What exactly is the R number and how useful is it?

Once you get past the astonishing mathematical illiteracy with which the UK government has tried to convey its Covid-19 “alert level” in preparation for an eventual exit from lockdown, it does make some sense. The two most relevant factors for the spread of the disease are indeed the level of infections and the reproduction number R, the latter being a measure of how readily the disease is being transmitted between people. (The government’s five criteria for relaxing the restrictions also take into account the availability of testing and protective equipment and the capacity of the health services.)

Suddenly everyone is talking about R. Is it rising or falling? Is it higher in cities, or in the north of England? What does it really mean?

The first thing to understand is that R does not quantify some fundamental feature of nature, like the strength of the Earth’s gravity or the speed of light in a vacuum. It emerges from specific epidemiological models, and depends on the context in which the virus is spreading. It will vary not just from country to country but from town to town, and is contingent on how individuals are acting in particular, how well they are socially distancing to reduce infection.

In effect R—the reproduction number—denotes the average number of people an infected individual passes the disease on to. If this value is less than 1, the disease cannot keep spreading, and an epidemic will die out. The larger it is, the more contagious the condition and the faster it spreads. The basic rate at which the virus is transmitted in a population in normal (pre-lockdown) circumstances, denoted R₀ (“R zero”—not to be confused with an R value of zero!), is a rough-and-ready measure of how intrinsically infectious the virus is—although it depends also on other factors such as the size of the population and how many people in it are susceptible in the first place. Strictly speaking, the national and regional R values now being estimated are “effective reproduction numbers,” more properly denoted R_e. These can and usually will vary over time and place, depending on—among other things—how people are behaving and what safety measures are in place.

At the outset of the Covid-19 pandemic there was a rush to pin down the virus’s R₀ so as to better understand what manner of challenge we were facing. Estimates tend to fall now around 2.5-2.7, compared to 1.4 or so for a typical seasonal flu. This might not sound like much of a difference—compared to the whopping R₀=15 for measles, say—but because infection increases exponentially, the outcomes certainly are. For an R₀ value of 3, an infected person will infect three others, they will each infect three more, and after just five transmission steps the number of people infected is 3⁵ or 243. After 10 steps it is about 60,000, compared to just over 1,000 for R₀=2.

From the number of people testing positive with coronavirus (or the number admitted to hospital or dying), epidemiological models enable a value of R for that population to be estimated. Because of the time taken for the symptoms to emerge and the data to be collected, these estimates generally lag behind the immediate situation by two or three weeks.

These R values are averages. Some people spread the virus more than others, perhaps because it reproduces more prolifically in their bodies or because they have more physical contacts in their daily life. By the same token, some spread it less—a big unknown at present is whether this is true of children, which will influence the decision of how and when to reopen schools. The R value doesn’t by any means impose a “quota” on how many people can be infected by someone with the virus, however—if you have it and you’re on a packed train or standing in a dense crowd, you will have more close contacts than usual and could infect many others. That’s why the comments of the government’s Chief Scientific Adviser Patrick Vallance, when questioned on 12th March about shutting down sporting events, made little sense: “On average, one person infects two or three others. You therefore have a very low probability of infecting a large number of people in a stadium and a rather higher probability of infecting people very close to you.”

R decreases if people have fewer social contacts through distancing and lockdown measures: the virus then simply has less opportunity to pass between individuals. That was reflected in a drop in R in the UK from around 4 to about 0.7 (as estimated in models run at Imperial College London) after lockdown began on 24th March.

The worry now is that, as the lockdown measures are relaxed, or as people take it upon themselves to ignore social distancing anyway, the value is creeping up again, creating the conditions that could unleash a second wave of infection that will necessitate a resumption of lockdown. Several studies suggest that R is around 0.75 for England as a whole but higher in the northeast and southwest (perhaps as much as 0.9), and also in Wales and Scotland. The value seems to be lowest in London (0.4-0.7), which initially had the highest incidence of Covid-19 and so might now be benefitting from higher levels of immunity acquired by infection. At any rate, the differences lead some scientists to conclude that relaxation of lockdown measures should not be a one-size-fits-all affair.

The fact that R is defined by a particular, relatively simple set of epidemiological models rather than by some kind of “truth of nature” means that it can hide a wealth of complications. One possible reason why R values might be rising even before the end of lockdown is that, if spreading declines in the general population, the contribution towards infections from care homes becomes more salient and visible. Another way to say this is that R is a blunt tool for understanding the detailed trajectory of an epidemic.

If and when a Covid-19 vaccine becomes available, it will—like all immunisation programmes—need to be used on a certain fraction of the population in order to create herd immunity, which prevents the disease from spreading. (This is the proper use of that now much vilified term: it was coined for understanding immunisation strategies, not the unimpeded progress of an infectious disease). If the estimates of R₀ for Covid-19 are correct, this implies that at least 60 per cent or so of the population would need to be vaccinated—although the number depends on how many people have already had the disease and are now immune, which won’t be known until reliable antibody tests become available and widely used. (Current estimates for the prevalence of infection in the UK are around 4 per cent, but this may be closer to 10 per cent in London.) At these levels, anti-vaccination sentiment might not be too great a hindrance to achieving a safe level of population immunity—but there is still good reason to try to head off that danger. At any rate, there’s good reason think that a second wave of infection is likely at some point—and that it could be equally devastating if the exit from lockdown is not orchestrated very carefully.