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

by Adam Kucharski

Rather than just viewing outbreaks in terms of whether they take off or not, we need to think about how to measure them and how to predict them.

Health agencies therefore needed to respond quickly: the longer it took them to tackle the epidemic, the larger their control efforts would need to be. In essence, opening one new treatment centre immediately was equivalent to opening four in a month’s time.

When it comes to contagion, history has shown that ideas about how things spread don’t always match reality.

In the fourth century, Chinese scholar Ge Hong described how the qinghao plant could reduce fevers. Extracts of this plant now form the basis for modern malaria treatments.

In 1902, Ross received the second ever Nobel Prize for medicine for his work on malaria.

Ross’s calculation showed that even if there were 48,000 mosquitoes in the area, on average they would generate only one new human infection.

Ross estimated that around 20 per cent of humans infected with malaria would recover each month. For malaria to remain endemic in the population, these two processes – infection and recovery – would need to balance each other out.

This was his crucial insight. It wasn’t necessary to get rid of every last mosquito to control malaria: there was a critical mosquito density, and once the mosquito population fell below this level, the disease would fade away by itself.

Farr’s method therefore focused on what shape epidemics take, not why they take that shape.[26] In contrast, Ross adopted a mechanistic approach. Rather than taking data and finding patterns that could describe the observed trends, he started by outlining the main processes that influenced transmission.

He showed how to examine epidemics in a dynamic way, treating them as a series of interacting processes rather than a set of static patterns.

It would take many more years for mosquito control to be widely adopted. Ross would not live to see the most dramatic reductions in malaria cases: the disease remained in England until the 1950s, and was only eliminated from continental Europe in 1975.[29]

Amazing fact

They focused their attention on one of the most important questions in infectious disease research: what causes epidemics to end? The pair noted that there were two popular explanations at the time. Either transmission ceased because there were no susceptible people left to infect, or because the pathogen itself became less infectious as the epidemic progressed. It would turn out that, in most situations, neither explanation was correct.[35]

Kermack and McKendick’s model suggests that this doesn’t happen: outbreaks can end before everyone picks up the infection. ‘An epidemic, in general, comes to an end before the susceptible population has been exhausted,’ as they put it.

When there are enough immune people to prevent transmission, we say that the population has acquired ‘herd immunity’.

Just as Ross suggested malaria could be controlled without removing every last mosquito, herd immunity makes it possible to control infections without vaccinating the entire population.

herd immunity allows vaccinated people to protect these vulnerable unvaccinated groups as well as themselves.

According to the SIR model, outbreaks need three things to take off: a sufficiently infectious pathogen, plenty of interactions between different people, and enough of the population who are susceptible. Near the critical herd immunity threshold, a small change in one of these factors can be the difference between a handful of cases and a major epidemic.

As more people adopt, it becomes harder and harder to meet someone who has not yet heard about the idea. Although the overall number of adopters continues to grow, there are fewer and fewer people adopting it at each point in time.

In contrast, Bass could use the early shape of the adoption curve to estimate the relative roles of innovators and everyone else, who he called ‘imitators’.

Based on outbreak dynamics, we can come up with a more precise definition for this take-off point. Specifically, we can work out when the number of new adoptions is growing fastest. After this point, a lack of susceptible people will start to slow the spread, causing the outbreak to eventually plateau.

The most familiar scenario that would create this outbreak shape is therefore a fictional one: a group of zombies hunting down the last few surviving humans.

Back in real life, there are a few infections that affect their hosts in a way that increases transmission. Animals infected with rabies are often more aggressive, which helps the virus to spread through bites,[55] and people who have malaria can give off an odour that makes them more attractive to mosquitoes.[56] But such effects generally aren’t large enough to overcome declining numbers of susceptibles in the later stages of an epidemic.

Although she matched the results of the male student who ranked seventh, her performance wasn’t included in the official listing (it wasn’t until 1948 that women were allowed to receive Cambridge degrees[60]

Some happenings simmered away over time, gradually affecting everyone. Others rose sharply then fell. Some caused large outbreaks then settled down to a lower endemic level. There were outbreaks that came in steady waves, rising and falling with the seasons, and outbreaks that recurred sporadically.

‘I can calculate the motion of heavenly bodies but not the madness of people.’ According to legend, Isaac Newton said this after losing a fortune investing in the South Sea Company.

Great academic minds have a mixed record when it comes to financial markets.

The Asian Financial Crisis was a prime example.[3] It wasn’t the crisis itself that hit funds like LTCM; it was the indirect shockwaves that propagated through other markets.

Correlation isn’t just some niche topic to keep a mathematically minded intern occupied. It turns out to be crucial for understanding why 2008 would end with a full-blown financial crisis. It can also help explain how contagion spreads more generally, from social behaviour to sexually transmitted infections.

CDOs were based on an idea borrowed from the life insurance industry. Insurers had noticed that people were more likely to die following the death of a spouse, a social effect known as ‘broken heart syndrome’. In the mid-1990s, they developed a way to account for this effect when calculating insurance costs.

‘Human beings have limited foresight and great imagination,’ financial mathematician Emanuel Derman once noted, ‘so that, inevitably, a model will be used in ways its creator never intended.’

What did Bernanke think the worst-case scenario was? What would happen if house prices dropped across the country? ‘It’s a pretty unlikely possibility,’ Bernanke said.[7] ‘We’ve never had a decline in house prices on a nationwide basis.’

In February 2007, a year before Bear Stearns collapsed, credit specialist Janet Tavakoli wrote about the rise of investment products like CDOs. She was particularly unimpressed with the models used to estimate correlations between mortgages. By making assumptions that were so far removed from reality, these models had in effect created a mathematical illusion, a way of making high-risk loans look like low-risk investments.[8] ‘Correlation trading has spread through the psyche of the financial markets like a highly infectious thought virus,’ Tavakoli noted. ‘So far, there have been few fatalities, but several victims have fallen ill, and the disease is rapidly spreading.’

The conference attendees came from a range of scientific fields. One was ecologist George Sugihara. His lab in San Diego focused on marine conservation, using models to understand the dynamics of fish populations. Sugihara was also familiar with the world of finance, having spent four years working for Deutsche Bank in the late 1990s.

Those years with Deutsche Bank would be highly profitable for both parties. Although the data involved financial stocks rather than fish stocks, Sugihara’s experience with predictive models successfully transferred across to his new field. ‘Basically, I modelled the fear and greed of mobs that trade,’ he later told Nature.[13]

In a 2013 piece for The Lancet medical journal, he noted the apparent similarity between disease outbreaks and financial bubbles. ‘The recent rise in financial assets and the subsequent crash have rather precisely the same shape as the typical rise and fall of cases in an outbreak of measles or other infection,’ he wrote.

‘When something is going up without a convincing explanation about why it’s going up, that really is an illustration of the foolishness of the people,’ as he put it.[14]

‘Newton did not just taste of the Bubble’s madness, but drank deeply of it.’ Some people timed their investments better. Bookseller Thomas Guy, an early investor, got out before the peak and used the profits to establish Guy’s Hospital in London.[20]

In China, some pyramid schemes – or ‘business cults’ as the authorities call them – have reached a huge scale. Since 2010, several schemes have managed to recruit over a million investors each.[23]

The four phases of a bubble Adapted from original graphic by Jean-Paul Rodrigue

One signature feature of a bubble is that it grows rapidly, with the rate of buying activity increasing over time. Bubbles often feature what’s known as ‘super-exponential’ growth;[25] not only does the buying activity accelerate, the acceleration itself accelerates.

Things aren’t so simple for financial bubbles. People can leverage their trades, borrowing money to cover additional investments. This makes it much harder to estimate how much susceptibility there is, and hence what phase of the bubble we’re in.

But the claim was nonsense. In 1998, Andrew Odlyzko, then a researcher at AT&T labs, realised the internet was growing at a much slower rate, taking about a year to double in size.

Arguably the greatest bubble of recent years has been Bitcoin, which uses a shared public transaction record with strong encryption to create a decentralised digital currency. Or as comedian John Oliver described it: ‘everything you don’t understand about money combined with everything you don’t understand about computers.’[28]

Each Bitcoin bubble involved a larger group of susceptible people, like an outbreak gradually making its way from a village into a town and finally into a city.

If susceptible populations are strongly connected, an epidemic will generally peak around the same time, rather than as a series of smaller outbreaks.

According to Jean-Paul Rodrigue, there is a dramatic shift during the main growth phase of a bubble. The amount of money available increases, while the average knowledge base decreases.

‘There is nothing so disturbing to one’s well being and judgment as to see a friend get rich’.[31] Investors’ desire to be part of a growing trend can even cause warnings about a bubble to backfire.

In the later stages of a bubble, fear can spread in much the same way as enthusiasm.

‘There was a lot of faith in the vast, complex financial system being self-correcting,’ he said. ‘The attitude was “we don’t need to know how the system works, instead we can concentrate on individual institutions”.’[36] Unfortunately, the events of 2008 would reveal the weakness in this approach. Surely there was a better way?