Some diseases have a relatively low R. For pandemic flu, R is generally around 1–2, which is about the same as Ebola during the early stages of the 2013–16 West Africa epidemic. On average, each Ebola case passed the virus onto a couple of other people. Other infections can spread more easily. The sars virus, which caused outbreaks in Asia in early 2003, had an R of 2–3. Smallpox, which is still the only human infection that’s been eradicated, had an R of 4–6 in an entirely susceptible population. Chickenpox is slightly higher, with an R around 6–8 if everyone is susceptible. Yet these numbers are low in comparison to what measles is capable of. In a fully susceptible community, a single measles case can generate more than 20 new infections on average.[42] Much of this is down to the incredible lingering power of the measles virus: if you sneeze in a room when you have the infection, there could still be virus floating around in the air a couple of hours later.[43]
As well as measuring transmission from a single infectious person, R can give clues about how quickly the epidemic will grow. Recall how the number of people in a pyramid scheme increased with each step. Using R, we can apply the same logic to disease outbreaks. If R is 2, an initial infected person will generate two cases on average. These two new cases will on average generate two more each, and so on. Carrying on doubling and by the fifth generation of the outbreak, we’d expect 32 new cases to appear; by the tenth, there would be 1,024 on average.
Because outbreaks often grow exponentially at first, a small change in R can have a big effect on the expected number of cases after a few generations. We’ve just seen that with an R of 2, we’d expect 32 new cases in the fifth generation of the outbreak. If R were 3 instead, we’d expect 243 at this same point.
Example of an outbreak where each case infects two other people. Circles are cases, arrows show route of transmission
One of the reasons R has become so popular is that it can be estimated from real-life data. From hiv to Ebola, R makes it possible to quantify and compare transmission for different diseases. Much of this popularity is down to Robert May and his long-standing collaborator Roy Anderson. During the late 1970s, the pair had helped bring epidemic research to a new audience. Both had a background in ecology, which gave them a more practical outlook than the mathematicians who’d preceded them. They were interested in data and how models could apply to real-life situations. In 1980, May read a draft paper by Paul Fine and Jacqueline Clarkson of the Ross Institute, who had used a reproduction number approach to analyse measles epidemics.[44] Realising its potential, May and Anderson quickly applied the idea to other problems, encouraging others to join them.
It soon became clear the reproduction number could vary a lot between different populations. For example, diseases like measles can spread to a lot of people if it hits a community with limited immunity, but we rarely see outbreaks in countries with high levels of vaccination. The R of measles can be 20 in populations where everyone is at risk, but in highly vaccinated populations, each infected person generates less than one secondary case on average. In other words, R is below one in these places.
We can therefore use the reproduction number to work out how many people we need to vaccinate to control an infection. Suppose an infection has an R of 5 in a fully susceptible population, as smallpox did, but we then vaccinate four out of every five people. Before vaccination, we’d have expected a typical infectious person to infect five other people. If the vaccine is 100 per cent effective, four of these people will now be immune on average. So each infectious person would be expected to generate only one additional case.
Comparison of transmission with and without 80 per cent vaccination, when R is 5 in a fully susceptible population
If we instead vaccinate more than four fifths of the population, the average number of secondary cases will drop below one. We’d therefore expect the number of infections to decline over time, which would bring the disease under control. We can use the same logic to work out vaccination targets for other infections. If R is 10 in a fully susceptible population, we’d need to vaccinate at least 9 in every 10 people. If R is 20, as it can be for measles, we need to vaccinate 19 out of every 20, or over 95 per cent of the population, to stop outbreaks. This percentage is commonly known as the ‘herd immunity threshold’. The idea follows from Kermack and McKendrick’s work: once this many people are immune, the infection won’t be able to spread effectively.
Reducing the susceptibility of a population is perhaps the most obvious way to bring down the reproduction number, but it’s not the only one. It turns out that there are four factors that influence the value of R. Uncovering them is the key to understanding how contagion works.
On 19 april 1987, Princess Diana opened a new treatment unit in London’s Middlesex Hospital. While there, she did something that surprised the accompanying media and even the hospital staff: she shook a patient’s hand. The unit was first in the country that was specifically built to care for people with aids. The handshake was significant because despite scientific evidence the disease could not spread through touch, there was still a common belief
