Coronavirus: How maths is helping to answer crucial covid-19 questions

Comment | Comment 13 February 2020
A person in a hazmat suit reads a piece of paper

Maths can help better understand the spread of the new corona virus

Xiao Yijiu/Xinhua News Agency/PA Images

Back in mid-January, the current coronavirus outbreak was merely an unusual cluster of pneumonia cases. At least, that is what the tally of 41 confirmed infections in the Chinese city of Wuhan suggested. But then cases started appearing in other countries: first one in Thailand, then one in Japan, then another in Thailand, all among people who had travelled from Wuhan.

There were some flights to these places from Wuhan, but for three cases to have already appeared internationally, there must have been a lot more infections in the city that hadn’t been picked up. When researchers used flight data to estimate how many unreported cases there must have been to generate these patterns, it suggested the total in Wuhan was more likely to be in the thousands than the dozens.


During an outbreak, we rarely see the full picture at first, and this is where mathematics is essential. As well as the question of how many cases there really are, we also need to know how severe the disease really is: if someone is diagnosed with the new coronavirus, what is the chance it will prove fatal?

As of 11 February, there had been 395 cases confirmed outside China and one death (which may be the most accurate picture of the outbreak). At first glance, it seems the chance of death must therefore be 1/395 or 0.3 per cent. However, this calculation makes a crucial error.

There is generally a delay of a couple of weeks between someone falling ill and dying or getting better, so we can’t include recent cases in the analysis, because we don’t yet know what will happen to them. If we adjust for this delay – and instead focus on the cases that occurred long enough ago to know what happened to them – we instead end up with a fatality risk of around 1 per cent. We saw a similar data illusion occur during the Ebola outbreak in West Africa in 2014: early reports put the chance of death much lower than it should have been, causing unnecessary speculation about why it seemed unusually low.

Maths isn’t only useful for understanding the extent of illness and infection. It can also help us to work out what to do about it. In my book, The Rules of Contagion, I outline how to tell whether disease-control measures are having an effect.

In 1854, English physician John Snow famously removed the handle from Broad Street’s water pump in London, apparently ending a huge cholera outbreak. There was just one problem: the outbreak had already peaked by the time he got to the handle.

In the current coronavirus outbreak, several unprecedented interventions were introduced in China in late January, from citywide travel restrictions to school closures. Mathematicians are now working to understand whether these measures have curbed transmission, or whether they are the pump handles removed after the situation has already changed.

One of the challenges again comes from the delays involved. Because it takes time for infected people to show symptoms, and further time for ill people to be reported as cases, any change in transmission that happens today may not show up in the data for another week or two. It means that if we put in a new control measure and cases decline immediately, we can be fairly confident that we shouldn’t be taking the credit.

Having helped us understand the past and present of an outbreak, maths can also provide clues about what might happen in future.

In reality, we only ever see one version of an outbreak. With mathematical models, we can simulate dozens of alternatives. We can forecast where the outbreak might go, how quickly it might grow and what new control measures might do. In just a few month, the new coronavirus has turned into a major outbreak. With some mathematical help, the hope is that before too long, we really will be counting a small number of cases.

Adam Kucharski’s The Rules of Contagion: Why things spread – and why they stop is published by Profile/Wellcome

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