The Rules of Contagion: Why Things Spread - and Why They Stop

by Adam Kucharski

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.

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?

In contrast to the rates and thresholds used by Kermack and McKendrick, R is a more intuitive – and general – way to think about contagion.

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.

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.

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.

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.

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.

In the past, smallpox and hiv have at times both had an R of around 5.[48]

Because R looks at the average level of transmission, it doesn’t capture some of the unusual events that can occur during outbreaks.

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.

In a susceptible population, smallpox has a reproduction number of around 4–6.

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]

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.

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’.

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.

The Erdős–Rényi model could capture the occasional long-range connections that occurred in real networks, but it couldn’t reproduce the clustering of interactions. This discrepancy was resolved in 1998, when mathematicians Duncan Watts and Steven Strogatz developed the concept of a ‘small-world’ network, in which most links were local but a few were long-range.

But a study of patients with gonorrhea had found that they’d had only 1.5 recent partners on average.[60] Even if the probability of transmission during sex was very high, it suggested that there simply weren’t enough encounters for the disease to persist. What was going on? If we just take the average number of partners, we are ignoring the fact that not everyone’s sex lives are the same. This variability is important: if someone has a lot of partners, we’d expect them to be both more likely to get infected and more likely to pass the infection on.

Anderson and May would later show that the more variation there was in the number of partners people had, the higher we’d expect the reproduction number to be.

By breaking contagion down into its basic DOTS components – duration, opportunities, transmission probability, susceptibility – and thinking about how network structure affects contagion, we can also estimate the risk posed by a new STI.

A closer look at the Colorado Springs data suggested that transmission was likely to be the result of delays in getting treatment among certain social groups, rather than an unusually high level of sexual activity.

The name emerged during the outbreak because media reports suggested Spain was the worst hit country in Europe. However, these reports weren’t quite what they seemed. At the time, Spain had no wartime censorship of news reports, unlike Germany, England and France, who quashed news of disease for fear that it might damage morale. The media blackout in these countries therefore made it appear that Spain had far more cases than anywhere else. (For their part, the Spanish media tried to blame the disease on the French.[76]

Haldane suggested that the public typically respond to an outbreak in one of two ways: flight or hide.

Alternatively, people may ‘hide’ during an outbreak, dodging situations that could potentially bring them into contact with the infection.

The idea was well-established in ecology: the structure of a network might make it resilient to minor shocks, but the same structure could also leave it vulnerable to complete collapse if put under enough stress.

‘The basic point was that all this integration did indeed reduce the probability of mini-crashes,’ Haldane said, ‘but increased the probability of a maxi-crash.’

Although each individual bank had diversified their investments, there was little diversity in the way they had collectively done it.

‘A sound banker, alas, is not one who foresees danger and avoids it,’ he once wrote, ‘but one who, when he is ruined, is ruined in a conventional way along with his fellows, so that no one can really blame him.’[88]

‘I was finding it a little hard to breathe. There was a bank run happening, in New York’s financial district. The people panicking were the Wall Streeters who best understood what was going on.’ Should he report what was happening? Given the severity of the crisis, Authers decided it would only make the situation worse. ‘Such a story on the FT’s front page might have been enough to push the system over the edge.’ His counterparts at other newspapers came to the same conclusion, and the news went uncovered.

The difference with finance is that firms don’t always need a direct exposure to fall ill.

During the early 1960s, US mathematician William Goffman suggested that the transfer of information between scientists worked much like an epidemic.[3] Just as diseases like malaria spread from person to person via mosquitoes, scientific research often passed from scientist to scientist via academic papers.

When the team plotted the data, the number of authors using the diagrams followed the familiar S-shaped adoption curve, rising exponentially before eventually plateauing.

Initially R was around 15 in the USA and potentially as high as 75 in Japan. It was one of the first times that researchers had tried to measure the reproduction number of an idea, putting a number on what had previously been a vague notion of contagiousness.

When researchers at Harvard University used digital trackers to monitor employees at two major companies, they found that the introduction of open-plan offices reduced face-to-face interactions by around 70 per cent.

During the past decade or so, researchers have increasingly tried to measure social contacts that are relevant for respiratory infections like flu. The best known is the polymod study, which asked over 7,000 participants in eight European countries who they interacted with.

Thanks to these studies, we now know that certain aspects of behaviour are fairly consistent around the world. People tend to mix with people of a similar age, with children having by far the most contacts.

Hong Kong residents typically have physical contact with around five other people each day; the UK is similar, but in Italy, the average is ten.[15]

At the start of this book, we saw that during the 2009 influenza pandemic, there were two outbreak peaks in the United Kingdom: one in the spring and one in the autumn. To understand what caused this pattern, we simply need to look at schools.

A few years ago, my colleagues and I looked at social contacts and infections during the 2009 flu pandemic in Hong Kong.[18] We found that it was the high number of social contacts among children that drove the pandemic, with a drop in contacts and infection after childhood.

Infections are spread by people, and there are more major flight routes linking Mexico and China – such as those via London – than those connecting Mexico with places like Barbados. China might be far away for a crow, but it’s relatively close for a human. It turns out that the spread of flu in 2009 is much easier to explain if we instead define distances according to airline passenger flows.

Once the 2009 flu pandemic arrived in a country, however, long travel distance seemed to be less important for transmission. In the US, the virus spread like a ripple, gradually travelling from the southeast outwards. It took about three months to move 2,000 kilometres across the eastern US, which works out at a speed of just under 1 km/h. On average, you could have outwalked it.[21]

To simulate these local movements, researchers often use what’s known as a ‘gravity model’. The idea is that we are drawn to places depending on how close and populous they are, much like larger, denser planets have a stronger gravitational pull.

To work out how people might actually travel, Belgian engineer Henri-Guillaume Desart designed the first ever gravity model in 1846. His analysis showed that there would be a lot of demand for local trips, an idea that was ignored by rail operators on the other side of the channel. The British railway network would probably have been far more efficient had it not been for this oversight.[23]

In 2007, physician Nicholas Christakis and social scientist James Fowler published a paper titled ‘The Spread of Obesity in a Large Social Network over 32 Years’.