In the early twentieth century, this approach was exactly what Ross needed. When he announced that Anopheles mosquitoes spread malaria, many of his peers were unconvinced that mosquito control would reduce the disease. This made descriptive analysis problematic: it’s tricky to assess a control measure if it’s not being used. Thanks to his new model, however, Ross had convinced himself that long-term mosquito reduction would work. The next challenge was convincing everyone else.
From a modern viewpoint, it might seem strange that there was so much opposition to Ross’s ideas. Although the science of epidemiology was expanding, creating new ways to analyse disease patterns, the medical community didn’t view malaria in the same way that Ross did. Fundamentally, it was a clash of philosophies. Most physicians thought about malaria in terms of descriptions: when looking at outbreaks, they dealt in classifications rather than calculus. But Ross was adamant that the processes behind disease epidemics needed to be quantified. ‘Epidemiology is in fact a mathematical subject,’ he wrote in 1911, ‘and fewer absurd mistakes would be made regarding it (for example, those regarding malaria) if more attention were given to the mathematical study of it.’[28]
It would take many more years for mosquito control to be widely adopted. Ross would not live to see the most dramatic reductions in malaria cases: the disease remained in England until the 1950s, and was only eliminated from continental Europe in 1975.[29] Although his ideas eventually started to catch on, he lamented the delay. ‘The world requires at least ten years to understand a new idea,’ he once wrote, ‘however important or simple it may be.’
It wasn’t just Ross’s practical efforts that would spread over time. One of the team on that 1901 expedition to Sierra Leone had been Anderson McKendrick, a newly qualified doctor from Glasgow. McKendrick had top-scored in the Indian Medical Service exams and was scheduled to start his new job in India after the Sierra Leone trip.[30] On the ship back to Britain, McKendrick and Ross talked at length about the mathematics of disease. The pair continued to exchange ideas over the following years. Eventually, McKendrick would pick up enough maths to try and build on Ross’s analysis. ‘I have read your work in your capital book,’ he told Ross in August 1911. ‘I am trying to reach the same conclusions from differential equations, but it is a very elusive business, and I am having to extend mathematics in new directions. I doubt whether I shall be able to get what I want, but “a man’s reach must exceed his grasp”.’[31]
McKendrick would develop a scathing view of statisticians like Karl Pearson, who relied heavily on descriptive analysis rather than adopting Ross’s mechanistic methods. ‘The Pearsonians have as usual made a frightful hash of the whole business,’ he told Ross after reading a flawed analysis of malaria infections. ‘I have no sympathy with them, or their methods.’[32] 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. McKendrick believed the future of outbreak analysis lay with a more dynamic way of thinking. Ross shared this view. ‘We shall end by establishing a new science,’ he once told McKendrick. ‘But first let you and me unlock the door and then anybody can go in who likes.’[33]
One summer evening in 1924, William Kermack’s experiment exploded, spraying corrosive alkali solution into his eyes. A chemist by training, Kermack had been investigating the methods commonly used to study spinal fluids. He was working alone in Edinburgh’s Royal College of Physicians Laboratory that evening, and would eventually spend two months in hospital with his injuries. The accident left the 26-year-old Kermack completely blind.[34]
During his stay in hospital, Kermack asked friends and nurses to read mathematics to him. Knowing that he could no longer see, he wanted to practise getting information another way. He had an exceptional memory and would work through mathematical problems in his head. ‘It was incredible to find how much he could do without being able to put anything down on paper,’ remarked William McCrea, one of his colleagues.
After leaving hospital, Kermack continued to work in science but shifted his focus to other topics. He left his chemical experiments behind, and began to develop new projects. In particular, he started to work on mathematical questions with Anderson McKendrick, who had risen to become head of the Edinburgh laboratory. Having served in India for almost two decades, McKendrick had left the Indian Medical Service in 1920 and moved to Scotland with his family.
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 transmi
ssion, 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 different. It was explosive, with most of the island getting infected, and completely unexpected. The little-known virus from the overgrown forest was apparently entering a new era. ‘Public health officials should be aware of the risk of further expansion of Zika virus transmission,’ concluded epidemiologist Mark Duffy and his colleagues in their outbreak report.[41]
In Yap, Zika had been a curiosity rather than a major threat. Despite lots of people getting a fever or rash, nobody ended up in hospital. That changed when the virus arrived on the much larger islands of French Polynesia in late 2013. During the resulting outbreak, forty-two people with Guillain-Barré Syndrome arrived at the main city hospital in Papeete, on the northern coast of Tahiti. The gbs cases cropped up slightly later than the main Zika outbreak, which is what we’d expect for a syndrome that takes a couple of weeks to appear after an infection. Speculation about a possible link was confirmed when local scientist Van-Mai Cao-Lormeau and her colleagues discovered that almost all the gbs cases had recently been infected with Zika.[42]
As in Yap, the French Polynesia outbreak had been huge, with the majority of the population infected. And like Yap, the outbreak had been very brief, with the bulk of cases appearing over a few weeks. Given that our team had spent 2014–15 developing mathematical models to analyse dengue in the Pacific, we decided to turn our attention to Zika as well. Unlike the plain-coloured Anophelines that can fly miles to spread malaria, dengue and Zika are both spread by Aedes mosquitoes, best known for being stripey and lazy (‘aedes’ means ‘house’ in Latin). As a result, the infection generally spreads when humans move from one place to another.[43]
When we tried to get our model simulations to reproduce the dynamics of Zika in French Polynesia, we realised there must have been a large, dengue-like rate of spread to generate such an explosive outbreak.[44] The short span of the outbreak stood out even more when we considered the delays involved in the infection process. During each cycle of transmission, the virus has to get from a human into a mosquito then back into another human.
While analysing transmission rates in French Polynesia, we also estimated how many people were already infected when the first cases were reported in October 2013. Our model suggested there had been several hundred infections by this point, meaning the virus probably arrived in the country weeks if not months earlier. This result would link into another mystery: how did the Zika virus reach Latin America? After the first cases were reported in Brazil during May 2015, there was a lot of speculation about when the infection had been introduced to the continent, and by whom. One early hypothesis pointed to the FIFA World Cup, held in Brazil during June/July 2014, which had attracted over three million football fans from around the globe. Another candidate was the Va’a sprint canoe championship, held in Rio de Janeiro during August 2014. Unlike the World Cup, this smaller event had included a team from French Polynesia. So which explanation was most plausible?
According to evolutionary biologist Nuno Faria and his colleagues, neither theory was particularly good.[45] Based on the genetic diversity of Zika viruses circulating in Latin America by 2016, they reckoned that the infection was introduced much earlier than previously thought. The virus probably hit the continent in mid-to-late 2013. Although too early for the canoe championship or World Cup, the time range coincided with the Confederations Cup, a regional football tournament held in June 2013. What’s more, French Polynesia was one of the countries competing.
There was just one gap in the theory: the Confederations Cup occurred five months before the first Zika cases were reported in French Polynesia. But if the outbreak in French Polynesia had in reality started earlier than October 2013 – as our analysis suggested – it was just about plausible that it could have spread to Latin America during that summer. (Of course, we should be cautious about trying too hard to find a sport-shaped prologue for the Zika story: there’s always a chance that it was just a random person in the Pacific taking a random flight to Brazil sometime in 2013.)
As well as analysing past outbreaks, we can use mathematical models to look at what might happen in future. This can be particularly useful for health agencies faced with difficult decisions during an outbreak. One such difficulty came in December 2015, when Zika reached the Caribbean island of Martinique. A big concern was the island’s ability to handle gbs cases: if patients’ lungs failed, they would need to be put on ventilators. At the time, Martinique only had eight ventilators for a population of 380,000. Would it be enough?
To find out, researchers at Institut Pasteur in Paris developed a model of Zika transmission on the island.[46] The crucial thing they wanted to know was the overall shape of the outbreak. gbs cases who required a ventilator typically stayed on it for several weeks, so a short outbreak with a large peak could overwhelm the health system, while a longer, flatter outbreak would not. At the very start of the Martinique outbreak, there hadn’t been many cases, so the team used data from French Polynesia as a starting point. Of the forty-two gbs cases reported there in 2013/14, twelve had required ventilators. According to the Pasteur model, this meant they could have a big problem. If the outbreak in Martinique followed the same pattern as French Polynesia, the island would probably need nine ventilators, one more than they had available.
Fortunately, the Martinique outbreak wouldn’t be the same. As new data came in, it became clear that the virus wasn’t spreading as quickly as it had in French Polynesia. At the peak of the outbreak, the researchers expected there would be around three gbs cases needing ventilators. Even in the worst-case scenario, they estimated that seven ventilators would be enough. Their conclusion about this upper limit turned out to be correct: at the peak of the outbreak, there were five gbs cases on ventilators. Overall, there were thirty cases of gbs during the outbreak, with two deaths. Without adequate medical facilities, the outcome could have been much worse.[47]
These Zika studies are just a few illustrations of how Ross’s methods have influenced our understanding of infectious disease
s. From predicting the shape of an outbreak to evaluating control measures, mechanistic models have become a fundamental part of how we study contagion today. Researchers are using models to help health agencies respond to a whole host of outbreaks, from malaria and Zika to hiv and Ebola, in locations ranging from remote islands to conflict zones.
Ross would no doubt be glad to see how influential his ideas have been. Despite winning a Nobel Prize for his discovery that mosquitoes transmit malaria, he did not view this as his biggest achievement. ‘In my own opinion my principal work has been to establish the general laws of epidemics,’ he once wrote.[48] And he didn’t just mean disease epidemics.
Although kermack and mckendrick would later extend Ross’s mosquito theorem to other types of infections, Ross had wider ambitions. ‘As infection is only one of many kinds of events which may happen to such organisms, we shall deal with “happenings” in general,’ he wrote in the second edition of The Prevention of Malaria. Ross proposed a ‘Theory of Happenings’ to describe how the number of people affected by something – whether a disease or another event – might change over time.
Ross suggested that there are two main types of happening. The first type affects people independently: if it happens to you, it generally won’t increase or decrease the chances of it happening to someone else afterwards. According to Ross, this could include things like non-infectious diseases, accidents or divorce.[49] For example, suppose there is a new condition that can randomly affect anyone, but at first nobody in the population has it. If each person has a certain chance of becoming affected every year – and remains affected from that point onwards – we’d expect to see a rising pattern over time.
The Rules of Contagion Page 3