Meet the Mathematicians 2010

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Welcome to Meet the Mathematicians 2010 (MTM10) at Royal Society of Edinburgh. This is the third meeting in the series that aims to give school-age students an insight into just what you can do with mathematics outside the confines of the classroom and what mathematicians actually do. MTM10 is funded by the Engineering and Physical Sciences Research Council (www.epsrc.ac.uk).

It may come as a surprise that mathematics is a living subject, with new advances each year in novel as well as classical areas of mathematics. Almost all university mathematicians spend a

good deal of time researching their favoured areas of mathematics, writing research articles and books, and attending conferences to discuss their latest discoveries. We hope to give you a chance to see the applications of mathematics beyond the ‘handle-turning’ constraints of the Highers or A-level syllabus. The MTM day is also designed to show students that university is not an extended sixth form, with lecturers playing the role of teachers, but is there for research and the advancement of knowledge. The MTM event will be recorded and will appear later on the website at www.meetmaths.org.uk or on YouTube (search for Meet the Mathematicians).

Welcome

As usual, MTM10 is being run in conjunction with the British Applied Mathematics Colloquium (BAMC), the largest annual applied mathematics research meeting in the UK. Each year it attracts over 300 professors, lecturers and research students to speak about their latest mathematical discoveries. Younger researchers are especially encouraged to attend and deliver their talks in a friendly and supportive environment. This year the BAMC and the British Mathematical Colloquium (BMC), its pure mathematical counterpart, have joined forces for an ‘Olympic’ event called Maths2010 hosted by the Maxwell Institute for Mathematical Sciences in Edinburgh.

Edinburgh continues to be a major UK centre for mathematics. The Maxwell Institute is a research collaboration between the Universities of Edinburgh and Heriot-Watt and is dedicated to research and training in mathematical sciences. It is named after James Clerk Maxwell, who first wrote down the mathematical equations underpinning electromagnetism, and therefore almost all of modern life. Maxwell was born in India Street and grew up in Heriot Row, both just a short walk away.

The idea of MTM days arose because we wanted students interested in studying mathematics, or a related subject, at university to interact with mathematicians in the context of a research meeting. The talks you will hear today cover

two highly topical themes for the application of mathematics: biological systems and risk.

Applications of mathematics to biological systems pose a great challenge, not least due to the complexity of living beings. In medicine, mathematical models can assess the dangers posed by epidemics, help design effective treatments and optimise the speed of their delivery. In some biological contexts, very simple mathematical approaches provide extremely effective descriptions of what is going on. However, in other biological areas, such as genetic encoding or consciousness, mathematical techniques don’t yet work very well. The multidisciplinary nature of the subject generates new results for both biologists and mathematicians. It is thus an exciting and extremely active area of worldwide research.

Risk, whether financial or safety related is playing a pivotal role in our lives. Politicians, journalists and lawyers often believe that all risk can be eliminated with only the ‘right’ set of often disproportionate, expensive and invasive controls. Conversely, bankers (not a million miles from this venue) often ignore the fundamental assumptions that underlie their models of financial risk, leading to catastrophic losses. Recent economic, social and legal history shows that there has never been a greater need for a balanced, rational and quantitative understanding of the

meet the mathematicians 2010

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multitude of threats facing us as individuals, companies or countries. A better understanding of the mathematics of risk and its assumptions is essential to that need and vital for all.

Some of you may think that mathematics has an intrinsic beauty and should be kept away from applications. That is a perfectly valid view, but if the beauty can be applied, so much the better. For example, pure mathematicians at Southampton, more often found analysing discrete groups and noncommutative geometry, are currently working with engineers from Edinburgh on a project to anticipate and control chain-reactions of country-wide power cuts across Europe.

Some of you may think that only the work of the best professors can be useful. More often than you may think, it is actually the work of younger researchers that turns out to be the most important! For example, the work of a student studying for an MSc at a UK university has recently managed to save an aircraft company many millions of pounds each year.

The future of the UK depends ever more on the application of skilled knowledge. Over the coming years the UK is likely to cut back the spending on science, engineering and mathematics, while our major competitors are increasing investment in these areas.

There is one glimmer of hope. Mathematicians are in great demand in all areas of employment, and the number of students entering university to do mathematics is booming. Mathematics is the most cost-effective science in that it doesn’t need large laboratories or huge colliders. All it needs is a decent brain and good ideas. (Ok, and perhaps some coffee.) Mathematics, through its application, through you, has a major opportunity to play a help the UK recover economically and diversify. To paraphrase John F. Kennedy, “ask not what your country can do for mathematics, but what mathematics can do for your country.”

We hope you have a great day, and look forward to seeing you at University soon.

Dave AbrahamsUniversity of Manchester

Chris HowlsUniversity of Southampton

MTM Grant Holders and BAMC Standing Committee



10.30am - 11.00am Registration and Refreshments

11.00am - 11.05am Welcome

11.05am - 11.45pm Do you feel lucky? Tim Johnson (Heriot Watt)

11.50am - 12.30pm Applications of mathematics to ecology, biology and medicine Christina Cobbold (Glasgow)

12.30pm - 2.00pm Lunch and Activities

2.00pm - 2.40pm Safety in numbers Paul Jowitt (ICE, Heriot Watt)

2.45pm - 3.35pm Why Mathematics? Alan Smith (Heriot Watt) and selected panellists

3.35pm - 4.05pm Refreshments

4.05pm - 4.45pm Patterns: the shape of nature and the nature of shape Professor L Mahadevan (Harvard)

4.50pm - 5.00pm Farewell

Programme

Do you feel lucky?Tim Johnson, Heriot-Watt UniversitySuccess in modern banking and investment is dependent on advanced mathematics; failure to understand mathematics leads to ruin, as demonstrated by the events leading to the collapse of Lehman brothers in 2008. Financial markets are random and complex systems, and science cannot perform experiments on them and needs to resort to mathematics to understand these systems. This talk will discuss the long relationship between finance and mathematics and explore how mathematicians are helping bankers understand the world in which they work.

Dr Tim Johnson is the UK Research Council’s Academic Fellow in Financial Mathematics and spends his time using maths to understand what you should do

when faced with an uncertain future. Dr Johnson studied physics at university, then worked in the oil industry where he realised guessing was not a good policy when risk and money are involved, and so became a mathematician. Tim has spent much of the last year explaining to people why one of the causes of the financial crisis was that there are not enough mathematicians who understand probability around, and banks that use science and mathematics have weathered the financial turmoil well.

Applications of Mathematics to Ecology, Biology and MedicineChristina Cobbold, University of GlasgowEcology, biology and medicine may not be the first subjects that come to mind when you think of how mathematics might be used, but

through some exciting examples we will see how vital mathematics is to these areas. These will include: predicting the spread of foot and mouth epidemics, understanding what impact a fish farm might have on the wild fish population, how mathematical modelling improves treatment both by suggesting better ways of using existing medicines and by predicting new types of treatment.

Christina has a degree in Mathematics from the University of Warwick and a PhD in Mathematical Biology from Heriot-Watt University, Edinburgh. She is currently a lecturer at the University of Glasgow where she uses mathematics to study a range of problems in ecology from understanding how warming temperatures might affect pest insect outbreaks to how movement and habitat affect population persistence. You can often find her hanging out in the biology department and when not doing mathematics Dr Cobbold enjoys hiking and camping and watching the latest movies at the cinema.


Speakers and Talks

Safety in NumbersPaul Jowitt, ICE, Heriot-WattWe all have to deal with uncertainty and manage risk in our daily lives. This talk will explain how engineers use mathematics to assess risk and make decisions about such things as flood defences, the safety of structures and water supply. These decisions affect society and the environment. How do we balance risk against cost? What is the chance that a reservoir will run dry before the end of the summer? And how strong should a bridge be?

Professor Paul Jowitt is a civil engineer, and Director of the Scottish Institute of Sustainable Technology at Heriot Watt University. He is also the 145th President of the Institution of Civil Engineers. Paul’s major interests are sustainable development, water resources, decision-making and risk.

Patterns: the shape of Nature and the nature of ShapeL. Mahadevan, Harvard UniversityThe world around us is replete with patterns. These patterns can be either natural or artificial, static or dynamic and occur on a range of scales, from the atomic to the galactic. It is particularly easy to observe these directly on the everyday scale – from the cracks in fine porcelain to the creases and wrinkles on our skin, from coffee stains left on the counter to the coiling of honey on toast, from the elastic ripples in a leaf to the fluid whorls in a bathtub. Each of the above examples involves characterizing the shape of things. Which raises the question: how do you define shape? And how can we understand their origin, regularity, irregularity and change? I will describe how researchers try to answer these questions using mathematics, physics, biology, experiment, theory, computation – in other words, whatever it takes… This leads to a wonderful interplay between disciplines that breaks down barriers (after all, nature does not know or care about these!) and allows us to answer some questions and sharpen others.

The talk will involve a number of demonstrations and movies of how shapes arise in various natural systems. Come armed with your curiosity about patterns that you are eager to explore and understand!

Professor Mahadevan began his studies with an undergraduate degree in engineering from the India Institute of Technology in 1986 and a master’s in engineering mechanics from the University of Texas-Austin, in 1987, before moving to Stanford, where he earned a master’s in mathematics in 1992 and a PhD in applied mathematics in 1995. His hobbies include reading “about everything,” but especially history, travel, and scientific biography; and “goofing around with the kids doing baby science experiments with them, and sometimes on them, but usually with them.”



Why Mathematics?Panel discussion led by Alan Smith, Careers Advisor, Heriot-Watt [There will be an opportunity during this session for you to ask questions about studying mathematics at university, career possibilities etc.]

Mathematics is fundamental to many aspects of modern life and mathematicians are highly sought after by employers. This session will give an insight into the vast range of career opportunities open to mathematics graduates and look at what recent graduates have done.

A degree in mathematics does not train you for a specifi c job. Rather it gives you a range of skills which enable you to enter a wide variety of careers. Employers across many sectors of industry and commerce are only too happy to receive applications from mathematics graduates. Whether it’s accountancy, computing, engineering or logistics, the highly numerate nature of your degree is a prized asset. Employers are aware

that maths graduates are more likely than most to be skilled in thinking clearly and logically, analysing and presenting data and effi ciently organising their workload. These skills are valued across the employment spectrum from weather forecasting to the statistical analysis of fi eld trials for a drug company. The sheer versatility of mathematics graduates makes them highly attractive to employers.

For an introduction to the many ways in which mathematics affects our everyday lives see: http://www.telegraph.co.uk/science/6439621/How-maths-makes-the-world-go-round.html

Why Mathematics?

What makes mathematics exciting? Julia Colllins, Postgraduate student, University of Edinburgh

For me, mathematics is the language of beauty. It’s about fi nding patterns and capturing symmetry in the world around us, then taking those ideas and letting our imaginations run wild in creating new patterns. I love the way that maths can explain human notions of beauty, such as music and architecture, and at the same time help us understand the deepest mysteries of physics and nature. In my outreach work I want to get across the idea that maths is not about mental arithmetic; it’s not just about getting solutions to complicated equations. More than anything, I want to show people that maths has a beauty all of its own and that you don’t need to be a genius to be able to appreciate it.

How do you use your mathematical skills?Alan Rankine, University of Edinburgh Graduate

Upon leaving university, I began a career as a trainee actuary at Standard Life. Actuarial work holds considerable appeal for me, as it allows me to maintain and develop my mathematical skills in a practical situation. Many of the courses that I studied at university have signifi cant crossover with the Faculty of Actuaries examinations - hopefully enough to gain me a few exemptions! Of at least equal importance, however, are the skills that I developed whilst taking my degree: an analytical approach to problem solving, the ability to study/learn independently and the confi dence to give presentations. Not only do I use these skills everyday at work, but I also found them very helpful during the recruitment process.

Some personal responses from mathematics students


Science has accomplished a lot; putting people on the moon, DNA fingerprinting, MRI scanning and the iPhone are a few of its achievements. But these are all products of ‘western science’ and raise the question why the science that developed in Europe is different to that of China, India or Africa. One of the characteristics of western science is that it used mathematics to describe natural phenomena in a way that others did not. Newton developed calculus, a branch of mathematics, to describe how the solar system works; James Clerk Maxwell used mathematics to understand electricity and magnetism, which lead him to predict radio.

Why did Europeans decide to use mathematics to describe the world? Some historians think an important factor was the complexity of Europe’s medieval financial markets. In Europe, unlike China or the Islamic world, there were hundreds of different types of money being used. In 1202 the son of an Italian merchant, Fibonacci, changed the face of mathematics by introducing Hindu numbers to enable the

complex calculations merchants needed to make [see Arbitrage, page 13]. Some fifty years later, a university scholar, St Albert the Great, influenced by what merchants were doing, combined science with mathematics and his ideas were developed in the next century by people like the Oxford Calculators, Jean Buridan and Nicolas Oresme, who all laid the foundations for scientists like Newton.

In 1564 an Italian scientist Girolamo Cardano wrote a book on gambling and in this book is the earliest record of the observation that the chance of rolling a six with a fair dice is 1 in 6, or the chance of tossing a head with a fair coin is 1 in 2. It might seem incredible that none of the clever people who lived before 1564 had noted this observation before. However, over a hundred years later Newton had to give a scientific definition of ‘time’ to explain the movements of planets, so Cardano’s scientific definition of ‘chance’ was not that late.

Do you feel lucky?


Tim Johnson, Heriot-Watt University

Cardano wrote his book on gambling because he wanted to work out when it was alright to gamble, he was addressing an ethical problem. He used mathematics to tackle this problem and his conclusion was that a gamble was justifiable when the gamblers were willing and knowledgeable, when they used science to guide their decision making. Over the next hundred years many scientists, including Galileo, considered problems relating to gambling and two mathematicians, Pascal and Fermat, are generally regarded as the founders of mathematical probability [see The Problem of Points, page 14] when they solved a gambling problem in 1654. Later the scientists Huygens, Jacob and Daniel Bernoulli, and even Newton, who left Cambridge to run the Royal Mint, all tackled problems related to finance.

While the foundations of financial mathematics were laid at the time of Newton, it took some two hundred and fifty years for mathematicians to work out what ‘probability’ actually is. The key was when maths worked out how to deal with infinity and developed analysis, the branch of maths that includes differentiation and integration. In 1933, using ideas relating to integration, a young Russian mathematician, Andrei Kolmogorov, defined the axioms of probability, as Euclid had identified the axioms of geometry in 300BC. Today, investment banks and hedge funds employ as many mathematicians as they can, who use Kolmogorov’s axioms to understand arbitrage, pricing (as in the Problem of Points) and decision making [see Birds and Berries, page 15].


Do you feel lucky?

In medieval Italy, each city minted its own currency, Genoa had livres (or pounds), Venice, ducats and Florence florins. Say a Pisan merchant, such as Fibonacci’s father, Guglielmo, observed that • 100 Genoese pounds buy 127 Venetian ducats • 100 Genoese pounds buy 192 Florentine florins • 100 Venetian ducats buy 153 Florentine florins

Using, what today we regard as, simple arithmetic, Guglielmo can work out that 100 pounds would buy 127 ducats, which could then be used to buy 127 x 153⁄100 = 194 florins. Since those same 100 pound would only buy 192 florins, directly, Guglielmo can use 100 pounds

to buy the ducats and then 194 florins and then convert these into 194⁄192 x 100 = 101 pounds. Gugliemo can make money at no risk and this is called an arbitrage. Fibonacci’s book Liber Abaci explained how, by using Hindu numbers, merchants could make these simple calculations.

Today modern financial institutions invest millions (billions?) in developing computer systems that can identify arbitrage opportunities, making trades in milliseconds. These banks often employ theoretical physicists because the mathematics behind string theory can help them identify arbitrages.

Arbitrage


Fibonacci’s Liber Abaci spawned a tradition of business schools that taught merchants by using simple example problems. One problem related to a (fair) game between two players, say Pascal (P) and Fermat (F). The game is made up of successive rounds and the winner, who will get 80 ducats, of the game is the first player to win 7 rounds. How should the stake be split if the game is forced to finish after P has won 5 rounds and F four?

Around 1500 people thought the answer was to split the winnings proportionally, 5 : 4. This is a statistical argument, and Cardano realised it was wrong – what would you do if P won 1 round and F none. It was unfair to give all the money to P. Cardano insisted it was not that important what had happened, you needed to focus on what would happen, given where you started from.

Cardano got the principles right but not the details, which was done by Pascal and Fermat in the summer of 1654. Pascal developed an algorithm, or a formula, which would give the value of the game to P for the general case, and this formula is often regarded as the ‘start’ of probability in maths.

Today, we can see Pascal’s formula as a special case of one of the most important derivative pricing formulae used by modern finance. In the 350 years that separate Pascal from us, maths figured out the idea of infinity, which then enabled Kolmogorov to understand probability. On the basis of this understanding banks use a version of Pascal’s formula. This is a bit like Fermat’s Last Theorem, Fermat came up with an idea 350 years ago but maths only understood it in 1993.

The Problem of Points

to P. Cardano insisted it was not that important what had happened, you needed to focus on what would happen, given where you started from.

Theorem, Fermat came up with an idea 350 years ago but maths only understood it in 1993.

Do you feel lucky?

Mathematics cannot predict an uncertain future, but it can be used to help make decisions when faced with risk. For example, birds need to eat up to 40% of their body weight in winter to survive the night and their survival depends on finding food. Let’s say a bird has 6 hours to find 9 berries and it has two choices:

Play it Safe The bird stays where it is, where it knows there are berries in the hope of finding a few. The chance of finding one or two berries in the hour is 50:50.

Take a Risk The bird flies off, in the hope of finding a berry-bonanza but with a high chance of only finding enough to replace the energy lost in flying. The energy cost of flying is one berry and the chance of only finding the one berry in the new field is 5 in 6, but there is a 1 in 6 chance of finding 10 berries (and getting an excess of 9).

What should the bird do? This is a problem in what is now known as ‘stochastic control’. The solution is fairly straightforward. If after 5 hours the bird has only found 5 berries by playing safe, its only rational option is to take a risk, and hope it hits a bonanza. However if it has found seven or eight berries after five hours, its best option is to play safe (it will certainly survive if it has found 8 berries and has a 50:50 chance if it has 7 berries). If the bird has found 9 berries, it can take the risk, certain of surviving the night. Using mathematics like this the bird can map out the best strategy given the time of day and number of berries found as follows

Birds and Berries

of only finding enough to replace the energy lost in flying. The energy cost of flying is one berry and the chance of only finding the one berry in the new field is 5 in 6, but there is a 1 in 6 chance of finding 10 berries (and getting an excess of 9).


Ber

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2 44 6Hour of DayHour of Day

No choice − take the riskNo choice − take the riskotherwise you will die.otherwise you will die.

Playing it safegives you thebest chance

Take the risk − if it doesTake the risk − if it doesnot work out, playingnot work out, playingsafe guarantees not work out, playingsafe guarantees not work out, playing

survivial.safe guarantees survivial.safe guarantees


Have you ever wondered why cats and dogs live longer than hamsters, or why insects live longer than the cells that they are composed of? There is a pattern here, between size and life-expectancy, or to be more precise there is a relationship between body size and metabolic rate. Larger organisms have slower specific metabolic rates and so live longer. There are many patterns and relationships in biology and the famous mathematically trained zoologist D’Arcy Wentworth Thompson (1860-1948) advocated that as the laws of physics govern the world in which we live the “pattern-seeking science of mathematics” can be used to understand these biological patterns.

Although people do not often think of mathematics and biology together there are many examples where uniting these two subjects has helped scientists solve important biological questions such as, “how do we stop a disease like foot and mouth from infecting all our farms?” In the radiotherapy treatment of cancer, “what dose should be used and how frequently should we treat a patient, daily, every 2 days?” Mathematics as simple as

multiplying and dividing to complex theories about mathematical objects known as dynamical systems can provide insight into these biological questions. So let us start with some multiplying and dividing and see what that can tell us about biology.

Biology and arithmetic “Hensen, who published so full and interesting an account of the habits of worms, calculates, from the number which he found in a measured space (his garden), that there must exist 53,767 living worms in an acre.” - Charles Darwin, The Origin of Species.

Whenever I go to websites like Wikipedia or read a science book, I often come across impressive statements like Hensen’s 53,767 worms per acre or the casual claim that “there are 1013 cells in the human body”. That is a big number and I am sure no one has actually counted the number of cells in the body, so one may ask how one can make such statements. At first glance it is

Applications of Mathematics to Ecology, Biology and MedicineChristina Cobbold, University of Glasgow

difficult to know if 1013 is even correct, but if we knew the total volume of cells in an average human body and knew the volume of a typical cell then some simple division would tell us how many cells were in the body. The first volume we can find relatively easily, the cells of a ‘typical 70-kg man’ contain 30 litres of water. Cells are more than just water so the volume of cells in the body must be larger, but it is not going to be 10 times larger, which is what matters in this calculation. So let us say that the volume of cells in the body is 40 litres. Now we just need to know the volume of a cell, unfortunately this is much harder to estimate. The problem is that this number varies massively depending on what cell we measure.

The diameter of human cells can range between a few hundred µm to about a 1µm. Many cells

have a volume of around 10−6µl, but cell volumes can be as small as 10−9µl to as large as 0.8µl (see Figure 1). This is a huge difference in volume, so we now need to know the proportion of cells in the body that are of each type. This is perhaps worse than our original question. Does this mean we have failed at the first hurdle and we cannot estimate the number of cells in the human body? Perhaps, but we can change our question to find out if 1013 is reasonable. If we assume that there are 1013 cells and we roughly know the total volume of cells is 40l (i.e. 4×107µl) then we can calculate what volume a typical cell would be and ask if we think this is reasonable. So 4 × 107⁄1013 = 4 × 10−6µ litre. It seems reasonable to me. Most cells are this size. The key to answering our question was leaving the parts of the calculation with the most uncertainty to last.


Figure 1: A picture of red blood cells with a typical volume of about 84 × 10−9µl, next to this is a picture of a nerve cell. Human nerve cells can be over a metre in length!


Scaling laws in biology We have seen that cell volume can vary considerably, by powers of 10, so trying to plot cell size on a graph requires a long piece of graph paper! If we take a little excursion into the world of logarithms we can plot cell size on a log scale: 10−1 , 100, 101 , 102 rather than 0.1, 1, 10, 100 which helps us to visualise data in a compact way (see Figure 2). The numbers indicating the power of 10 that appear on the log scale are logarithms. For example, the logarithm of 10,000 (i.e. 104) is 4. In symbols log(104) = 4. With the notion of logarithm in place we can ask what the

function Y = aXb might look like on a log scale (here a and b are known numbers). If we take logarithms of both sides of the function the rules of logarithms give

which is a straight line on a log-log plot, with slope b and intercept log a. We call such a straight line relationship a power law because Y is related to a ‘power’ of X. There are many examples of power laws in nature, for example the relationship between body mass and optimal

Figure 2: A log-log plot of mass against metabolic rate. The dots are measurements from a range or organisms, and the line fitted through this data has slope 3/4, illustrating Kleiber’s rule.

visualise data in a compact way (see Figure ). The numbers indicating thepower of 10 that appear on the log scale are logarithms. For example, thelogarithm of 10,000 (i.e. 104) is 4. In symbols log(104) = 4. With the notionof logarithm in place we can ask what the function Y = aXb might look likeon a log scale (here a and b are known numbers). If we take logarithms ofboth sides of the function the rules of logarithms give

log Y = log a + b log X,

which is a straight line on a log-log plot, with slope b and intercept log a.We call such a straight line relationship a power law because Y is relatedto a ‘power’ of X. There are many examples of power laws in nature, forexample the relationship between body mass and optimal cruising speed offlying insects, birds and airplanes follow a power law. One important exampleis Kleiber’s rule which relates body mass, M , to metabolic rate. The ruleaddresses the observations we made at the start of this article, that hamstersdo not live as long as dogs. Kleiber found that metabolic rate varies as M 3/4.We see in Figure in a log-log plot of mass against metabolic rate the graphis a straight line with slope 3/4. So as mass increases, the metabolic rate perunit mass decreases and so the heart beats slower and generally the animallives longer. Amazingly this relationship holds not only for mammals, butfor insects and even unicellular organisms like algae. Power laws like this areuseful, scientists use Kleiber’s relationship to adjust drug doses in rats for usein humans for example. If we know what happens at one scale we can predictwhat will happen at another scale. In the next section we introduce anotheruse of logarithms, this time to predict cell survival probabilities following theuse of radiation to treat cancer.

Radiotherapy treatment schedules and quadratic equa-tions

Radiotherapy is used in the treatment of over half of all cancer patients. Itworks by damaging the DNA in the cells. If there is enough damage thecell dies. A mathematical equation known as the Linear-Quadratic model isused to predict the fraction of cells that survive a given radiation dose. Thismodel has played a vital role in developing the treatment schedules used byradiologists.

4

Applications of Mathematics


cruising speed of flying insects, birds and airplanes follows a power law. One important example is Kleiber’s rule which relates body mass, M, to metabolic rate. The rule addresses the observations we made at the start of this article, that hamsters do not live as long as dogs. Kleiber found that metabolic rate varies as M3/4. We see in Figure 2, in a log-log plot of mass against metabolic rate the graph is a straight line with slope 3/4. So as mass increases, the metabolic rate per unit mass decreases and so the heart beats slower and generally the animal lives longer. Amazingly this relationship holds not only for mammals, but for insects and even unicellular organisms like algae. Power laws like this are useful, scientists use Kleiber’s relationship to adjust drug doses in rats for use in humans for example. If we know what happens at one scale we can predict what will happen at another scale. In the next section we introduce another use of logarithms, this time to predict cell survival probabilities following the use of radiation to treat cancer.

Radiotherapy treatment schedules and quadratic equations Radiotherapy is used in the treatment of over half of all cancer patients. It works by damaging the DNA in the cells. If there is enough damage the cell dies. A mathematical equation known as the Linear-Quadratic model is used to predict the fraction of cells that survive a given radiation dose. This model has played a vital role in developing the treatment schedules used by radiologists.

σ = Fraction of cells that survive, D = nd = Total radiation dose measured in Gy. D is made up of n doses of strength d, given daily. α/β is an inverse measure of a tissue’s sensitivity to the radiation. Small values of α/β correspond to tissue with a high repair capacity like normal cells (α/β = 3) and high values correspond to tissue with a low repair capacity like advanced head and neck cancer (α/β = 20).

Linear-Quadratic Model:

log(σ) = −α n d cell death from singlestrand DNA breaks

−β n d2

cell death from doublestrand DNA breaks


Successful radiotherapy treatment needs to maximise the number of cancer cells that are killed, but also minimise the number of normal healthy cells that are killed. By rearranging the equation above we can calculate the dose that achieves this.

Biologically Effective Dose = BDE = Total dose × Relative effectiveness

�

From this we can calculate BDE for normal and cancerous cells. Using the example of head and neck cancer we find that BDEnormal = D(1 + d/3) and BDEcancer = D(1 + d/20). We want to choose the dose, d, that maximises BDEnormal

and minimises BDEcancer, this is the same as maximising

This tells us that the aim of a successful treatment protocol is to make d, the daily dose size as small as possible. So lots of small doses is better than one big dose. This is not the end of the story, mathematical modellers have extended this idea to also take into account that the cells reproduce between treatments. They discovered that how cells repopulate the radiated tissue can influence the outcome of a particular treatment schedule. Knowing when a treatment schedule is not suitable is important information a model can give to clinicians.

Figure 3: A log plot of the Linear-Quadratic model for a range of cell sensitivities. The plot shows the percentage of cells that survive a given radiotherapy dose.

BDE =−log(σ)

α= D

1 +

d

α/β

.

From this we can calculate BDE for normal and cancerous cells. Using theexample of head and neck cancer we find that BDEnormal = D(1 + d/3) andBDEcancer = D(1 + d/20). We want to choose the dose, d, that maximisesBDEnormal and minimises BDEcancer, this is the same as maximising

BDEcancer −BDEnormal = (d

20− d

3) = −D17d

60.

This tells us that the aim of a successful treatment protocol is to make d,the daily dose size as small as possible. So lots of small doses is better thanone big dose. This is not the end of the story, mathematical modellers haveextended this idea to also take into account that the cells reproduce betweentreatments. They discovered that how cells repopulate the radiated tissuecan influence the outcome of a particular treatment schedule. Knowing whena treatment schedule is not suitable is important information a model cangive to clinicians.

We have talked about cells and groups of cells, now let’s go up a scaleand talk about groups of animals. We also advance our mathematics a stepfurther to consider a mathematical object known as a matrix. It turns outthat a matrix is useful in helping us understand the spatial spread of diseases,as we demonstrate in the next example.

Foot and mouth epidemics and matrices

Foot-and-mouth is a highly infectious livestock disease and in 2001 an epi-demic spread across the UK which resulted in the culling of over half a millioncattle and over 3 million sheep and cost the national economy over 2 billionpounds. The size of this epidemic could have been even worse had controlpolicies not been implemented to prevent the spread of the disease. Mathe-matical modelling played an important role in guiding the control effort andassessing the performance and timing of control strategies, like movementrestrictions, culling and vaccination.

One of the models (see Figure ) used during the Foot and Mouth epidemicdescribes the farm network and probability of transmitting the disease fromone farm to another. Mathematical objects known as matrices were used tocalculate the probability that a given farm will become infected tomorrow.A matrix is just a grid of numbers, but as with numbers there are rules for

6

BDE =−log(σ)

α= D

1 +

d

α/β

.

From this we can calculate BDE for normal and cancerous cells. Using theexample of head and neck cancer we find that BDEnormal = D(1 + d/3) andBDEcancer = D(1 + d/20). We want to choose the dose, d, that maximisesBDEnormal and minimises BDEcancer, this is the same as maximising

BDEcancer −BDEnormal = (d

20− d

3) = −D17d

60.

This tells us that the aim of a successful treatment protocol is to make d,the daily dose size as small as possible. So lots of small doses is better thanone big dose. This is not the end of the story, mathematical modellers haveextended this idea to also take into account that the cells reproduce betweentreatments. They discovered that how cells repopulate the radiated tissuecan influence the outcome of a particular treatment schedule. Knowing whena treatment schedule is not suitable is important information a model cangive to clinicians.

We have talked about cells and groups of cells, now let’s go up a scaleand talk about groups of animals. We also advance our mathematics a stepfurther to consider a mathematical object known as a matrix. It turns outthat a matrix is useful in helping us understand the spatial spread of diseases,as we demonstrate in the next example.

Foot and mouth epidemics and matrices

Foot-and-mouth is a highly infectious livestock disease and in 2001 an epi-demic spread across the UK which resulted in the culling of over half a millioncattle and over 3 million sheep and cost the national economy over 2 billionpounds. The size of this epidemic could have been even worse had controlpolicies not been implemented to prevent the spread of the disease. Mathe-matical modelling played an important role in guiding the control effort andassessing the performance and timing of control strategies, like movementrestrictions, culling and vaccination.

One of the models (see Figure ) used during the Foot and Mouth epidemicdescribes the farm network and probability of transmitting the disease fromone farm to another. Mathematical objects known as matrices were used tocalculate the probability that a given farm will become infected tomorrow.A matrix is just a grid of numbers, but as with numbers there are rules for

6

Applications of Mathematics


We have talked about cells and groups of cells, now let’s go up a scale and talk about groups of animals. We also advance our mathematics a step further to consider a mathematical object known as a matrix. It turns out that a matrix is useful in helping us understand the spatial spread of diseases, as we demonstrate in the next example.

Foot and mouth epidemics and matrices Foot and mouth is a highly infectious livestock disease and in 2001 an epidemic spread across the UK which resulted in the culling of over half a million cattle and over 3 million sheep and cost the national economy over 2 billion pounds. The size of this epidemic could have been even worse had control policies not been implemented to prevent the spread of the disease. Mathematical

modelling played an important role in guiding the control effort and assessing the performance and timing of control strategies, like movement restrictions, culling and vaccination.

One of the models (see Figure 4) used during the foot and mouth epidemic describes the farm network and probability of transmitting the disease from one farm to another. Mathematical objects known as matrices were used to calculate the probability that a given farm will become infected tomorrow. A matrix is just a grid of numbers, but as with numbers there are rules for multiplying, dividing, adding and subtracting matrices from one another. Starting with a matrix containing all the distances between farms, the entry in row i and column j is the distance

Figure 4: An illustration of one of the mathematical models used to advise the foot and mouth control effort.

between farm i and j. We then feed this into the function that tells us the probability of infection given two farms are a given distance apart, which produces a matrix containing the probability farm i will infect farm j. We are not done yet, we need to also consider how infectious farm i is and how susceptible farm j is, for example the size of the farm and the types of animals on the farm determine this. Putting all this information together we get a matrix that tells us the probability a farm will be infected tomorrow.

One of the predictions from the model was that if only infected farms were culled then the 2001 foot and mouth epidemic would have been much larger and would have infected around 20,000 farms instead of the 9900 that were culled during the epidemic. In 2001 infected farms and those farms within a 3km were culled and this was important in limiting the spread of the disease. The model was also able to identify “hot spots” of infection. In Figure 5 we see how well the model did at predicting the spread of foot and mouth.


Figure 5: Cattle culled and burned on a Scottish farm during the 2001 Foot and Mouth outbreak. The graph shows the number of reported cases of Foot and Mouth, the pink dots are results from the model and the red line is the average of the model results.

Careers Resources for Schools/Collegeshttp://www.ima.org.uk/Careers/school19.htm

Mathematics Departments in the UKhttp://www.ma.hw.ac.uk/uk_maths.html

MacTutor History of MathematicsA comprehisive resource indexed by mathematician, topic, culture and datehttp://www-groups.dcs.st-and.ac.uk/~history/

Eric Weinstein’s World of MathematicsWell designed reference sitehttp://mathworld.wolfram.com

iSquared MagazineOnline home of the magazine in your packhttp://www.isquaredmagazine.co.uk

Plus MagazineOnline publication with a huge diversity of articles by mathematicians and science writers.http://plus.maths.org

Mathematical Quotationshttp://math.furman.edu/~mwoodard/mqs/mquot.shtml

Maxwell’s Demona mathematical bloghttp://maxwelldemon.com/

Travels in a Mathematical WorldPodcasts and blog posts about mathshttp://travelsinamathematicalworld.blogspot.com/

Maxwell Institute for Mathematical Scienceshttp://www.maxwell.ac.uk

International Centre for Mathematical Scienceshttp://www.icms.org.uk

Engineering and Physical Sciences Research Councilhttp://www.epsrc.ac.uk

Links and Resources


Where Next?A small selection of links to useful and interesting resources. You’ll find more like this on the Meet the Mathematicians website at http://www.meetmaths.org.uk

“ Hensen, who published so full and interesting an account of the habits of worms, calculates, from the number which he found in a measured space (his garden), that there must exist 53,767 living worms in an acre.”

Charles Darwin, The Origin of Species

www.meetmaths.org.uk

AcknowledgementsMTM was created by Professor David Abrahams (Manchester) and Dr Chris Howls (Southampton). The MTM days are sponsored by the Engineering and Physical Sciences Research Council (EPSRC). The EPSRC are one of the main civilian funders of mathematical research in the UK.

We are indebted to the effi ciency and professionalism of the local organising team

in Edinburgh: Professor Jack Carr (Heriot-Watt), Irene Moore (International Centre for Mathematical Sciences), Lois Rollings (Edinburgh), Madeleine Shepherd (International Centre for Mathematical Sciences). We also extend our grateful thanks to Calum Wilson (Media Services, Heriot-Watt), Sarah Shepherd (editor of iSquared Magazine) and to Dr Wendy Sadler (Science Made Simple) for invaluable advice.

Documents

Meet the Mathematicians 2010