Then progress stuttered. The obstacle was a 1957 textbook written by mathematician Norman Bailey. Continuing the theme of the preceding years, it was almost entirely theoretical, with hardly any real-life data. The textbook was an impressive survey of epidemic theory, which would help lure several young researchers into the field. But there was a problem: Bailey had left out a crucial idea, which would turn out to be one of the most important concepts in outbreak analysis.[38]
The idea in question had originated with George MacDonald, a malaria researcher based in the Ross Institute at the London School of Hygiene & Tropical Medicine. In the early 1950s, MacDonald had refined Ronald Ross’s mosquito model, making it possible to incorporate real-life data about things like mosquito lifespan and feeding rates. By tailoring the model to actual scenarios, MacDonald was able to spot which part of the transmission process was most vulnerable to control measures. Whereas Ross had focused on the mosquito larvae that lived in water, MacDonald realised that to tackle malaria, agencies would be better off targeting the adult mosquitoes. They were the weakest link in the chain of transmission.[39]
In 1955, the World Health Organization announced plans to eradicate a disease for the first time. Inspired by MacDonald’s analysis, they had chosen malaria. Eradication meant getting rid of all infections globally, something that would eventually prove harder to achieve than hoped; some mosquitoes became resistant to pesticides, and control measures targeting mosquitoes were less effective in some areas than others. As a result, who would later shift its focus to smallpox, eradicating the disease in 1980.[40]
MacDonald’s idea to target adult mosquitos had been a crucial piece of research, but it wasn’t the one that Bailey had omitted in his textbook. The truly groundbreaking idea had been nestled in the appendix of MacDonald’s paper.[41] Almost as an afterthought, he had proposed a new way of thinking about infections. Rather than looking at critical mosquito densities, he suggested thinking about what would happen if a single infectious person arrived in the population. How many more infections would follow?
Twenty years later, mathematician Klaus Dietz would finally pick up on the idea in MacDonald’s appendix. In doing so, he would help bring the theory of epidemics out of its mathematical niche and into the wider world of public health. Dietz outlined a quantity that would become known as the ‘reproduction number’, or R for short. R represented the number of new infections we’d expect a typical infectious person to generate on average.
In contrast to the rates and thresholds used by Kermack and McKendrick, R is a more intuitive – and general – way to think about contagion. It simply asks: how many people would we expect a case to pass the infection on to? As we shall see in later chapters, it’s an idea that we can apply to a wide range of outbreaks, from gun violence to online memes.
R is particularly useful because it tells us whether to expect a large outbreak or not. If R is below one, each infectious person will on average generate less than one additional infection. We’d therefore expect the number of cases to decline over time. However, if R is above one, the level of infection will rise on average, creating the potential for a large epidemic.
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 th
e 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 that it could.[45]
The rise of hiv/aids in the 1980s created an urgent need to uncover how the epidemic was spreading. What features of the disease were driving transmission? The month before Diana visited Middlesex Hospital, Robert May and Roy Anderson had published a paper that broke down the reproduction number for hiv.[46] They noted that R was influenced by a number of different things. First, it depends on how long a person is infectious: the shorter an infection is, the less time there is to give it to someone else. As well as the duration of infection, R will depend on how many people someone interacts with while infectious. If they have a lot of contact with others, it will provide plenty of opportunities for the infection to spread. Finally, it depends on the probability that the infection is passed on during each of these encounters, assuming the other person is susceptible.
R therefore depends on four factors: the duration of time a person is infectious; the average number of opportunities they have to spread the infection each day they’re infectious; the probability an opportunity results in transmission; and the average susceptibility of the population. I like to call these the ‘DOTS’ for short. Joining them together gives us the value of the reproduction number:
R = Duration × Opportunities × Transmission probability × Susceptibility
Breaking the reproduction number down into these DOTS components, we can see how different aspects of transmission trade off against each other. This can help us work out the best way to control an epidemic, because some aspects of the reproduction number will be easier to change than others. For example, widespread sexual abstinence would reduce the number of opportunities for hiv transmission, but it’s not an appealing or practical option for most people. Health agencies have therefore focused on getting people to use condoms, which reduce the probability of transmission during sex. In recent years, there has also been a lot of success with so-called pre-exposure prophylaxis (PrEP), whereby hiv-negative people take anti-hiv drugs to reduce their susceptibility to the infection.[47]
The type of transmission opportunities we’re interested in will depend on the infection. For influenza or smallpox, transmission can occur during face-to-face conversations, while infections like hiv and gonorrhea are spread mostly through sexual encounters. The trade-off in the DOTS means that if someone is infectious for twice as long, in transmission terms it’s equivalent to them making twice as many contacts. In the past, smallpox and hiv have at times both had an R of around 5.[48] However, people are generally infectious with smallpox for a shorter period, which means that there must be more opportunities to spread infection per day, or a higher transmission probability during each opportunity, to compensate.
The reproduction number has become a crucial part of modern outbreak research, but there’s another feature of contagion we also need to consider. Because R looks at the average level of transmission, it doesn’t capture some of the unusual events that can occur during outbreaks. One such event happened in March 1972, when a Serbian teacher arrived at Belgrade’s main hospital with an unusual mix of symptoms. He’d been given penicillin at his local medical centre to treat a rash, but severe haemorrhaging had followed. Dozens of students and staff in the hospital gathered to see what they presumed was a strange reaction to the drug. But it was no allergy. After the man’s brother also fell ill, staff realised what the real problem was, and what they had exposed themselves to. The man had been infected with smallpox, and there would be 38 more cases – all traceable to him – before the infections in Belgrade subsided.[49]
Although smallpox wouldn’t be eradicated globally until 1980, it was already gone from Europe, with no cases reported in Serbia since 1930. The teacher had likely caught the disease from a local clergyman who’d recently returned from Iraq. Several similar flare-ups had happened in Europe during the 1960s and 1970s, most of them travel-related. In 1961, a girl returned from Karachi, Pakistan to Bradford, England, bringing the smallpox virus with her and unwittingly infecting ten other people. An outbreak in Meschede, Germany, in 1969 also started with a visitor to Karachi. This time it was a German electrician who’d travelled there; he would pass the infection on to seventeen others.[50] However, these events weren’t typical: most cases who returned to Europe didn’t infect anyone.
In a susceptible population, smallpox has a reproduction number of around 4–6. This represents the number of secondary cases we’d expect to see, but it’s still just an average value: in reality there can be a lot of variation between individuals and outbreaks. Although the reproduction number provides a useful summary of overall transmission, it doesn’t tell us how much of this transmission comes from a handful of what epidemiologists call ‘superspreading’ events.
A common misconception about disease outbreaks is that they grow steadily generation-by-generation, with each case infecting a similar number of people. If an infection spreads from person-to-person, creating a chain of cases, we refer to it as ‘propagated transmission’. However, propagated outbreaks don’t necessarily follow the clockwork pattern of the reproduction number, growing by the exact same amount each generation. In 1997, a group of epidemiologists proposed the ‘20/80 rule’ to describe disease transmission. For diseases like hiv and malaria they’d found that 20 per cent of cases were responsible for around 80 per cent of transmission.[51] But like most biological rules, there were some exceptions to the 20/80 rule of transmission. The researchers had focused on sexually transmitted infections (STIs) and mosquito-borne infections. Other outbreaks didn’t always follow this pattern. After the 2003 sars epidemic – which had involved several instances of mass infection – there was renewed interest in the notion of superspreading. For sars, it seemed to be particularly important: 20 per cent of cases caused almost 90 per cent of transmission. In contrast, diseases like plague have fewer superspreading events, with the top 20 per cent of cases responsible for only 50 per cent of transmission.[52]
In other situations, an outbreak may not be propagated at all. It may be the result of ‘common source transmission’, with all cases coming from the same place. One example is food poisoning: outbreaks can often be traced to a specific meal or person. The most infamous case is that of Mary Mallon – often referred to as ‘Typhoid Mary’ – who carried a typhoid infection without symptoms. In the early twentieth century, Mallon was employed as a cook for several families around New York City, leading to multiple outbreaks of the disease and several deaths.[53]
During a common source outbreak, cases often appear within a short period of time. In May 1916, there was a typhoid outbreak in California a few days after a school picnic. Like Mallon, the cook who’d made the ice cream for the picnic had been carrying the infection without knowing.
Typhoid outbreak following a picnic in California, 1916[54]
We can therefore think of disease transmission as a continuum. At one end, we have a situation where a single person – such as Mary Mallon – generates all of the cases. This is the most extreme example of superspreading, with one source responsible for 100 per cent of transmission. At the other end, we have a clockwork epidemic where each case generates exactly the same number of secondary cases. In most cases, an outbreak will lie somewhere between these two extremes.
If there is potential for superspreading events during an outbreak, it implies that some groups of people might be particularly important. When researchers realised that 80 per cent of hiv transmission came from 20 per cent of cases, they suggested targeting control measures at these ‘core groups’. For such approaches to be effective, though, we need to think about how individuals are connected in a network – and why some people might be more at risk than others.
The most prolific mathematician in history was an academic nomad. Paul Erdős spent his care
er travelling the world, living from two half-full suitcases without a credit card or chequebook. ‘Property is a nuisance,’ as he put it. Far from being a recluse, though, he used his trips to accumulate a vast network of research collaborations. Fuelled by coffee and amphetemines, he’d turn up at colleagues’ houses, announcing that ‘my brain is open’. By the time he died in 1996, he’d published about 1,500 papers, with over eight thousand co-authors.[55]
As well as building networks, Erdős was interested in researching them. Along with Alfréd Rényi, he pioneered a way of analysing networks in which individual ‘nodes’ were linked together at random. The pair were particularly interested in the chance these networks would end up being fully connected – with a possible route between any two nodes – rather than split into distinct pieces. Such connectedness matters for outbreaks. Suppose a network represents sexual partnerships. If it’s fully connected, a single infected person could in theory spread an STI to everyone else. But if the network is split into many pieces, there’s no way for a person in one component to infect somebody in another.
It can also make a difference if there is a single path across the network, or several. If networks contain closed loops of contacts, it can increase STI transmission.[56] When there’s a loop, the infection can spread across the network in two different ways; even if one of the social links breaks, there’s still another route left. For STIs, outbreaks are therefore more likely to spread if there are several loops present in the network.
Although the randomness of Erdős–Rényi networks is convenient from a mathematical point of view, real life can look very different. Friends cluster together. Researchers collaborate with the same group of co-authors. People often have only one sexual partner at a time. There are also links that go beyond such clusters. In 1994, epidemiologists Mirjam Kretzschmar and Martina Morris modelled how STIs might spread if some people had multiple sexual partners at the same time. Perhaps unsurprisingly, they found that these partnerships could lead to a much faster outbreak, because they created links between very different parts of the network.
The Rules of Contagion Page 6