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By Roger Highfield on

Coronavirus: From R to lifting lockdown

Roger Highfield, Science Director, on why the fate of the nation rests on R, the ‘reproduction number’, and a novel ‘stringency index’ that compares the pandemic responses of 160 nations.

The pandemic response of the nation rests on a seemingly simple number: R is the cornerstone of the Government’s lockdown policies.‘Driving down the R is, of course, our collective endeavour and the better we can do it locally, nationally, and across the country, the faster we will be out of it,’ said the Prime Minister, at a UK coronavirus briefing.

With Swapnil Mishra of Imperial College London, a leading author on multiple COVID-19 modelling reports, I delve into the nuanced world of R. His edited answers are in italics to distinguish them from my questions and commentary.

if r is more than one, the epidemic is out of control?

That’s right. R, the reproduction number, is a measure of transmission, that is the rate of increase or decrease in the rate of COVID-19 infection. Above one, the spread of COVID-19 is going to accelerate. Under one, the pandemic will decelerate.

But the value of R is volatile and depends what the Government does and how society behaves – even if the Government ended lockdown tomorrow, society would not behave as it did before, from wearing masks to avoiding crowded trains.

Though we can try to model these changes in behaviour, no-one really knows how human behaviour will change, so predicting R is difficult.

what is the uk’s r number?

R is between 0.7 and 1.0, based on estimates from various academic groups, summarised on 15 May by the Scientific Pandemic Influenza Group on Modelling, a sub-group of the Scientific Advisory Group for Emergencies (SAGE).

SAGE is confident that, overall, R is not above 1, so the number of infections is very likely decreasing.

However, R should always be considered alongside the number of people who are infected. A value of 1.0 is more troubling when 100,000 people are infected compared with 100 infected people.

Importantly, the R value gives a snapshot of what was happening in the community two or three weeks ago, for reasons explained below.

what is r (0), ‘r zero’, then?

That’s the value of R at the start of the pandemic, when there’s no immunity and people behave as they did before. Computer modellers like me refer to R (t), where t stands for time, so R zero is the value of R when the time is zero, or t=0.

You need data over a long period to estimate R (0) but we have a good idea from studies in the epicentre of the epidemic, in China.

From what we’ve seen, R(0) is around three for SARS-CoV-2. That R (0) value means on average one person infects three others, those three infect nine others overall, those nine infect another twenty seven and so on. This is so-called exponential growth, where the rate grows in proportion to the total number of infected people.

By comparison, each case of measles causes 12 to 18 new cases, compared to about six for polio, smallpox and rubella. The 1918 influenza R ranged from three to four.

Swab used in measles frequency studies, London, England, 1996
Salivette used in measels prevalence surveys by the Communicable Diseases Surveillance Centre (PHLS), complete with postal packaging, 1996. Black, matt background.

However, R (0) is ‘easily misrepresented, misinterpreted, and misapplied.’

what else can r tell us?

You can use R(0) to work out the ‘herd immunity’ threshold. Herd Immunity is the percentage of the population who need to be immune to the virus, either from being vaccinated or having already been infected, for the epidemic to fizzle out. We put it at around 67 per cent (the formula is (1 – 1/R(0))*100% so for R(0)=3 that’s 67%).

But we’re a long way from achieving herd immunity. Data suggest that no more than a fraction of a per cent of the population is infected at any one time. By 7th May, we estimated that no more than seven per cent of the population had been infected over the course of the pandemic.

Our current estimates range from 4.2 to 6.9 per cent, which are larger than some, though smaller than others, but these other estimates are based on older data, and other assumptions.

Our estimates are in line with surveys of the percentage of the population who were infected in Spain and Austria.

Our models are only as good as the data we put into them and our basic understanding of COVID-19 – for example, we do not know how much immunity they have or how long it will last.

Also bear in mind the old joke and aphorism: ‘all models are wrong, but some are useful’.

how big is the danger of a second wave among the 90 plus per cent of uninfected people in the uk?

Huge. If you look at our report on Italy (p10-11) on a partial return of mobility to pre lockdown levels, from 20 to 40 per cent, you can see that a second wave of infections is predicted in most of the regions. This is one that is much higher than the first they have already experienced.

Though we don’t know for sure how people will behave, the models suggest we have to monitor the relaxation of measures extremely carefully.

By the time you pick up a rise in people admitted to intensive care, which occurs around 10 days after infection, you could have a lot of cases.

how do you measure the r (t) for the uk?

In an ideal world, you could calculate R (t) at a given time if you knew exactly how many people are currently infected, how long they have been infected (they can spread the disease before symptoms develop, or not develop symptoms at all) and how many people they encounter while infectious.

Because no-one really knows how many people are actually infected, there are various ways to figure out how many people go on to contract SARS-CoV-2 from an infected person and, from that, work out R (t).

This transmission electron microscope image shows SARS-CoV-2, the virus that causes COVID-19, emerging from the surface of cells
This transmission electron microscope image shows SARS-CoV-2, the virus that causes COVID-19, emerging from the surface of cells.
The image was captured and colorized at NIAID’s Rocky Mountain Laboratories (RML) in Hamilton, Montana.

We estimate how many infections result for each death for example,  and do this over different age groups and on a daily basis using a so-called recursive formula to figure out how the overall number builds up.

We measure from when a person is infected on day zero to day 23, this is the average time from infection to death.

Once I have done this, I can work out R from the number of deaths.

The most precise way one could in theory calculate R would be to determine, through virus sequencing and behavioural links, exactly who infected whom in every case.  But this isn’t feasible in virtually any setting.

so, your r number does not tell us what’s happening now but over the past 23 days?

Yes – deaths will not fall as soon as R (t) drops because the deaths that you see now are the result of infections in the past three weeks or so. Hence a drop in the actual reproduction of the virus today can only really be seen in death rates after a period of two or three weeks.   

can you see how r (t) has been cut by social distancing, lockdowns and so on?

When control measures are in place, R falls. If R (t) were pushed down to 0.7, for example, then 100 infected people would infect another 70 people, who would go on to infect 49, then 34 and so on it would eventually fizzle out.

could you see r fall as the government introduced measures, from asking over-70s to isolate to lockdown on 23 march?

Because these measures came one after the other, and in short order, we can’t easily tell how each one affected R.

i presume r is different between different cities or counties?

Yes, For example in Italy, the region of Veneto had a higher R (t) and you can see differences between north and south, with many more deaths in the north.

Variations in R across England published on 10 May by the Medical Research Council’s Biostatistics Unit in Cambridge, working with Public Health England, which reveal that the median (‘middle value’) of R in England is 0.75, varying from 0.4 in London to 0.8 in the northeast and Yorkshire.

The Centre for the Mathematical Modelling of Infectious Diseases at the London School of Hygiene & Tropical Medicine published regional variations on 7 May suggesting that, for example, R is still around 1.0 in Scotland and Northern Ireland.

does r vary between different age groups?

R (t) will also vary depending on the age of the people you consider, for example. In France, older people tend to socialise with more diverse age groups, but they also have a much higher mortality rate.

These more fine-grained views of R show how other factors come into play. If coronavirus took hold in a densely-populated city that hadn’t experienced an outbreak and didn’t have any social distancing rules in place, it’d likely have a much higher R than a place where social distancing had been implemented for a long time.

can trends in r change when going from smaller regions to the uk overall?

You have to watch out for Simpson’s paradox, when a trend that seems obvious in smaller groups disappears or even reverses when the data are combined to give the bigger picture.

Once you aggregate groups you make an assumption that they are similar. However, it could be that one group has more older people than the other, more care homes, more who live close together, poorer, and more diverse groups or another difference that can skew the results.

can you have an individual r(t)?

No. The reproduction number, R, describes average behaviour so it does not really translate to an individual person. Even if someone has the infection, and the value R suggests they will infect two more people, the reality could be quite different depending on if that person lives like a hermit or goes clubbing (as happened with one infected person recently in Seoul, so thousands of contacts had to be traced).

What is the K number?

K is the so-called dispersion parameter, such that a low value suggests that a small number of infected people are responsible for a relatively large number of cases.

For the 1918 influenza pandemic, the number K is thought to be around 1 but for COVID-19 the K number is as low as 0.1, so that one infected person can trigger many new cases, as observed recently in South Korea.

how is the world reacting to covid-19?

Governments are taking a wide range of measures in response to the COVID-19 outbreak. Just as epidemiologists measure the behaviour of the virus through the reproduction number, R, the Oxford COVID-19 Government Response Tracker systematically collects information on how 160 nations are responding to the pandemic.

The edited comments of the head of the project, Thomas Hale, associate professor in public policy at the Blavatnik School of Government at Oxford University, are in italic.

how do you weigh up how a government is responding to covid-19?

With our team of more than 100 students, staff and alumni, we gather information on the policy responses such as school closures, banning public events and restrictions in public transport, 17 in all

We also combine nine closure and containment indicators into what we call a Stringency Index, a snapshot to quickly summarise what a country is doing. You can see the data here.

It’s important to note that this index reflects the number and strictness of government policies and doesn’t necessarily mean that one country’s response is ‘better’ than another that’s lower on the index. We are also comparing official policy and have no insight into enforcement and implementation.

what’s the big picture?

Overall, the world remains under lock and key. The average stringency score per country as of May 7 is 79, where 0 means no measures, and 100 means every single measure is in place.

what about the speed of response?

Looking at the number of deaths a country had recorded by the time their lockdown went above 50, or half of the maximum stringency on our index, you can see New Zealand took action while the deaths were still in single figures.

For example, the Caribbean nation of Trinidad and Tobago enforced lockdown on confirming its first case. In Europe, Germany and Austria stand out as nations that adopted aggressive and early control strategies.

But the UK took action at 281 deaths, higher than the US (150), France (127), Spain (121) and China (56). At its most restricted, after 700 deaths, the UK registered a score of 82.3 – ranking it 71st out of 109 countries on the index for which there is comparable data. Seven countries, including Argentina and India, scored 100.

Some have commented that the UK response was slow.

are countries ready to relax restrictions?

In one way they are not. If you take the World Health Organization guidance on six criteria for easing some restrictions —   intended to minimize a risk of a resurgence  – we find no country has met even four of the criteria, though New Zealand is close, along with China.

In fact, only 20 countries and territories have come close, including Trinidad and Tobago, Croatia, Hong Kong and South Korea.  

what is the most common shortcoming?

Most countries have not yet met the first requirement – for COVID-19 to be reduced to sporadic cases and identifiable clusters – and many lack adequate policies for testing, contact-tracing and isolating infected individuals. 

what about lockdown measures?

Because our data only measure four of the six WHO recommended actions, we should be cautious about inferring what countries are ready to roll back lockdown from this measure.

However, we can see some countries are beginning to relax measures as nearly a quarter of the 160 countries that we have tracked have now seen a reduction in their stringency score in the past two weeks. New Zealand, for example, has seen its stringency score fall by nearly ten points as it started to allow residents to travel to work and school at the end of April.

We see a systematic pattern among nations where information campaigns come first, then travel restrictions, limiting public events and then school closures.

are rich countries are doing better than poor?

Not necessarily. If you look at Europe, for example, poorer nations were quicker at imposing tough restrictions and have fared better. For example, Croatia has one of the world’s top stringency scores, at 100, and, based on European data, fewer than 100 people had died of COVID-19 there by mid-May, compared with the UK’s index of 80.95 and more than 33,000 deaths.

Overall, stringency is higher in Eastern Europe than in Western Europe. Moreover, three Caribbean countries rank in the top 10 list of nations globally: Trinidad and Tobago, Belize and Barbados.

It is striking that poorer countries have moved to higher levels of stringency more quickly. There are various reasons that might be, such as they are more risk-averse, lacking health infrastructure, or had more time to prepare, as the disease took hold elsewhere.

When it comes to the known number of cases, which is fraught and much more difficult to record than deaths, we think there’s a correlation between countries that don’t take strong measures and under reporting.

how does the uk response look?

We put it in the whiplash category, where there’s not much action for a long time and then, once the death rate goes up, there’s a fast move to lockdown. That’s not far different from what was seen in the US, Canada, Australia and so on.

what else is striking?

What bothers me most about COVID-19 is not the disease itself but how it’s shown the fragility of some of our national governance systems.

Moreover, how ill-prepared countries are to work together on global threats – a concern as the impact of climate change is increasingly felt.

what makes a country more stringent or faster when introducing measures?

This is complicated but some fascinating insights are emerging, notably, if people trust their government. We see that governments that are less trusted tend to be more stringent. We have also seen governments that act more stringently and quickly earn more trust. It will be interesting to see how this develops.

where can i find out more about r and the pandemic?

Find more technical discussions of R from the resources below:

  • Graham Medley, London School of Hygiene and Tropical Medicine and chair of the Scientific Pandemic Influenza Group on Modelling.
  • Tanja Stadler at ETH’s Department of Biosystems Science in Switzerland.
  • The ‘alternative’ to the government’s Scientific Advisory Group for Emergencies (SAGE).

The latest picture of how far the pandemic has spread can be seen on the Johns Hopkins Coronavirus Resource Center or Robert Koch-Institute. You can check the number of UK COVID-19 lab-confirmed cases and deaths, and figures from the Office of National Statistics.

There is more information in my earlier blog posts, from the UKRI, on this COVID-19 portal and Our World in Data.

The Science Museum Group is collecting objects and ephemera to document this public health emergency for future generations.