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

by Adam Kucharski

But outbreaks can come in many forms. They might involve things that bring harm – like malware, violence or financial crises – or benefits, like innovations and culture.

During 1918 and 1919, the infection would become a global epidemic – otherwise known as a pandemic – and would kill over fifty million people.

Over the following century, there would be four more flu pandemics.

Along the way, we’ll see the connections that are emerging between seemingly unrelated problems: from banking crises, gun violence and fake news to disease evolution, opioid addiction and social inequality.

Although the shape can vary a lot, it will typically include four main stages: the spark, growth, peak, and decline.

Most things don’t spark: for every influenza virus that jumps from animals to humans and spreads worldwide as a pandemic, there are millions that fail to infect any people at all.

Some outbreaks grow on even faster timescales. In May 2017, the WannaCry computer virus hit machines around the world, including crucial nhs systems. In its early stages, the attack was doubling in size almost every hour, eventually affecting more than 200,000 computers in 150 countries.[6]

As well as speed, there’s also the question of size: contagion that spreads quickly won’t necessarily cause a larger overall outbreak.

Researchers found increasing evidence of a link between Zika infection and neurological conditions: as well as gbs, Zika seemed to lead to pregnancy complications. The main concern was microcephaly, where babies develop a smaller brain than usual, resulting in a smaller skull.[6]

For me, the first challenge was to understand the dynamics of these Zika outbreaks. How easily did the infection spread? Were the outbreaks similar to dengue ones? How many cases should we expect? To answer these questions, our research group started to develop mathematical models of the outbreaks.

Malaria is one of the oldest diseases known to humanity. In fact, it may have been with us for our entire history as a species.[12]

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

If the recoveries outpaced the rate of new infections, the level of disease eventually would decline to zero.

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.

As Ross saw it, there were two ways to approach disease analysis. Let’s call them ‘descriptive’ and ‘mechanistic’ methods.

In Ross’s era, most studies used descriptive reasoning. This involved starting with real-life data and working backwards to identify predictable patterns.

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.

Descriptive and mechanistic methods – one looking back and the other forward – should in theory converge to the same answer.

Mechanistic models are particularly useful for questions that we can’t answer with experiments.

Models give us the ability to examine outbreaks without interfering with reality.

Most physicians thought about malaria in terms of descriptions: when looking at outbreaks, they dealt in classifications rather than calculus.

Traditional descriptive approaches were an important part of medicine – and still are – but they have limitations when it comes to understanding the process of transmission.

‘An epidemic, in general, comes to an end before the susceptible population has been exhausted,’ as they put it.

It’s because of a transition that happens mid-outbreak.

In the early stages of an epidemic, there are lots of s...

Over time, however, the pool of susceptible people shrinks. When this pool gets small enough, the situation flips around: there are more recoveries than new infections each day, so the epidemic begins to decline.

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

Likewise, herd immunity meant that the population as a whole could block transmission, even if some individuals were still susceptible.

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.

According to Rogers, four different types of people are responsible for the growth of a product: initial uptake comes from ‘innovators’, followed by ‘early adopters’, then the majority of the population, and finally ‘laggards’.

His research into innovations mostly followed this descriptive approach, starting with the S-curve and trying to find possible explanations.

In Ross’s simple model, the fastest growth occurs when just over 21 per cent of the potential audience have adopted the idea.

For the epidemic to keep increasing faster and faster, infectious people would have to actively start seeking out the remaining susceptibles in the later stages of the epidemic.

Back in real life, there are a few infections that affect their hosts in a way that increases transmission.

From innovations to infections, epidemics almost inevitably slow down as susceptibles become harder to find.

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.

Although some had speculated that housing was a bubble set to burst, many remained optimistic.

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]

The rise of complex financial products – and fall of funds like Long Term Capital Management – had persuaded central banks that they needed to understand the tangled web of financial trading.

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]

Bubbles generally involve a situation where investors pile in, leading to a rapid rise in price, followed by a crash when the bubble bursts.

This can lead to what is known as the ‘greater fool theory’: people may know it’s foolish to buy something expensive, but believe there is a greater fool out there, who will later buy it off them at a higher price.[22]

Given their unsustainable nature, pyramid schemes are generally illegal. But the potential for rapid growth, and the money it brings for the people at the top, means that they remain a popular option for scammers, particularly if there is a large pool of potential participants.

Unlike pyramid schemes, which follow a rigid structure, financial bubbles can be harder to analyse.

These bubble stages are analogous to the four stages of an outbreak: spark, growth, peak, decline.[24]

Things aren’t so simple for financial bubbles. People can leverage their trades, borrowing money to cover additional investments.

Still, it is sometimes possible to spot the signals of unsustainable growth. As the dot-com bubble grew in the late 1990s, a common justification for rising prices was the claim that internet traffic was doubling every 100 days.

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.[26]