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

by Adam Kucharski

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.

Economist Charles Kindleberger, who wrote the landmark book Manias, Panics, and Crashes in 1978, along with Robert Aliber, emphasised the role of social contagion during this phase of a bubble: ‘There is nothing so disturbing to one’s well being and judgment as to see a friend get rich’.[31]

Arinaminpathy recalled that, pre-2007, it was unusual to study the financial system as a whole. ‘There was a lot of faith in the vast, complex financial system being self-correcting,’ he said.

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]

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.

R is particularly useful because it tells us whether to expect a large outbreak or not.

However, if R is above one, the level of infection will rise on average, creating the potential for a large epidemic.

If R is 2, an initial infected person will generate two cases 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.

Much of this popularity is down to Robert May and his long-standing collaborator Roy Anderson.

Both had a background in ecology, which gave them a more practical outlook than the mathematicians who’d preceded them.

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’.

It turns out that there are four factors that influence the value of R. Uncovering them is the key to understanding how contagion works.

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.

R = Duration × Opportunities × Transmission probability × Susceptibility

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...

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]

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.

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 career travelling the world, living from two half-full suitcases without a credit card or chequebook.

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]

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.

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.

The small-world idea had addressed the issue of clustering and long-range links, but physicists Albert-László Barabási and Réka Albert spotted something else unusual about real-life networks. From film collaborations to the World Wide Web, they’d noticed that some nodes in the network had a huge number of connections, far more than typically appeared in the Erdős–Rényi or small-world networks.

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.

Identifying people who are at higher risk – and finding ways to reduce this risk – can help stop an outbreak in its early stages.

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.

Based on the genetic diversity of these viruses and the rate of hiv evolution, the team estimated that hiv had arrived into North America in 1970 or 1971.

This is true of many outbreaks: it’s often difficult to predict in advance what role a specific individual will play.

Put simply, the people driving the epidemic were generally the ones the health authorities didn’t know about. These people went undetected until they sparked a new set of infections, making it near impossible to predict superspreading events.[70]

Philosophers call it ‘moral luck’: the idea that we tend to view actions with unfortunate consequences as worse than equal actions without any repercussions.[71]

Viewing at-risk people as special or different can encourage a ‘them and us’ attitude, leading to segregation and stigma.

Blaming certain groups for outbreaks is not a new phenomenon.

In the sixteenth century, the English believed syphilis came from France, so referred to it as the ‘French pox’.

The sars epidemic would result in over eight thousand cases and several hundred deaths, across multiple continents.

It wasn’t just the direct cost of treating disease cases; it was the economic impact of closed workplaces, empty hotels and cancelled trade.

‘An external event strikes. Fear grips the system, which, in consequence, seizes. The resulting collateral damage is wide and deep.’

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

The flight response can also happen in finance. Faced with a crash, investors may cut their losses and sell off assets, driving prices even lower.

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

Hiding behaviour will generally help reduce disease transmission, even if it incurs a cost in the process.

In contrast, when banks hoard money it can amplify problems, as happened with the ‘credit crunch’ that hit economies in the run up to the 2008 crisis.

‘I remember back in 2004/5, writing a note about us having entered the era of “super-systemic risk” as a result of these sorts of infections.’

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.’

After Lehman Brothers collapsed, people across the banking industry started thinking in terms of epidemics.