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 career 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.
Illustration of fully-connected and broken Erdős–Rényi networks
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. They found that such networks cropped up in all sorts of places: the electricity grid, neurons in worm brains, co-stars in film casts, even Erdős’s academic collaborations.[57] It was a remarkable finding, and more discoveries were about to follow.
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. In 1999, the pair proposed a simple mechanism to explain this extreme variability in connections: new nodes that joined the network would preferentially attach to already popular ones.[58] It was a case of the ‘rich get richer’.
The following year, a team at the University of Stockholm showed that the number of sexual partnerships in Sweden also appeared to follow this rule: the vast majority of people had slept with at most one person in the past year, whereas some reported dozens of partners. Researchers have since found similar patterns of sexual behaviour in countries ranging from Burkina Faso to the United Kingdom.[59]
What effect does this extreme variability in number of partners have on outbreaks? In the 1970s, mathematician James Yorke and his colleagues noticed there was a problem with the ongoing gonorrhea epidemic in the United States. Namely, it didn’t seem possible. For the disease to keep spreading, the reproduction number needed to
