that, there are a number of ways to calculate a range of R values, as Wired notes. One is by monitoring hospitalisation and death figures to get a sense of how many people have the virus – but the problem with that is, since the virus’s incubation period is so long, it only gives an accurate picture of a few weeks ago. To check transmission rates in a more accurate way, scientists at Imperial College London in the UK have started testing randomised 25,000 groups of the population to see how many are ill.

The R number is not fixed. Instead, it changes as our behaviour changes, or as immunity in the population develops.

Mathematical modellers at Imperial College London are attempting to track how the R number has changed as isolation, social distancing and full lockdown have been introduced.

Before any measures came in, the R number was well above one and the conditions were ripe for a large outbreak. Successive restrictions brought it down, but it was not until full lockdown that it was driven below one.

However, in all the research I have done on the science of the R number, there are just estimated values for R, using phrases such as “somewhere between 2 and 2.5” or “roughly X ” or “estimated as between X and Y”.

Even the World Health Organisation quotes estimated R number ranges rather than a precise number for the Wuhan coronavirus outbreak :

- WHO’s estimated (on Jan. 23) Ro to be between 1.4 and 2.5.
- Other studies have estimated a Ro between 3.6 and 4.0, and between 2.24 to 3.58. .
- Preliminary studies had estimated Ro to be between 1.5 and 3.5
**.** - For comparison, the Ro for the common flu is 1.3 and for SARS it was 2.0.

At the April 30 press conference, the UK’s chief scientific officer Patrick Vallence said that the UK’s R number was between 0.6 and 0.9 while the R number in London was between 0.5 and 0.7.

So why are we using such an indicator as a measure that cannot easily be calculated from the wealth of data available ? A simple and easy methodology needs to be developed to calculate a practical and precise R number that can be used to manage the national pandemic and also at a local level for regional and city epidemics in future waves of infection.

**Update 13th May** – ” We should be very wary of the R number” – Unherd.com

I came across another article on the R number, which makes interesting if not incomprehensible reading :

Let’s imagine that we had two epidemics, of equal size, one in the community and one in care homes. Say 1,000 people are infected in each, and in the community each person on average infects two people, while in the care homes on average each person infects three. The total R is 2.5^{[1]}.

But now imagine you lock down and reduce both the R and the number of people infected, but by more in the community than in the care homes. Say that now there are 100 people infected in the community, and they each pass it on to an average of one person; and there are 900 people infected in the care homes, and they pass it on to an average of 2.8 people.

Now your average R is 2.62^{[2]}; it’s gone up! But — just as with the Berkeley graduate students above — when you divide up the data into its constituent parts, it’s actually gone down in each category.

Even something as apparently simple as the R value has to be treated with immense caution.

^{[1]} ((1,000*2)+(1,000*3))/2000=2.5

^{[2]} ((100*1)+(900*2.8))/1000=2.62

**Further update 15th May **– further evidence that the R number is so difficult to calculate that it does not have a great deal of value as the main index for key decision making :

**How do models calculate the R value?**

The R value tells you how many infections each infected person passes the virus on to, on average. For an R above 1, the epidemic increases exponentially; below 1 it will eventually fizzle out. In the absence of direct measures of this elusive number, modellers have to rely on what firm data we have and extrapolate from there.

Some models, such as one by Public Health England (PHE) and Cambridge University, rely on the numbers of reported deaths, and the steep, steady decline in deaths in London translates into an estimate that community transmission must have also rapidly dropped off.

Deaths are reliable figures, less subject to biases than some other measures, but they reflect the infection rates that were occurring about three weeks previously and so do not give a dynamic reflection of where things are.

Policy changes around working and socialising introduced in the past week may have influenced R, but a model based on deaths will not give any insights into this. The PHE model projected that there might be as few as 24 new cases in the capital per day on 10 May. This reflects the strong downward trend London has been on, but the model is not designed to give razor-sharp predictions of day-by-day infection statistics. On Thursday, 49 people with new infections were admitted to hospitals in London.

Some models rely on measures such as symptoms reported to phone apps, which have a shorter lag time. However, these data are more “noisy” and so give a less robust measure of R overall, although they can give better insight into short-range changes.

**How sure are we about regional differences?**

Almost all available data and models suggest there are strong regional differences in terms of the number of infections and current transmission rates. Serology surveys suggest that more than 10% of people in London have been infected with Covid-19, compared with around 4% in the rest of the country. In general, urban centres have had more infections than rural areas.

The data from hospital admissions and deaths also suggest that R has come down far more steeply in London than in other regions. The PHE and Cambridge model suggests that R in London is 0.4, compared with 0.8 in the north-east and Yorkshire and 0.75 for the country as a whole.

Click for full Guardian article

**Update 20th May**

Phased approach to unlocking during the COVID‐19 pandemic—Lessons from trend analysis

This an excellent paper which provides a precise R number formula to calculate daily.