Together, the pair extended Ross’s ideas to look at epidemics in general. They focused their attention on one of the most important questions in infectious disease research: what causes epidemics to end? The pair noted that there were two popular explanations at the time. Either transmission ceased because there were no susceptible people left to infect, or because the pathogen itself became less infectious as the epidemic progressed. It would turn out that, in most situations, neither explanation was correct.[35]
Like Ross, Kermack and McKendrick started by developing a mathematical model of disease transmission. For simplicity, they assumed the population mixed randomly in their model. Like marbles being shaken in a jar, everyone in the population has an equal chance of meeting everyone else. In the model, the epidemic sparks with a certain number of infectious people, and everyone else susceptible to infection. Once someone has recovered from infection, they are immune to the disease. We can therefore put the population into one of three groups, based on their disease status:
Given the names of the three groups, this is commonly known as the ‘SIR model’. Say a single influenza case arrives in a population of 10,000 people. If we simulate a flu-like epidemic using the SIR model, we get the following pattern:
Simulated influenza outbreak using the SIR model
The simulated epidemic takes a while to grow because only one person is infectious at the start, but it still peaks within fifty days. And by eighty days, it’s all but over. Notice that at the end of the epidemic, there are still some susceptible people left. If everyone had been infected, then all 10,000 people would have eventually ended up in the ‘recovered’ group. Kermack and McKendick’s model suggests that this doesn’t happen: outbreaks can end before everyone picks up the infection. ‘An epidemic, in general, comes to an end before the susceptible population has been exhausted,’ as they put it.
Why doesn’t everyone get infected? It’s because of a transition that happens mid-outbreak. In the early stages of an epidemic, there are lots of susceptible people. As a result, the number of people who become infected each day is larger than the number who recover, and the epidemic grows. 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. There are still susceptible people out there who could be infected, but there are so few left that an infectious person is more likely to recover than meet one.
To illustrate the effect, Kermack and McKendrick showed how the SIR model could reproduce the dynamics of a 1906 outbreak of plague in Bombay (now Mumbai). In the model, the pathogen remains equally infectious over time; it is the shifting numbers of susceptible and infectious people that lead to the rise and fall.
The 1906 plague outbreak in Bombay, with SIR model shown alongside real data
The crucial change happens at the peak of the epidemic. At this point, there are so many immune people – and so few susceptible – that the epidemic cannot continue to grow. The epidemic will therefore turn over and start its decline.
When there are enough immune people to prevent transmission, we say that the population has acquired ‘herd immunity’. The phrase was originally coined by statistician Major Greenwood in the early twentieth century (Major was his first name, his army rank was actually captain).[36] Psychologists had previously used ‘herd instinct’ to describe groups that acted as a collective rather than as individuals.[37] Likewise, herd immunity meant that the population as a whole could block transmission, even if some individuals were still susceptible.
The concept of herd immunity would find popularity several decades later, when people realised it could be a powerful tool for disease control. During an epidemic, people naturally move out of the susceptible group as they become infected. But for many infections, health agencies can move people out of this group deliberately, by vaccinating them. Just as Ross suggested malaria could be controlled without removing every last mosquito, herd immunity makes it possible to control infections without vaccinating the entire population. There are often people who cannot be vaccinated – such as newborn babies or those with compromised immune systems – but herd immunity allows vaccinated people to protect these vulnerable unvaccinated groups as well as themselves.[38] And if diseases can be controlled through vaccination, they can potentially be eliminated from a population. This is why herd immunity has found its way into the heart of epidemic theory. ‘The concept has a special aura,’ as epidemiologist Paul Fine once put it.[39]
As well as looking at why epidemics end, Kermack and McKendrick were also interested in the apparently random occurrence of outbreaks. Analysing their model, they found that transmission was highly sensitive to small differences in the characteristics of the pathogen or human population. This explains why large outbreaks can seemingly appear from nowhere. 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. Near the critical herd immunity threshold, a small change in one of these factors can be the difference between a handful of cases and a major epidemic.
Zika and Guillain-Barré syndrome cases in French Polynesia, 2013/14
Data: French Polynesia Ministry of Health[40]
The first reported outbreak of Zika began on the Micronesian island of Yap in early 2007. Before then, only fourteen human cases of Zika had ever been spotted, scattered across Uganda, Nigeria, and Senegal. But the Yap outbreak was
