By Robin Thompson
Junior Research Fellow, Christ Church, University of Oxford
π = 3.14… 
22/7
14th March and 22nd July are closely linked. These are Pi Day, and Pi Approximation Day, based on the representations of the numbers above in the month/day and day/month formats. Every Pi Day, mathematicians are commissioned to write articles like this one, some people recite the digits of π, and others use the day as an excuse to eat as much pie as possible. But why does π matter to mathematicians and mathematical biologists?
π is the ratio of any circle’s circumference to its diameter. This is what we are taught at school, and we learn a number of formulae and facts involving π, such as that the area of a circle is πr2, and that the trigonometric functions sine and cosine oscillate with period 2π. If we go on to study mathematics at university, we learn about Euler’s identity: e^iπ+1=0, and there are a large number of other expressions that involve π.
π just keeps appearing in mathematics, again and again! But this does not explain why π is particularly important for mathematical biologists. As explained by Professor Santiago Schnell two years ago, π is related to the formation of patterns.π is encoded in a zebra’s stripes, and in the spots on a leopard.
The number π is of great practical importance too. I study the mathematics of infectious disease epidemics, and use epidemiological models for forecasting how outbreaks are likely to progress. Models of the spread of many pathogens have involved π. As described above, the sine and cosine functions oscillate with period 2π. So, if we want to describe oscillating features of epidemics, such as changes in influenza infection rates within each year, then a natural simple assumption might be to use an expression involving π:
Infection rate,β(t)=β(1+σcos(2πt)),
where σ sets the amplitude of the seasonal variations in transmission.
But infection rates are not the only epidemiological quantities that oscillate seasonally. Following infection, hosts do not cause new infections immediately. Instead there is a latent period, during which an individual is infected but not yet causing new infections. Outbreaks of pathogens with short latent periods might be expected to progress more quickly than outbreaks of pathogens with long latent periods. Latent periods are a feature of infections with pathogens of humans, but also pathogens of animals and plants. An important plant disease – citrus greening – is devastating citrus groves in Florida and is responsible for drastic recent increases in the price of orange juice. A recent model representing the spread of citrus greening included a latent period that varied within each season. And, of course, π appeared again in the formula describing seasonal variations in the length of the latent period.
So, as you enjoy π day – maybe you too will be involved in a competition to recite π, or you will eat as much pie as possible – I will be reflecting about the huge practical importance of π. This number appears everywhere throughout mathematics and throughout mathematical biology.
-- Robin Thompson is a Junior Research Fellow at Christ Church, one of the constituent colleges of the University of Oxford. His research involves using mathematical models for forecasting during outbreaks of infectious disease in human, animal and plant populations. For more information about Dr Thompson or his most recent publications, or to collaborate with him on a project, see www.robin-thompson.co.uk/publications or email [email protected].