VCR ownership over time in the United States
Data: Consumer Electronics Association
In the 1960s, marketing researcher Frank Bass developed what was essentially an extended version of Ross’s model.[52] Unlike Rogers’s descriptive analysis, Bass used his model to look at the timescale of adoption as well as the overall shape. By thinking about the way people might adopt innovations, Bass was able to make predictions about the uptake of new technology. In Rogers’s curve, innovators are responsible for the first 2.5 per cent of adoptions, with everyone else in the remaining 97.5 per cent. These values are somewhat arbitrary: because Rogers relied on a descriptive method, he needed to know the full shape of the S-curve; it was only possible to categorise people once an idea had been fully adopted. 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’. In a 1966 working paper, he predicted that new colour television sales – then still rising – would peak in 1968. ‘Industry forecasts were much more optimistic than mine,’ Bass later noted,[53] ‘and it was perhaps to be expected that my forecast would not be well received.’ Bass’ prediction wasn’t popular, but it ended up being much closer to reality. New sales indeed slowed then peaked, just as the model suggested they would.
As well as looking at how interest plateaus, we can also examine the early stages of adoption. When Everett Rogers published the S-curve in the early 1960s, he suggested that a new idea had ‘taken off’ once 20–25 per cent of people had adopted it. ‘After that point, it is probably impossible to stop the further diffusion of a new idea,’ he argued, ‘even if one wishes to do so’. 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. In Ross’s simple model, the fastest growth occurs when just over 21 per cent of the potential audience have adopted the idea. Remarkably, this is the case regardless of how easily the innovation spreads.[54]
Ross’s mechanistic approach is useful because it shows us what different types of happenings might look like in real life. Think about how the VCR adoption curve compares with the home ownership one: both eventually plateau, but the VCR curve grows exponentially at first. Simple models of contagion will usually predict this kind of growth, because each new adoption creates even more adoptions, whereas models of independent happenings will not. It doesn’t mean that exponential growth is always a sign that something is contagious – there might be other reasons why people increasingly adopt a technology – but it does show how different infection processes can affect the shape of an outbreak.
If we think about the dynamics of an outbreak, we can also identify shapes that would be very unlikely in reality. Imagine a disease epidemic that increases exponentially until all of the population is affected. What would be required to generate this shape?
In large epidemics, transmission generally slows down because there aren’t many susceptible people left to infect. 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. It’s the equivalent of you catching a cold, finding all your friends who hadn’t got it yet and deliberately coughing on them until they got infected. 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.
Illustration of an outbreak curve that grows exponentially until everyone is affected
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. What’s more, many infections have the opposite effect on behaviour, causing lethargy or inactivity, which reduces the potential for transmission.[57] From innovations to infections, epidemics almost inevitably slow down as susceptibles become harder to find.
Ronald ross had planned to study a whole range of outbreaks, but as his models became more complicated, the mathematics became trickier. He could outline the transmission processes, but he couldn’t analyse the resulting dynamics. That’s when he turned to Hilda Hudson, a lecturer at London’s West Ham Technical Institute.[58] The daughter of a mathematician, Hudson had published her first piece of research in the journal Nature when she was ten years old.[59] She later studied at the University of Cambridge, where she was the only woman in her year to get first class marks in mathematics. 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]).
Hudson’s expertise made it possible to expand the Theory of Happenings, visualising the patterns the different models could produce. 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
