1 + R + R2 + R3 + … = 1/(1–R)
In other words, as long as the reproduction number is below 1, the expected outbreak size is equal to 1/(1–R). Even if you’re not especially interested in nineteenth-century mathematics, it’s worth taking a moment to appreciate how useful this shortcut is. Rather than having to simulate how an infection might stutter along from one generation to the next until it eventually fizzles out, we can instead estimate the final outbreak size directly from the reproduction number.[43] If R is 0.8, for example, we’d expect an outbreak with 1/(1–0.8) = 5 cases in total. And that’s not all we can do. We can also work backwards to estimate the reproduction number from the average outbreak size. If outbreaks consist of five cases on average, it means R is 0.8.
In my field, we regularly use this back-of-the-envelope calculation to estimate the reproduction number of new disease threats. During the early months of 2013, there were 130 human cases of H7N9 bird flu in China. Although most of these picked up the disease from contact with poultry, there were four clusters of infection that were likely to be the result of transmission between humans.[44] Because most people didn’t infect anyone else, the average size of a human H7N9 outbreak was 1.04 cases, suggesting that R in humans was a paltry 0.04.
This idea isn’t only useful for diseases. During the mid-2000s, Jonah Peretti and Duncan Watts applied the same method to marketing campaigns. It meant they could get at the underlying transmissibility of an idea, rather than just describing what a campaign had looked like. In 2004, for example, anti gun violence group The Brady Campaign had sent out e-mails asking people to support new gun control measures. They encouraged recipients to forward the e-mails to their friends; some of these friends then forwarded the messages to their friends, and so on. For each e-mail that was sent, on average around 2.4 people ended up seeing the message. Based on this typical outbreak size, the reproduction number of the campaign was about 0.58. A subsequent e-mail campaign aimed to raise money for Hurricane Katrina relief efforts; this time R was 0.77. However, there wasn’t always so much transmission. Spare a thought for the marketing executives trying to spread messages about cleaning products: Peretti and Watts found that e-mails promoting Tide Coldwater detergent had an R of only 0.04 (i.e. the same as H7N9 bird flu). Whereas most of the Katrina e-mails had spread between multiple people, over 99 per cent of the Tide outbreaks stuttered to an end after only one transmission event.[45]
Why do we care about measuring an infection if it won’t lead to a large outbreak? For biological pathogens, a big concern is that these infections will adapt to their new hosts. During a small outbreak, viruses could pick up mutations that enable them to transmit more easily. The more people that get infected, the more chances for such adaptation. Before sars sparked a major outbreak in Hong Kong in February 2003, there were a series of small clusters of infection in Guangdong province, in southern China.[46] Between November 2002 and January 2003, seven outbreaks were reported in Guangdong, with between one and nine cases in each. The average outbreak size was five cases, suggesting that R may have been around 0.8 during this period. But by the time of the Hong Kong outbreak a couple of months later, sars had a far more troubling R of more than 2.
There are several reasons the reproduction number of an infection may increase. Recall that R depends on the four DOTS: duration of infection, opportunities for transmission, transmission probability during each opportunity, and average susceptibility. For biological viruses, all of these features can influence transmission. Of the viruses that can spread among humans, the most successful tend to cause longer infections (i.e. larger duration) and spread directly from one person to another rather than via an intermediate source (i.e. more opportunities).[47] Transmission probability can also make a difference: bird flu viruses struggle to spread among people because they can’t latch onto the cells in our airway as easily as human viruses can.[48]
The same sort of adaptation can happen with online content. There are many examples of online memes – such as posts and images – evolving to increase their catchiness. When Facebook researcher Lada Adamic and her colleagues analysed the spread of memes on the social network, they noticed that content would often change over time.[49] One example was a post that read: ‘No one should die because they cannot afford health care and no one should go broke because they get sick.’ In its original form, the meme was shared almost half a million times. But variants soon emerged, with one in every ten posts adding a mutation to the wording. Some of these edits helped the meme propagate; when people included phrases like ‘post if you agree’, the meme was almost twice as likely to spread. The meme was also highly resilient. After an initial peak in popularity, it persisted in one form or another for at least two years.
Even so, there seems to be a limit to the potential contagiousness of online content. The most popular trends on Facebook during 2014–2016 all had a reproduction number of around 2. This limit seems to occur because the different components of transmission trade off against each other. Some trends – like the ice bucket challenge – involved only a few nominations per person, but came with a high probability of transmission during each nomination. Other content, such as videos and links, had far more opportunities to spread, but in reality only a few friends who saw the post reshared it.[50] Remarkably, there were no examples of Facebook content that reached lots of
