I am an applied mathematician. Like, very applied. I use mathematics, sure, but I’m far more interested in what maths can do to solve real problems. Specifically, how can mathematics be used to tackle issues in infectious diseases?

The best thing about maths, from my point of view, is that it has one incredible superpower: it can predict the future. That’s amazing. And very useful, of course.

When teaching my undergraduate students the required details to make these predictions, I stumbled upon a very profound realisation. Namely, that the Jacobian matrix – a technical thing from linear algebra – is a) just about the most massively useful thing ever and b) a glorious way to reconcile two apparently disparate strands of mathematics.

Don’t believe me? Read on and hopefully you too will become a convert to the church of the Jacobian…

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If you take two identical cups, fill one with warm water and one with cold and put them in the freezer, you’d expect the cooler one to freeze first, but it doesn’t always. In fact, in many circumstances, it is the warm water that freezes first.

This is the Mpemba paradox, named after Erasto Mpemba, who observed it as a schoolboy in Tanzania in the 1960s. When physicist Dr Denis G. Osbourne visited his school, Mpemba took the opportunity to ask about his strange observation. Although initially skeptical, Osbourne later reproduced the observations and several years later, in 1969, they jointly published the result. Since then, it has been reproduced in many experimental studies and its origins have been debated extensively.

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The prisoner’s dilemma often features in television programmes (such as ITV’s Golden Balls) where two contestants have to decide whether they want to share or steal a pot of money. They make their choices in secret from one another and then their decisions are simultaneously revealed.

Let the tuple $(a, b)$ mean that you get $a$ and your opponent gets $b$: so $(3, -1)$ represents you winning £3 and your opponent losing £1. We introduce the pay-off matrix for a non-iterated (played only once) prisoner’s dilemma:

Each player is given the opportunity either to defect or to cooperate. In the original set-up with prisoners the option of defecting meant betraying the other and testifying whilst cooperation represented remaining silent.

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Imagine that we have a group of 20 volunteers. We give all 20 people identical pills, and measure a response in each of the people. The responses would not all be the same—there is always some variability. If we divide the 20 responses randomly into two groups of 10, the means of the two groups will therefore not be identical.

If we had instead given each group of 10 people different pills (say drug A and drug B) then we would also find that the means of the two groups differed. If drug A was better than B then the mean response of the 10 people given A would be bigger than the mean of the 10 responses to B. But of course the response of group A might well have been bigger, even if drugs A and B were actually identical pills.

It is one of the jobs of applied statisticians to tell us how to distinguish between random variability and real effects. They can tell us how big the difference between the means for A and B must be before we believe that A is really better than B and not just the result of random variability.

It is the aim of this article to persuade you that the ways of doing this that are commonly taught give rise to far more wrong decisions than most people realise. This is not trivial. It gives rise to the publication of discoveries that are untrue. For example, it may result in the approval of medical treatments that don’t work.

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