The Rules of Contagion

by Adam Kucharski

  To answer these questions, our research group started to develop mathematical models of the outbreaks. Such approaches are now commonly used in public health, as well as appearing in several other fields of research. But where do these models originally come from? And how do they actually work? It’s a story that starts in 1883 with a young army surgeon, a water tank and an angry staff officer.

  Ronald ross had wanted to be a writer, but his father pushed him into medical school. His studies at St Bartholomew’s in London struggled to compete with his poems, plays and music, and when Ross took his two qualifying exams in 1879, he passed only the surgery one. This meant he could not join the colonial Indian Medical Service, his father’s preferred career path.[10]

  Unable to practice general medicine, Ross spent the next year sailing the Atlantic as a ship’s surgeon. Eventually he passed his remaining medical exam and scraped into the Indian Medical Service in 1881. After two years in Madras, Ross moved to Bangalore to take up a post as Garrison Surgeon in September 1883. From his comfortable colonial viewpoint, he claimed it was a ‘picture of pleasure’, a city of sun, gardens and pillared villas. The only problem, as he saw it, was the mosquitoes. His new bungalow seemed to attract far more than the other army rooms. He suspected it was something to do with the water barrel sitting outside his window, which was surrounded by the insects.

  Ross’s solution was to tip over the tank, destroying the mosquitoes’ breeding ground. It seemed to work: without the stagnant water, the insects left him alone. Spurred on by his successful experiment, he asked his staff officer if they could remove the other water tanks too. And while they were at it, why not also get rid of the vases and tins that lay scattered around the mess? If the mosquitoes had nowhere to breed, they would have little option but to move on. The officer wasn’t interested. ‘He was very scornful and refused to allow men to deal with them,’ Ross later wrote, ‘for he said it would be upsetting to the order of nature, and as mosquitoes were created for some purpose it was our duty to bear with them.’

  The experiment would turn out to be the first in a lifelong analysis of mosquitoes. The second study would come over a decade later, inspired by a conversation in London. In 1894, Ross had travelled back to England for a one-year sabbatical. The city had changed a lot since his last visit: Tower Bridge had been completed, Prime Minister William Gladstone had just resigned, and the country was about to get its first film parlour.[11] When Ross arrived, though, his mind was focused elsewhere. He wanted to catch up on the latest malaria research. In India, people regularly fell ill with the disease, which could lead to fever, vomiting, and sometimes death.

  Malaria is one of the oldest diseases known to humanity. In fact, it may have been with us for our entire history as a species.[12] However, its name comes from Medieval Italy. Those who caught a fever would often blame ‘mala aria’: bad air.[13] The name stuck, as did the blame. Although the disease was eventually traced to a parasite called Plasmodium, when Ross arrived back in England the cause of its spread was still a mystery.

  In London, Ross called on biologist Alfredo Kanthack at St Bartholomew’s, hoping to learn about developments he may have missed while in India. Kanthack said that if Ross wanted to know more about parasites like malaria, he should go and speak to a doctor called Patrick Manson. For several years, Manson had researched parasites in southeastern China. While there, he had discovered how people get infected with a particularly nasty family of microscopic worms called filariae. These parasites were small enough to get into a person’s bloodstream and infect their lymph nodes, causing fluid to accumulate within the body. In severe cases, a person’s limbs could swell to many times their natural size, a condition known as elephantiasis. As well as identifying how the filariae caused disease, Manson had shown that when mosquitoes fed on infected humans, they could also suck up the worms.[14]

  Manson invited Ross into his lab, teaching him how to find parasites like malaria in infected patients. He also pointed Ross to recent academic papers he’d missed while out in India. ‘I visited him often and learnt all he had to tell me,’ Ross later recalled. One winter afternoon, they were walking down Oxford Street, when Manson made a comment that would transform Ross’s career. ‘Do you know,’ he said, ‘I have formed the theory that mosquitoes carry malaria just as they carry filariae.’

  Other cultures had long speculated about a potential link between mosquitoes and malaria. British geographer Richard Burton noted that in Somalia, it was often said that mosquito bites brought on deadly fevers, though Burton himself dismissed the idea. ‘The superstition probably arises from the fact that mosquitoes and fevers become formidable about the same time,’ he wrote in 1856.[15] Some people had even developed treatments for malaria, despite not knowing what caused the disease. In the fourth century, Chinese scholar Ge Hong described how the qinghao plant could reduce fevers. Extracts of this plant now form the basis for modern malaria treatments.[16] (Other attempts were less successful: the word ‘abracadabra’ originated as a Roman spell to ward off the disease.[17])

  Ross had heard the speculation linking mosquitoes and malaria, but Manson’s argument was the first to really convince him. Just as mosquitoes ingested those tiny worms when they fed on human blood, Manson reckoned that they could also pick up malaria parasites. These parasites then reproduced within the mosquito before somehow making their way back into humans. Manson suggested that drinking water might be the source of infection. When Ross returned to India, he set out to test the idea, with an experiment that would be unlikely to pass a modern ethics board.[18] He got mosquitoes to feed on an infected patient then lay eggs in a bottle of water; once the eggs had hatched, he paid three people to drink the water. To his disappointment, none of them got malaria. So how did the parasites get into people?

  Ross eventually wrote to Manson with a new theory, suggesting that the infection might spread through mosquito bites. The mosquitoes injected some saliva with each bite: maybe this was enough to let the parasites in? Unable to recruit enough human volunteers for another study, Ross experimented with birds. First, he collected some mosquitoes and got them to feed on the blood of an infected bird. Then he let these mosquitoes bite healthy birds, which soon came down with the disease as well. Finally, he dissected the saliva glands of the infected mosquitoes, where he found malaria parasites. Having discovered the true route of transmission, he realised just how absurd their previous theories had been. ‘Men and birds don’t go about eating dead mosquitoes,’ he told Manson.

  In 1902, Ross received the second ever Nobel Prize for medicine for his work on malaria. Despite contributing to the discovery, Manson did not share the award. He only found out that Ross had won when he saw it in a newspaper.[19] The once close friendship between mentor and student gradually splintered into a sharp animosity. Though he was a brilliant scientist, Ross could be a divisive colleague. He got into a series of disputes with his rivals, often involving legal action. In 1912, he even threatened to sue Manson for libel.[20] The offence? Manson had written a complimentary reference letter for another researcher, who was taking up a professorship that Ross had recently vacated. Manson did not rise to the argument, choosing to apologise instead. ‘It takes two fools to make a quarrel,’ as he later put it.[21]

  Ross would continue to work on malaria without Manson. In the process, he’d find a new outlet for his single-minded stubbornness, and a new set of opponents. Having discovered how malaria spread, he wanted to demonstrate that it could be stopped.

  Malaria once had a much broader reach than it does today. For centuries, the disease stretched across Europe and North America, from Oslo to Ontario. Even as temperatures dropped during the so-called Little Ice Age in the seventeenth and eighteenth centuries, the biting cold of winter would still be followed by the biting mosquitoes of summer.[22] Malaria was endemic in many temperate countries, with ongoing transmission and a regular stream of new cases from one year to the next. Eight of Shakespeare’s plays include mentions o
f ‘ague’, a medieval term for malarial fever. The salt marshes of Essex, northeast of London, had been a notorious source of disease for centuries; when Ronald Ross was a student, he’d treated a woman who picked up malaria there.

  Having made the link between insects and infections, Ross argued that removing mosquitoes was the key to controlling malaria. His experiences in India – like the experiment with the water tank in Bangalore – had persuaded him that mosquito numbers could be reduced. But the idea went against popular wisdom. It was impossible to get rid of every last mosquito, went the argument, which meant there would always be some insects left, and hence potential for malaria to spread. Ross acknowledged that some mosquitoes would remain, but he believed that malaria transmission could still be stopped. From Freetown to Calcutta, his suggestions were at best ignored and at worst derided. ‘Everywhere, my proposal to reduce mosquitoes in towns was treated only with ridicule,’ he later recalled.

  In 1901, Ross had led a team to Sierra Leone to try and put his mosquito control ideas into practice. They cleared away cartloads of tins and bottles. They poisoned the standing water mosquitoes loved to breed in. And they filled potholes so ‘death-dealing street-puddles’, as Ross called them, couldn’t form on the roads. The results were promising: when Ross visited again a year later, there were far fewer mosquitoes. However, he had warned health authorities the effect would only last if the control measures continued. Funding for the clean up had come from a wealthy Glaswegian donor. When the money ran out, enthusiasm waned, and mosquito numbers increased once again.

  Ross had more success advising the Suez Canal Company the following year. They’d been seeing around 2,000 malaria cases a year in the Egyptian city of Ismailia. After intensive mosquito reduction efforts, this number fell below a hundred. Mosquito control was also proving effective elsewhere. When the French had attempted to build a canal in Panama during the 1880s, thousands of workers had died from malaria, as well as yellow fever, another mosquito-borne infection. In 1905, with the Americans now leading the Panama project, US Army Colonel William Gorgas oversaw an intensive mosquito control campaign, making it possible to complete the canal.[23] Meanwhile further south, physicians Oswaldo Cruz and Carlos Chagas were spearheading anti-malaria programmes in Brazil, helping to reduce cases among construction workers.[24]

  Despite these projects, many remained sceptical about mosquito control. Ross would need a stronger argument to persuade his peers. To make his point, he would eventually turn to mathematics. During those early years in the Indian Medical Service, he’d taught himself the subject to a fairly advanced level. The artist in him admired its elegance. ‘A proved proposition was like a perfectly balanced picture,’ he later suggested. ‘An infinite series died away into the future like the long-drawn variations of a sonata.’ Realising how much he liked the subject, he regretted not studying it properly at school. He was now too far into his career to change direction; what use was mathematics to someone working in medicine? ‘It was the unfortunate passion of a married man for some beautiful but inaccessible lady,’ as he put it.

  Ross put the intellectual affair behind him for a while, but returned to the subject after his mosquito discovery. This time, he found a way to make his mathematical hobby useful to his professional work. There was a vital question he needed to answer: was it really possible to control malaria without removing every mosquito? To find out, he developed a simple conceptual model of malaria transmission. He started by calc­ulating how many new human malaria infections there might be each month, on average, in a given geographic area. This meant breaking down the process of transmission into its basic components. For transmission to occur, he reasoned, there first needs to be at least one human in the area who is infectious with malaria. As an example, he picked a scenario where there was one infectious person in a village of 1,000. For the infection to pass to another human, an Anopheles mosquito would have to bite this infectious human. Ross reckoned only 1 in 4 mosquitoes would manage to bite someone. So if there were 48,000 mosquitoes in an area, he’d expect only 12,000 to bite a person. And because only 1 person in 1,000 was initially infectious, on average only 12 of those 12,000 mosquitoes would bite that one infectious person and pick up the parasite.

  It takes some time for the malaria parasite to reproduce within a mosquito, so these insects would also have to survive long enough to become infectious. Ross assumed only 1 in every 3 mosquitoes would make it this far, which meant that of the 12 mosquitoes with the parasite, only 4 would eventually become infectious. Finally, these mosquitoes would need to bite another human to pass on the infection. If, again, only 1 in 4 of them successfully fed off a human, this would leave a single infectious mosquito to transmit the virus. Ross’s calculation showed that even if there were 48,000 mosquitoes in the area, on average they would generate only one new human infection.

  If there were more mosquitoes, or more infected humans, by the above logic we’d expect more new infections per month. However, there is a second process that counteracts this effect: Ross estimated that around 20 per cent of humans infected with malaria would recover each month. For malaria to remain endemic in the population, these two processes – infection and recovery – would need to balance each other out. If the recoveries outpaced the rate of new infections, the level of disease eventually would decline to zero.

  This was his crucial insight. It wasn’t necessary to get rid of every last mosquito to control malaria: there was a critical mosquito density, and once the mosquito population fell below this level, the disease would fade away by itself. As Ross put it, ‘malaria cannot persist in a community unless the Anophelines are so numerous that the number of new infections compensates for the number of recoveries.’

  Ross calculated that even if there were 48,000 mosquitoes in a village that contained someone infected with malaria, it might only result in one additional human case

  When he wrote up the analysis in his 1910 book The Prevention of Malaria, Ross acknowledged that his readers might not follow all of his calculations. Still, he believed that they would be able to appreciate the implications. ‘The reader should make a careful study of those ideas,’ he wrote, ‘and will, I think, have little difficulty in understanding them, though he may have forgotten most of his mathematics’. Keeping with the mathematical theme, he called his discovery the ‘mosquito theorem’.

  The analysis showed how malaria could be controlled, but it also included a much deeper insight, which would revolutionise how we look at contagion. As Ross saw it, there were two ways to approach disease analysis. Let’s call them ‘descriptive’ and ‘mechanistic’ methods. In Ross’s era, most studies used descriptive reasoning. This involved starting with real-life data and working backwards to identify predictable patterns. Take William Farr’s analysis of a London smallpox outbreak in the late 1830s. A government statistician, Farr had noticed that the epidemic grew rapidly at first, but eventually this growth slowed until the outbreak peaked, then started to decline. This decline was almost a mirror image of the growth phase. Farr plotted a curve through case data to capture the general shape; when another outbreak started in 1840, he found it followed much the same path.[25] In his analysis, Farr didn’t account for the mechanics of disease transmission. There were no rates of infection or rates of recovery. This isn’t that surprising: at the time nobody knew that smallpox was a virus. Farr’s method therefore focused on what shape epidemics take, not why they take that shape.[26]

  In contrast, Ross adopted a mechanistic approach. Rather than taking data and finding patterns that could describe the observed trends, he started by outlining the main processes that influenced transmission. Using his knowledge of malaria, he specified how people became infected, how they infected others, and how quickly they recovered. He summarised this conceptual model of transmission using mathematical equations, which he then analysed to make conclusions about likely outbreak patterns.

  Because his analysis included specific assumptions about the t
ransmission process, Ross could tweak these assumptions to see what might happen if the situation changed. What effect might mosquito reduction have? How quickly would the disease disappear if transmission declined? Ross’s approach meant he could look forward and ask ‘what if?’, rather than just searching for patterns in existing data. Although other researchers had made rough attempts at this type of analysis before, Ross brought the ideas together into a clear, comprehensive theory.[27] He showed how to examine epidemics in a dynamic way, treating them as a series of interacting processes rather than a set of static patterns.

  Descriptive and mechanistic methods – one looking back and the other forward – should in theory converge to the same answer. Take the descriptive approach. With enough real-life data, it would be possible to estimate the effect of mosquito control: tip over a water tank, or remove mosquitoes in some other way, and we can observe what happens. Conversely, the predicted effect of mosquito control in Ross’s mathematical analysis should ideally match the real impact of such measures. If a control strategy genuinely works, both methods should tell us that it does. The difference is that with Ross’s mechanistic approach, we don’t need to knock over water tanks to estimate what effect it might have.

  Mathematical models like Ross’s often have a reputation for being opaque or complicated. But in essence, a model is just a simplification of the world, designed to help us understand what might happen in a given situation. Mechanistic models are particularly useful for questions that we can’t answer with experiments. If a health agency wants to know how effective their disease control strategy was, they can’t go back and rerun the same epidemic without it. Likewise, if we want to know what a future pandemic might look like, we can’t deliberately release a new virus and see how it spreads. Models give us the ability to examine outbreaks without interfering with reality. We can explore how things like transmission and recovery affect the spread of infection. We can introduce different control measures – from mosquito removal to vaccination – and see how effective they might be in different situations.