CONTENTSIntroduction: why maths matters 1

Staying safe: mathematics in security 2

Keeping in touch: mathematics in telecommunications and the internet 4

Nature by numbers: mathematics and the environment 6

The money programme: mathematics of finance and economics 8

Counting on commerce: mathematics in industry 10

On the go: mathematics in transport and travel 12

A healthy outlook: mathematics in medicine and biology 14

Conclusion: keep the flame burning 17

ENGAGING MATHS 1

You might not realise it, but the mathematical sciences play a pervasive,direct and fundamental role in our lives. Many disciplines such as economicsand computer science have always depended on a bedrock of mathematicsbut as technology begins to play an increasingly vital part in virtually everyaspect of modern society from healthcare to telecommunications themathematical sciences are becoming indispensable.

Complex phenomena can be conceptualised by mathematicians by constructing sophisticatedmodels. These models are abstract but they can have immense power in demonstratingcausal links between processes and in predicting how complicated systems with manyinteracting components will behave under different circumstances. They can also tease out the most important aspects of a system and give scientists new ideas about how to direct their experiments.

Many different issues and concerns impact on todays society. Some of these include:increased regulation; awareness of all kinds of risk; concern over the environment and climate;knowledge of the natural environment, especially in relationto energy reserves and nuclear waste disposal; the growthof the financial sector; the next generation of advancedmanufacturing and engineering; the service sector; post-genome informatics and medicine; the growth in IT systems,secure communications, software, intelligent systems,defence research, security, imaging, and so the list goes on.The mathematical sciences have a role to play in all these areas indeed, for many they are the only way of addressing and solving the deep problems posed.

This brochure contains a number of case studies showing how the mathematical scienceshave a direct impact on topics of public interest. These examples clearly demonstrate that a society without mathematicians would be technologically and economically impoverished.

Intellectually and culturally, mathematicians have an important role to play in society bycontinuing to think creatively and radically, by being innovative and by utilising their skills to support the knowledge economy. The application of mathematics provides tangiblebenefits to, and enriches all areas of life and therefore it is crucial that the mathematicalsciences are exploited further so that the UK extends its strong intellectual and competitiveeconomic position within many sectors.

Introduction:

why maths matters

The mathematical sciences

play a direct and fundamental

role in all our lives

2 ENGAGING MATHS

Security is becoming an increasingly important issue in society. From guarding against

online fraud to ensuring that baggage being checked in at airports is safe, vigilance is the

watchword. Mathematicians are helping in this effort by providing the underlying knowledge

for new and improved security systems. A new type of airport X-ray scanner relies on novel

mathematical techniques to enable it to scan luggage quickly and accurately; automatic

face-recognition systems that can be used as security checks or to identify suspects from

CCTV footage require sophisticated statistical analysis; and the private information, such

as our credit card details, that we send across the internet for shopping or banking must

be appropriately encoded so that it cannot be intercepted and read by criminals.

Staying safe:

IF THE FACE FITS

Identification is a key issue in a security-conscious society.Companies often restrict access to their premises to employeesonly, who are required to demonstrate that they have authorityto enter. This could be by a swipe card or by keying in a code,for example. But these methods are relatively insecure. Whiletechniques such as iris scans or fingerprint recognition arebecoming more widespread, a more versatile approach is the automatic recognition of peoples faces.

Automatic face recognition systems are at various stages ofdevelopment, but problems remain with their reliability andconsistency. At the University of Nottingham, statistician ProfessorIan Dryden is working closely with computer scientist Dr Bai Li to provide new and reliable methods for face recognition.

There are two main areas where the methods can be useful face recognition and face identification, Professor Dryden says.For face recognition you might have a database of individualsfaces on your system and you want to match up an image thatis presented to you for example from a CCTV camera locatedat the entrance to a building so that only an authorised personcan enter. The other issue is face identification in criminalinvestigations, where you have a CCTV image in which it isdifficult to make out the face of a suspect clearly. Can we get some kind of statistical measure that could be used asevidence that the person in the image is indeed the suspect?

To analyse faces automatically, the Nottingham system takes asingle image and breaks it down into 40 separate transformedimages, filtered at eight different angles and at five differentscales, from large scale concentrating on the big features, to much smaller scales that home in on smaller features.

Each image is made up of tens of thousands of tiny elements of light or shade called pixels, and values are ascribed to each of these. They correspond to features such as the angle of theedges of the eye, or the edges of other features such as the nose and the overall shape of the face.

We use a statistical model to pick out the most importantfeatures, says Professor Dryden. The key thing is to identifythose features that vary widely across the population and focus on these.

It is crucial, says Professor Dryden, to develop systems that arerobust and not prone to changes in lighting or expression, forexample. As well as being able to analyse the image quickly, it isimportant to have a sound underlying statistical basis to understandwhy the system works well in some cases and not so well in others. If such recognition systems are eventually to be widely used in security and criminal investigations, they must have a solid statistical footing taking into account all sources of variabilityand uncertainty, which is what we are helping to provide.

Contact: Professor I Dryden, email: ian.dryden@nottingham.ac.uk

mathematicsin security

KEEPING ONE STEP AHEAD OF THE HACKERS

With billions of pounds of business transacted over the interneteach day, the security of information sent through cyberspacehas become an issue of huge importance. Each time weprovide our credit card details to an internet shopping site or online bank we must be confident that the informationcannot be intercepted and used fraudulently. The internet is, after all, essentially insecure. Information travels throughvarious hubs around the world, any one of which could,potentially, be accessed by someone with criminal intent.

For this reason, confidential information must be scrambled into an unreadable gobbledegook as it flies across the internet, to be unscrambled at its destination. This process of scramblinginformation is called encryption, and mathematics lies at its heart.

Two big questions in the world of internet security are: howresistant to attack are existing encryption systems and can new systems be developed that make encryption easier toapply and more versatile without compromising security?

Both of these questions are being addressed by UK researchers.At Royal Holloway, University of London Dr Steven Galbraith issubjecting existing encryption systems to rigorous mathematicalanalysis. The mathematical techniques used to developencryption codes are well established and have been shown inpractice to work, Dr Galbraith says. But underlying theoriesabout why they remain secure or if indeed they might bevulnerable in a way that has not yet been considered have not been studied closely. What I am doing is developing a deeper understanding of the mathematics underpinningcertain encryption systems. Thistheoretical work will lead to moreconfidence in the security of thepractical systems being used today.

Meanwhile at the University of Bristol,Professor Nigel Smarts team isinvestigating more efficient and versatileways to send encrypted messagesacross the internet. A cornerstone of cryptology is the use of keys,information that is held by both thesender and receiver in order to encryptand decrypt messages. In the early days of electronic encryption,both parties needed to hold the same key. This meant that if, forexample, a party wished to communicate with a thousand people,it would require a thousand pairs of keys. In the 1970s the conceptof a public key was developed, where a key can be deposited on the internet to be collected by any number of other parties.It means, for example, that when you log on to Amazon, you donot have to somehow meet beforehand and arrange a swap ofkeys, says Professor Smart. You automatically download theirkey when you log in and can then encrypt messages to them.

The next advance is to do away even with the need to obtain a public key. There is some very clever mathematics peoplecall it magic maths which could allow someone to send andreceive encrypted messages without having to know the key of the recipient, says Professor Smart. It will make the wholebusiness of sending and receiving encrypted messages a loteasier and allow much more flexibility.

Contacts: Dr S Galbraith, email: steven.galbraith@rhul.ac.uk;Professor N Smart, email: nigel@cs.bris.ac.uk

GIVING AIRPORT X-RAYS AN EXTRA DIMENSION

In recent years security at airports has become more importantthan ever. A key part of the security check is the X-ray scan,and UK mathematicians are helping to develop a new, rapidmethod of producing three-dimensional X-ray images of thecontents of luggage. The new system will provide much moredetailed information than is currently possible and will reducethe number of manual checks necessary on peoples luggage which will speed up the processing of baggage.

While most of us are familiar with the X-ray scans carried outon hand luggage, behind the scenes every piece of baggagethat has been checked in is also scanned. These scans take atwo-dimensional flat picture of the contents of the baggage,which allows some items inside the luggage to be concealed by denser objects. When this is suspected, the baggage mustbe searched manually, which results in many false positives,slowing down the process.

A 3-D scan, similar to CT scans in hospitals, provides muchmore information and reduces the number of false positives. A handful of these are in use in airports, but because themachines rely on an X-ray transmitter being rotated on a gantry around the target object, the scanning takes three or four times longer than the conventional system.

Now, CXR Ltd, the research subsidiary of global airport security company Rapiscan Systems, is developing a novelsystem where a number of fixed X-ray sources are placedaround the moving baggage to enable a 3-D image to beobtained much more rapidly. Rapiscan has teamed up with

mathematicians from the University of Manchester, led by Professor Bill Lionheart, to solve some of thefundamental questions necessary to make the system work efficiently.

As CXR director Ed Morton explains,We have developed specialisedcomputer hardware and software toprocess the information, but we needto achieve the fastest, most accurateresults possible. We called in expertsfrom the University of Manchesters

School of Mathematics to help us develop the novel maths and computer algorithms required.

Professor Lionheart says the project represents a fascinatingmathematical challenge. For example, because the X-raysources are fixed, unlike with a conventional CT scanner, there are restrictions on where the detectors that receive the signals once they have passed through the object can be located. The X-rays can be switched on and off in anysequence we need to understand which is the best way to do it. We are working on new theories to account for these different geometries. The key thing is that the systemmust be fast and accurate we have to be able to reconstructan image from the data as they are gathered in real time. You cannot have a bottleneck.

Contact: Professor W Lionheart, email: bill.lionheart@manchester.ac.uk

ENGAGING MATHS 3

What I am doing is developing

a deeper understanding of the

mathematics underpinning certain

encryption codes [which] will

lead to more confidence in the

security of practical systems

being used today

as a constant, steady stream, but rather comes in short,concentrated bursts. One thing wed like to know is how this burstiness affects queuing.

Dr Clegg is representing the flow of data packets by a series of equations that mathematically have the same characteristicsas the data. If we can model the data in this way, we are then in a position to see what is causing the problem and where we might be able to intervene to make the traffic flow more smoothly and avoid queues, he says.

For example, the mathematical models might show thatparticular patterns of data flow are more likely than others to result in congestion. In such cases it might be possible todisrupt that pattern by slowing down or speeding up thetraffic at that point to reduce the amount of queuing atsubsequent nodes.

Its a delicate balancing act, says Dr Clegg. Clearly onewouldnt want to slow the data stream so much that it results in similar delays to the queuing that you are trying to avoid. So we need to work out smart, subtle ways of doing this andmathematical models should help. Ultimately if we are successful,then we could incorporate our findings into internet trafficmanagement schemes that make more efficient use of theexisting infrastructure and help reduce those irritating delays.

Contact: Dr R Clegg, email: richard@richardclegg.org

Weve all had that sinking feeling when our internetconnection grinds to a painful crawl halfway throughdownloading a file. Often the reason for this is simply an excess of internet traffic: so many people are using thesystem at once that the intersections on the informationsuperhighway get clogged up just as roads leading up to traffic lights and roundabouts do in the rush hour.

Dr Richard Clegg, a mathematician at the University of York, is using his expertise to discover why this internet congestionoccurs and how best it can be managed to make more efficient use of the existing infrastructure.

Information travels across the internet along copper wires,optical fibres and even to and from satellites as discretepackets of data. On its journey from someones computer to a web server and back again, this information will passthrough several intermediate machines which are sometimesreferred to as hubs or, in mathematical terms, nodes.

Each time the packets of information enter one of thesenodes, the node needs to work out where to send it, says Dr Clegg. This takes time, and if a lot of packets arrive atonce they are forced to queue up in something called a buffer.I am interested in trying to understand what it is about thepattern of traffic that makes it queue badly and slow thesystem down. For example, internet traffic seldom flows

Keeping in touch:

There has been a remarkable revolution in telecommunications over the past few years.

The internet has expanded hugely and mobile phones are now ubiquitous. None of us gives

a second thought to sending an email or making a telephone call to the other side of the

planet. As telecommunications networks grow and carry more traffic, they must adapt quickly

and efficiently if they are to avoid meltdown. Mathematicians are expert at understanding

networks and traffic. Mathematicians can model telecommunications networks to work out

how best to share the available capacity, and how to simplify complex networks to allow

managers to make strategic decisions about the best way to deal with expansion.

Mathematics can also answer key questions about how traffic accumulates on

telecommunications networks and how to avoid traffic jams.

4 ENGAGING MATHS

TRAFFIC JAMS ON THE INFORMATION SUPERHIGHWAY

mathematicsin telecommunicationsand the internet

A FAIR SHARE

Over the past few years the internet has grown dramatically.There are now more than a billion internet users around theworld almost one in six of the planets population and athree-fold increase since 2000. The volume of traffic on theinternet is also increasing rapidly. It is now commonplace for users to download music and video clips, and even to use the internet for phone calls.

These changes are posing a significant challenge to the smoothfunctioning of this global network of inter-linked computers:how best to share the available capacity among users. Thiscapacity is termed bandwidth and is a measure of the speedthat information is downloaded through the networks pipes the optical fibres, satellite links and so on that carry thedigital information.

Professor Frank Kelly at the University of Cambridge and Dr Damon Wischik at University College London have beendeveloping novel mathematical models of the way thatbandwidth is distributed on the internet. Their goal is toimprove the computer software that assigns bandwidth, sothat everyone gets a fair share and the internet continues to run smoothly.

One of the remarkable features of the internet is that it relieson cooperation between users computers, says Dr Wischik.There is no central control system that decides how muchbandwidth any individual user should have it sorts itself out and shares available bandwidth as fairly as it can.

The key to bandwidth sharing is a computerprogram called TCP transmission controlprotocol that is present in every computer thatuses the internet. TCP was written some 20years ago and has been remarkably successfuluntil now. But it is showing signs of strain under the demand for ever-faster downloads.

The innovative approach that Professor Kellyhas devised is to construct models of thenetwork as a whole, rather than a series ofindividual computers each running the software.Conventionally, engineers and computer scientists tend toregard networks as a discrete number of individual elementsbolted together. The problem with this is that interactions can make the whole network behave in surprising ways, wayswhich one wouldnt expect from looking at the components on their own.

What we are doing is taking the TCP code and simplifying itmathematically, then trying to derive equations which model theentire network, says Dr Wischik. Then we imagine changingTCP, and predict what the effect might be on the network.

In this way it is possible to test a range of modifications to the code. Ultimately the aim is to provide a code that is moreappropriate to the internet as it is used today. We are lookingfor the best way to share bandwidth between users, says Dr Wischik. If we get it right, it will mean faster downloadsand better quality audio and video over the internet.

Contact: Professor F Kelly, email: f.p.kelly@statslab.cam.ac.uk

UNTANGLING TELECOMS NETWORKS

Telecommunications networks are in a state of constant change.New technologies such as broadband require new infrastructure.As the number of users on the network increases and thepatterns of use change, so the network must be expanded and altered to accommodate the new circumstances.

In order to adapt the network in an efficient and economical way,network managers must have a good idea of the existing structure the networks architecture. This information is not always easyto come by. Telecommunication networks have grown over theyears into hugely complex structures consisting of thousands of miles of cabling, vast numbers of junctions, switches, servicepoints and so on. For network managers making strategicdecisions about future developments of the network, the sheer complexity of the system can present a big headache.

But mathematicians such as Dr Sergei Zuyev at the University of Strathclyde are coming to the rescue. Dr Zuyev and hiscolleagues are developing ways of using statistical andgeometrical techniques to produce simplified yet realisticmodels of telecommunications architectures. These contain the essential information needed by planners, withoutoverloading the system with extraneous complexities.

Telecommunications systems must be able to adapt to providenew services to users as technology advances, says Dr Zuyev.For example telecommunication managers need to forecastand plan for future demand, or to predict where bottlenecks in the system might arise and plan the network accordingly. But if your starting point is a detailed plan of the network,

you will find yourself having to deal with a problem that has millions of parameters and this is unmanageable.

Instead, it is possible to pull out a smallernumber of key factors and use these in amathematical model to construct a more usefulrepresentation of the network. There will becertain statistical features of the system thatare central to the networks architecture, says

Dr Zuyev. For example there will be a certain proportion ofconnections that are a relatively long distance from the users,and a proportion that are closer to the user. One can take a smaller number of key parameters such as these and apply probabilistic and geometrical techniques to produce arepresentation of the network that is useable by the planners.

The efficiency of this approach was demonstrated when Dr Zuyev worked on a project to estimate the total length oftelecommunications cabling in an entire country. By modellingthe density of the population in different parts of the country,and including the major elements of the telecommunicationinfrastructure we were able to estimate within a few percent the total amount of wiring, he says. The only other way of doing it would be to go through individual plans of thenetwork, which would take hundreds of man hours.

Contact: Dr S Zuyev, email: sergei@stams.strath.ac.uk

ENGAGING MATHS 5

If we get it right,

it will mean faster

downloads and better

quality audio and video

over the internet

I have not got that equation yet, says Professor Dold. But wehave started by studying more normal forms of fire behaviour.Collaborations that have developed with bushfire experts inAustralia and in Portugal, which also suffers from severe fires,have highlighted a lot of experimentally observed phenomenathat I believe we can usefully model mathematically.

For example, one phenomenon that can prove dangerous for firefighters is that a narrow head fire with trailing flanking firesgenerally advances more slowly than a fire with a wide headunder the same atmospheric conditions. If the wind changesdirection, however, a long flanking fire can suddenly become a wide head fire that spreads faster than the original fire.

Professor Dold has found that the plume of hot gases createdby the fire seems to be responsible for this behaviour, becauseit draws in colder air as it rises. As the fire advances, each partof it is sucking in cold air from its immediate surroundings. Thisdraws air away from other parts of the fire and so influencesthe way it spreads. The effect is weaker for larger fires andthis explains why a fire with a wider head is able to advancemore quickly.

I am very hopeful that this mathematical research will help us to develop a better understanding of how and whybushfires behave as they do, which should help when it comesto managing or trying to control them, Professor Dold says.

Contact: Professor J Dold, email: dold@man.ac.uk

A BURNING ISSUE

Many parts of the world experience bushfires. Sometimes these fires are mild and they may even help the local ecology.But under dry and windy conditions they can become extremelydangerous and difficult to extinguish. Whether started bylightning or by human negligence, bushfires can spread rapidly, destroying everything in their path.

At the University of Manchester, Professor John Dold is applyingmathematical approaches to understanding the behaviour ofbushfires how they burn and the factors that can influencethe ways in which they spread. The work should, ultimately,help to improve the tools used by fire fighters for predicting and hopefully controlling the course of a fire.

Professor Dold first became interested in the subject when on a visit to Australia he was invited by an expert on bushfiresto provide the equation for various mysterious events thatoccurred when a large bushfire reached parts of Canberra in January 2003, destroying some 500 houses and killing four people.

Some strange things happened during this fire, ProfessorDold says. For example at a point 10 miles west of Canberra a powerful whirlwind developed within the fire. It initiallylevelled a 100 metre belt of mature pine trees and thentravelled all the way into a western suburb, adding to thedevastation. Elsewhere, the fire crossed a large area offarmland that had been grazed almost to bare earth therewas nothing there to burn and yet flames up to six metres high were seen by experienced fire fighters.

The environment sits at the top of the agenda

of global politics. The Earth is warming up

with potentially catastrophic consequences.

Species are under threat from mans activities

across the planet. Mathematics and statistics

can provide vital information about many

aspects of the natural world, from how

populations of animals are changing over

time, to better and more accurate forecasts

of climate change. Mathematics can also

give insights into important environmental

phenomena, such as bushfires.

6 ENGAGING MATHS

Nature by numbers:

mathematics and the environment

COUNTING ON BETTER CONSERVATIONMANAGEMENT

Noah had it easy. He had to count only two of each animal that stepped onto the Ark. But counting animals in the wildis a fiendishly complicated business. Accurately estimating thesizes of populations of different species, and how these are likely to change over time, is crucial to the success of wildlifeconservation. And the issue extends beyond a few exotic animalsin remote parts of the world it can have a direct impact on all our lifestyles and livelihoods. International agreements onfishing quotas, for example, depend on accurate estimates of fish stocks and how these stocks might decline or recoverunder various circumstances.

The National Centre for Statistical Ecology is a bold initiativeaimed at improving our understanding of how populations of wild animals develop and predicting how changes in the environment will affect species. The centre is a virtualorganisation, consisting of a partnership between the universities of Kent, Cambridge and St Andrews, with respectiveco-directors Byron Morgan, Steve Brooks and Steve Buckland.

An important part of what we are trying to do here is come upwith new methods for dealing with conservation management,says Professor Morgan. Hardly a week goes by without usreading about another species becoming endangered even to the point of extinction.

Ecology the study of organisms in their environment reliescrucially on the collection of reliable data and the correctanalysis of those data. It is absolutely vital to measure things,says Professor Morgan. You cannot rely on anecdote or simple observation. Inferences and predictions must be rooted firmly in fact.

Accurate estimation of the characteristicsof wild animal populations is a difficulttask. Scientists cannot count every single member of a population, and so samples must be taken and analysedwith appropriate statistical tools to give reliable information about the

population as a whole. This requires a sophisticated fusion of mathematics, statistics and biology. It is important to factorin aspects of the biology of the organism in question, saysProfessor Morgan. For example, if you are modelling how the members of many populations of wild mammals move and survive, it is essential to realise that males and females can behave very differently; it is often the case that males do not live as long as females. Models that do not takeaccount of sex differences would not be very useful.

Increasingly powerful computers are enabling scientists todevelop better statistical tools for ecologists to carry out this kind of complicated analysis. We can factor in all kinds of complexity that allows us to do things we could not dobefore, says Professor Morgan. You could say that this kind of modelling is an idea whose time has now come.

Contacts: Professor B Morgan, email: b.j.t.morgan@kent.ac.uk

A QUESTION OF TRUST

Computer modelling is playing an increasingly important role in modern life. As well as being vital in many areas of researchand development, modelling has a direct impact on most of us in our daily lives. The weather forecasts we see on thetelevision each evening are produced by some of the mostcomplex and sophisticated computer models that have beendeveloped, running on the worlds most powerful computers.

Predictions of long-term changes in the Earths climate alsodepend on computer models. Clearly the reliability of suchmodels is crucial their forecasts can form the basis for keypolicy decisions made by governments.

People run these models by feeding in data based on the bestavailable scientific knowledge, says Professor Tony OHagan ofthe University of Sheffield. However, we never know for sureall the science that we need. For example in climate modellingmany of the numbers used to describe atmospheric processesare estimates. Clearly if we are uncertain about what goes intothe model, we will necessarily be uncertain about what comesout. So the question arises: how trustworthy are the models?

This is a fundamental question that needs answering for any computer model, which is why Research Councils UK issupporting a major programme of research called ManagingUncertainty in Complex Models, led by Professor OHagan.

The predictions made by computer models can have hugeimplications for national policy, so it is vital to know the rangeof uncertainty, says Professor OHagan. What we would liketo see ultimately is that instead of providing a given answer toa problem, a model routinely provides a probabilistic answer:there is a 65 per cent chance that this will be the case, that sort of thing.

Working out the degree of uncertainty is not trivial, however. A particular modelcould have hundreds or thousands ofdifferent inputs, each with its own degreeof uncertainty. One way of calculating the trustworthiness of the model wouldbe to run it with very many differentpermutations of the input data, chosen to exhibit all theuncertainty in those inputs. But this could take, literally, years or even decades to complete.

Instead, Professor OHagan and his colleagues are devising newapproaches using cutting-edge statistical and mathematicaltechniques. Essentially, the researchers are able to construct a statistical facsimile of a given computer model called anemulator which is able to simulate the functions of the model but can provide information about the uncertainty of the output data extremely quickly.

I believe it is vital that this kind of uncertainty calculation isdone routinely and robustly for all modelling, says ProfessorOHagan. Once we understand the uncertainties in modelpredictions, and their various sources, we can see the best waysto reduce those uncertainties. The ultimate goal of the project is to provide the tools to target future research efforts.

Contact: Professor A OHagan, email: a.ohagan@sheffield.ac.uk

ENGAGING MATHS 7

You could say that this kind

of modelling is an idea whose

time has now come

or the banks interest for loans to be made inappropriately.Furthermore, it has been demonstrated that such objectiveassessments provide a more reliable risk assessment than thesubjective view of an individual employee of the bank.

Similarly, sophisticated statistical models can be constructed of peoples spending behaviour. Many of us have beenphoned by our card supplier to ask about a particular recenttransaction, Professor Hand says. This is because the securitysystem has noticed an anomaly in the pattern of spending. We are working on new and more efficient statistical modelsthat can identify any sudden blips in spending patterns thatmight indicate a fraudulent transaction.

Statistics is also a vital tool in gaining a big picture of a banksfinancial position. If the bank has a portfolio of 100,000accounts, how much should it keep in reserve, how much can it expect to lose through people defaulting on payments? saysProfessor Hand. Statistical modelling allows a risk assessment ofthe entire portfolio. Statisticians are now playing an increasinglycentral role in the banking system in all sorts of ways.

Contact: Professor D Hand, email: d.j.hand@imperial.ac.uk

BANKING ON STATISTICS

Forty years ago credit cards were virtually unheard of. Then the Barclaycard appeared, heralding a revolution in personalfinance. Now there are around 74 million credit and debit cardsin circulation in the UK. The ubiquity of credit cards presentshigh street banks with two big problems: how to assess aclients creditworthiness and how to protect against fraud.

Statistics can come to the rescue in both of these cases. At Imperial College London, Professor David Hand worksclosely with banks to apply his statistical expertise to problemssuch as these. Because there are such large numbers ofpeople who have loans and credit cards, it presents an obviousapplication area for statistical analysis, he says. With so manypeople taking out loans and cards, it is no longer feasible forthe old-fashioned meeting with the bank manager to discussevery application. This means that the creditworthiness of a client needs to be assessed automatically and objectively.

By analysing data from thousands of previous customers,statisticians can create computerised models that can predictwhether a person applying for a loan is likely to be able to meetrepayments or not. Based on the information that the applicantprovides, he or she can be automatically classified as high- orlow-risk, says Professor Hand. It is not in the clients interest

Mathematics has always played a central role

in the world of money, from the relatively simple

arithmetic of balancing a companys books, to

complicated forecasts of national economies.

Unsurprisingly, maths also lies at the heart

of the worlds banking and finance systems.

As mathematical and statistical modelling

techniques become increasingly sophisticated,

researchers are providing innovative solutions

to a range of important issues, from how to spot

credit card fraudsters to better ways of assessing

risk on the financial markets and more efficient

techniques for solving hugely complex problems

faced by institutions that trade on the worlds

stock exchanges.

8 ENGAGING MATHS

The money programme:

mathematicsof finance and economics

HOW TO TAKE A BETTER RISK

In the world of finance, the lending and borrowing of moneyplays a key role. Investors can lend money to a new venture inthe expectation that at some point in the future this money willbe returned with interest. Often these loans can be traded asbonds essentially a promise that the money will be repaid.

However, such transactions carry an inherent risk: the borrowermight default. How this risk is assessed is crucial to the price ofsuch financial products. But the way that traders assess theserisks is inconsistent and often fails to take into account importantinformation. This can lead to bad investment decisions eithertaking an excessive risk, or being too conservative.

At the London School of Economics, Professor Ron Anderson is developing new mathematically based techniques to enablefinancial institutions to make more informed decisions on creditrisk. The work also has potentially important implications forother areas of finance, such as the management of the assetsand liabilities of pension funds.

Essentially there are two sources of information that give the market an assessment of the risk associated with a financialproduct, Professor Anderson says. First there is historicalinformation. For example in a case where credit has beenadvanced to a company in a particular emerging market, therewill be information about how lenders have previously faredhere whether there has been a high rate of default and if so, what the level of debt recovery has been. Secondly there is the markets future assessment of the risk, which will bereflected in the market price of the product.

The question that Professor Anderson isseeking to answer is how can these twosources of information best be interpretedconsistently and objectively to provide a better assessment of risk?

The situation is complicated, ProfessorAnderson says. For example the marketprice of a product reflects not only themarkets assessment of the underlying risk,but also the markets appetite for risk.

Often the appetite for risk can change while the assessment of the risk stays the same, and this could cause the price of the product to change in a way that could easily be misinterpreted.

Similarly, historical data need to be analysed carefully. For example suppose in the past lenders have lost 100 per centon defaulted loans in an emerging market because of poorbankruptcy laws, says Professor Anderson. If the laws haveimproved subsequently, making loans based on past evaluationsmight lead lenders to refuse credit to good borrowers.

The LSE researchers are looking at new mathematical models to marry together this complex information from historical dataand market prices to produce a more accurate assessment ofrisk to help investors make better decisions. The work also hasimplications for examining the assets and liabilities of pensionfunds, to ensure that regulators and investors have a betteridea of how the movement of prices in the stock market can affect the solvency of a pension fund.

Contact: Professor R Anderson,email: r.w.anderson@lse.ac.uk

BUY! SELL! MATHS ON THE STOCK MARKET

The worlds financial markets are vital engines of the globaleconomy, trading vast quantities of assets daily. The Bank for International Settlements estimated that in 2004 the daily turnover of foreign exchange amounted to 1 trillion equivalent to about 150 for each person on Earth. But the daily turnover in currency and interest-rate derivatives was almost as large, at 600 million. A derivative is a financialcontract which promises certain payments depending on the behaviour of the prices of simpler underlying assets, and this business in effect did not exist until the appropriatemathematical tools were evolved, some 20 years ago.

In the Statistical Laboratory at the University of Cambridge,Professor Chris Rogers investigates new mathematicalapproaches to understanding how asset prices move over time, and to the science of pricing and hedging derivatives.

For example, one simple derivative that is traded is theEuropean call option. A trader who buys one of these has the option to buy (at some fixed future date, the expiry date)one unit of the underlying asset (typically a share) at a fixed(strike) price, agreed when the option was bought.

At expiry, if the share is trading above the strike price, the trader buys the share for the strike price and immediatelysells it for the higher market price. On the other hand, if atexpiry the share is trading for less than the strike price, thetrader declines to exercise his option. For the trader, the optionis an insurance against a possible rise in price, but for the bankit is a source of risk which must be managed. It does this byholding an offsetting (hedging) position inshares and cash, adjusting the amounts asthe share price varies, so that in theory at expiry the value of the hedge exactlyequals the value of the option.

The bank writing the option constantlyevaluates the price of the option, and the amount of stock it should hold in its hedge, says Professor Rogers. Thisrequires a certain amount of sophistication there are unbelievable complications inoption pricing and hedging.

Some of these are evident in the case of so-called Americanoptions, where the buyer can exercise the option at any timeup to expiry, or just at expiry itself. If the value of the optionwhen exercised depends on the prices of many assets, thingsget complicated.

Trying to model such problems in conventional mathematicalways can present, among other things, huge problems of datastorage, says Professor Rogers. The obvious approach wouldneed billions of gigabytes of storage it is simply not feasible.

Professor Rogers has developed novel approaches to thesehugely complicated problems which for the first time provide a way of dealing with American options on many assets, at the same time providing new perspectives on hedging.Long before these results had appeared in print, I was beingcontacted by mathematicians in the City, and the methodologyis now quite widely used in financial institutions, he says.

Contact: Professor C Rogers, email: l.c.g.rogers@statslab.cam.ac.uk

ENGAGING MATHS 9

Long before these results

had appeared in print, I was

being contacted by the

City and the methodology

is now quite widely used

in financial institutions

The models show how a range of factors can interact to resultin survival or failure. For a production company, for example,you need to consider how much stock to hold and what levelof production to set, says Professor Thomas. Other factorssuch as the size of the factory, the number of employees,overhead costs and so forth also come into play. A big issue is how much money is available in the first place. All thesefactors can be put into the model to get an idea of whichstrategy is likely to enable the company to survive in thedifficult first few years.

Often the strategy for survival is very different from the one aimed at producing maximum profit. The models for maximisingprofit can be fairly simplistic and can assume that the companyhas access to unlimited resources, which is clearly not usually the case, Professor Thomas says. Our models include morecomplex interactions and the outcome usually is that during thefirst few years the company should adopt a relatively conservativestrategy which might not realise the most profit but which shouldhelp the company to survive. Once it has established itself, it canthen be more adventurous.

Such models could help budding entrepreneurs develop a realistic business plan when seeking funds. Similarly themodels could help investors decide if a plan is feasible or not.This is an interesting issue, says Professor Thomas. For theentrepreneur the first priority is to survive, for the investor thename of the game is the return on his investment. Can wemodel this tension and arrive at some kind of compromise?

Contact: Professor L Thomas, email: l.thomas@soton.ac.uk

A MATTER OF SURVIVAL

Small businesses form the backbone of the British economy,employing more than half the countrys private sectorworkforce some 12 million people and contributing 40 per cent to the UKs gross national product.

Unfortunately, however, one out of every two small firms goesbust within its first two years. Could mathematics help toreduce this rate of failure? At the University of SouthamptonsSchool of Management, Professor Lyn Thomas thinks it could.

Models of business have traditionally had the objective ofmaximising profit, says Professor Thomas. What a start-upfirm should be more interested in during its early life ismaximising the chances of survival and what we need ismodels that look at this objective.

Together with Dr Tom Archibald at the University of Edinburgh,Professor Thomas has been developing novel mathematicalmodels of the operation of a business aimed squarely atreducing the chances of failure in the firms early years.

Thriving industry and commerce are crucial to the economic health and growth of a country.

New businesses must be able to start up, survive and expand. Existing companies must be

able to adapt to changing circumstances to ensure that their order books stay full. And all

this must be done against a backdrop of ensuring the safety of the workforce and public.

Mathematics can play a role in each of these areas, providing new ideas about how to ensure

that small businesses have the best chance of surviving, how production lines can cope with

sudden and unexpected changes to their schedules, and how assessments of risk, which

could affect peoples wellbeing, can be made more accurately.

10 ENGAGING MATHS

Counting on commerce:

mathematicsin industry

A RISKY BUSINESS

In a modern industrial economy, issues of health and safety areparamount. Every industry must assess the risk to its workersand to the public of all its products and processes. Often thisentails the creation of sophisticated risk models that simulatea variety of situations and their potential outcomes the failureof a mechanical component or the release of a chemical intothe environment, for example. But such models are neverperfect and a key question that must be answered is: howuncertain is your model?

At the University of Strathclyde, mathematician Professor Tim Bedford and his colleagues in the department ofmanagement science are developing new, multidisciplinarytechniques for quantifying the uncertainties associated withassessments of risks of unwelcome consequences in safety and operations.

Some types of risk are relatively easy to assess. For example a metal component in a piece of movingmachinery could be subjected to manytrials in the laboratory to determineaccurately its typical lifespan, assuming itwas known how the equipment was goingto be used. Other risks are much moredifficult to predict. If a cloud of hazardousgas is released from a chemical plant, forexample, it is difficult to know how it willdisperse: it is not possible to carry outexperiments to test every conceivablescenario. Therefore, the company running the plant mustproduce risk models, but the inherent uncertainties of the models are often not stated.

In the case of the dispersion of a gas cloud, the simplest kindof model might just consider the rate that the height and widthof the plume changes as the gas is blown downwind, saysProfessor Bedford. But there are many factors that can affectthis and it is difficult to incorporate all of these into a model.

So instead of relying on relatively scant experimental data toassess the trustworthiness of the model, the researchers turn toexperts in the field and ask them to quantify the uncertaintiesthat might arise. For example we can go to a number ofphysicists with particular expertise in this area, present a rangeof scenarios and ask them to make assessments of the crediblerange of possible outcomes, including a measure of uncertaintyon those outcomes, Professor Bedford says.

By accumulating experts information of this kind and analysingit in an appropriate way, the researchers are able to presentdecision-makers, and other interested parties such as industryregulators, with sound information about the uncertaintiesattached to a given risk model. Crucially, the techniques are applicable across the entire range of risk modelling.

Predictive models are never perfect however much detail you put in, says Professor Bedford. If you want to make hard decisions using a given model, it is vital that you knowjust how good a representation of reality it is. What we aredoing is developing new methods that will let key decision-makers know how much faith they can put in the informationthat these models are providing.

Contact: Professor T Bedford, email: tim@mansci.strath.ac.uk

KEEPING THE CUSTOMER HAPPY

A successful manufacturer needs to have tight control overproduction schedules. Customers value reliability above all else,and it is crucial that orders are delivered at the agreed time.

However unforeseen spanners will, on occasion, inevitably be thrown into the works: a machine might break down, or a regular and valuable customer might put in a big orderunexpectedly that needs completing in double-quick time. Such events require the production schedule to be rejigged at short notice, presenting the production manager withdifficult decisions about which customers should take priority.

At the University of Southampton, Professor Chris Potts has been investigating novel mathematical approaches toproduction rescheduling to try to find innovative ways ofensuring that there is minimum disruption to the scheduleand that customers orders are completed as close to theiroriginal deadlines as possible.

People need appropriate tools to usewhen these disruptions occur, saysProfessor Potts. What we are trying to do is design new mathematically based ways to do the rescheduling when problems arise.

The key to any rescheduling is to try tokeep as close as possible to the originalschedule. A lot of these problems revolvearound the sequence in which various

jobs should be done, says Professor Potts. Usually the most important thing is to keep your customer happy. So themanufacturer must assign a weighting to different customers an important, long-standing customer who provides a lot ofbusiness could have a higher priority than a smaller, one-offcustomer, for example.

Other factors must then be taken into account, such as the need to hire more people or source another piece of machinery,or contract some of the work to another company. The costsassociated with each of these options are added into theequation. The cost functions included in the models can be relatively simple or much more complicated depending on the circumstances, Professor Potts says.

When all these factors and their associated costs are considered,there can be a huge number of potential production schedules.Using a field of mathematics called operational research, thesedifferent options can be searched and assessed. In the morecomplicated models we would be searching over a large numberof possible schedules and move from one to another by alteringthe parameters very slightly to see if this yields a better result,says Professor Potts. In this way we can soon home in on themost efficient schedules.

The Southampton-based project is believed to have been thefirst to examine production rescheduling in a systematic way.Results showed that our approaches can result in significantimprovements, Professor Potts says.

Contact: Professor C Potts, email: c.n.potts@maths.soton.ac.uk

ENGAGING MATHS 11

What we are doing is

developing new methods that

will let key decision-makers

know how much faith they

can put in the information that

these models are providing

So the planners need ways of anticipating how travellers will respond and so how their decisions will work in practice. And it is here that the mathematician can help.

The travelling public are faced with a large number of decisions how far to travel to work each day, whether they go by publictransport or by car or bicycle, at what time they travel, howmuch they are prepared to pay for a ticket, and so on, saysProfessor Heydecker. By using information gained from surveysto support our models, we can estimate how usage of how thetransport system will change. When planners make a decisionabout new bus services, or levying a congestion charge, we canfactor these into our models to get a range of scenarios abouthow this will affect travelling patterns and how this mightimpact on the transport system as a whole.

For example on a congested route, planners might decide to addan extra road lane. If the level of traffic remains the same, thiscould ease the congestion. However, once the traffic is movingmore freely, other drivers could decide to use this route, increasingthe traffic flow and ultimately re-introducing congestion. The mathematical models can provide information about thepotential costs, benefits and pitfalls of such planning decisions.

Contact: Professor B Heydecker, email: ben@transport.ucl.ac.uk

KEEPING THE FLOW GOING

Planning and running a transport system in society is notsimple. How to schedule bus timetables, whether to widenroads, the impact of congestion charges, the price of traintickets all of these things and many more must be taken into account to produce a smoothly functioning system that satisfies the needs of its users: thats all of us.

Mathematicians have for many years played a key role inunderstanding how the various strands of the transportnetwork come together and impact on each other. Experts suchas Professor Benjamin Heydecker at University College Londonare applying novel mathematical techniques to give transportplanners new tools to develop appropriate strategies for thecomplex and ever-changing transport requirements of a modern society.

A key aspect of the transport system is that it operates on twolevels, says Professor Heydecker. On the one hand there arecentral agencies that plan, manage and run the system, and onthe other there are the users of the system, travellers who maketheir own decisions as individuals about when to travel, whichroute to take, which mode of transport to use and so on.

This has an important consequence. It means that the decisionsthat transport planners make must work in the real world populated by hundreds of thousands of individuals.

12 ENGAGING MATHS

Transport is becoming an increasingly important

issue in modern society. As more and more

people own cars, roads are becoming busier

and congestion is a serious problem in many

towns and cities. Transport planners are trying

desperately to come up with new ideas to keep

traffic moving, by measures such as congestion

charging. Meanwhile air travel is also increasing,

and efficient ways need to be designed to

ensure that airports can cope with extra traffic.

Mathematics has a long-standing record of

providing solutions for problems such as these, and EPSRC-supported mathematicians

are working on new methods of modelling transport networks to help ensure that traffic

flows smoothly, and new ways to schedule take-offs from airports to minimise delays.

On the go:

mathematics in transport and travel

COMBATING CONGESTION

Traffic congestion is one of the major frustrations of modernlife. But not only is congestion irritating for travellers, it coststhe economy millions of pounds each year in wasted hours, as well as resulting in increased pollution.

The challenge of tackling congestion is inherently difficult. Forexample, people wont take the bus because on a congestedroute the bus is slowed down more than a car. But if more peopledid take the bus, the congestion would decrease. But if thathappened, people would start using their cars again.

There are these chicken-and-egg situations which are difficultto resolve, says mathematician Professor Mike Smith ofUniversity of York. There are some obvious ways of improvingthe attractiveness of the bus by introducing bus lanes for example but we are looking at more subtle ways ofchanging travellers perceptions of the transport network so that they make choices that will ultimately reduce theamount of congestion.

Professor Smith and his colleagues usemathematical models of the road network to test how various control mechanisms,such as traffic signalling and charging driversto use roads, might affect traffic flow.

These models are difficult to construct,Professor Smith says. Users of the networkcan make many different choices: whether totravel or not to travel, to go by car or by bus,

to choose route A or route B. Controls at various points on thenetwork will influence these choices. We end up with what istermed mathematically as an optimisation problem: how do I best choose the controls to make the network run smoothly,taking into account the reactions of the users and non-users?

A number of good models have been developed, but theserequire the operator to feed in various control factors position of signals, where road widening could be introduced, pricingcontrols at various points. The computer then provides anindication of what the consequences of these measures might be.

What we really want is for the computer itself to work out theoptimum control strategy for a network, says Professor Smith.You would like a big button on the front of the computer thatsays push me, and the computer would work through all thevarious permutations of controls, tweaking here and nudgingthere, until it arrived at the best solution. But this is provinghugely challenging. It will require three distinct groups of people to come together in a highly multidisciplinary way and to develop a common language: mathematicians, transport engineers and computer scientists.

New mathematical techniques necessary to achieve this goal arebeing developed, and cross-disciplinary research programmesare beginning to emerge. Things will take a while, but I thinkwe are getting there. Professor Smith says.

Contact: Professor M Smith, email: mjs7@york.ac.uk

MATHS PREPARES FOR TAKE-OFF

Heathrow is the worlds busiest international airport, withnearly 70 million passengers a year. Each day more than 600aircraft take off from Heathrow, and managing the order inwhich the waiting aircraft are allocated a take-off slot is centralto the smooth running of the entire operation and in ensuringthat delays are minimised.

At any one time at Heathrow there can be up to ten aircraftwaiting for clearance to take-off on the single runway available(there is one runway for take-off and one for landing), and a further ten planes taxiing on the circuit before entering the holding area. Air traffic controllers must decide the order in which to let the aircraft go something which is not asstraightforward as it might seem.

Edmund Burke, Professor of Computer Science at the University of Nottingham, explains. Upon take-off aircraftleave turbulence in their wake and this needs time to settle. If an aircraft of the same size or larger is next in the queue, it must wait for one minute; if a smalleraircraft is next, it has to wait for two minutes.Furthermore, different aircraft take off indifferent directions, and the delay betweentake-offs is increased if the aircraft are flyingin the same direction. Another factor is thatseveral of the aircraft will have been allocateda 15-minute window for their take-off, so thecontrollers must ensure that the plane leaveswithin that time.

Professor Burke and research student Jason Atkin are workingwith National Air Traffic Services, who are funding the projecttogether with EPSRC, through the Smith Institute, to developthe underlying search mechanism the brain for acomputerised system that automatically suggests the mostefficient take-off schedule for waiting aircraft.

If you have 20 aircraft waiting for a slot, there are literallybillions of possible permutations for scheduling them, says Mr Atkin. The aim is to produce a sequence with as manyone-minute gaps and as few two-minute gaps as possible.

Currently this is done manually by an air traffic controller usingground radar to pinpoint where particular aircraft might be in the system and using paper strips to represent the aircraft in the constantly changing take-off schedule.

With our system we feed in all the relevant information suchas how many planes are waiting, where they are in the system,how big they are, which direction they will be flying and so forth, says Professor Burke. By using the branch ofmathematics called operational research we can modelmathematically what the situation is, then navigate through thevast number of permutations to arrive at improved solutions.

The Nottingham researchers are working closely with air trafficcontrollers to ensure that the systems they are developing will be practical to use in busy airports. Our system will not replace an air traffic controller, Professor Burke stresses.Rather it will provide a valuable tool and allow the controllerto do his or her job more efficiently.

Contact: Professor E Burke, email: ekb@cs.nott.ac.uk

ENGAGING MATHS 13

By using operational

research we can model

mathematically what the

situation is, to arrive at

improved solutions

that is released and the number of cells that die. What we asmathematicians will do is model this three-way relationship, saysDr Davidson. Ultimately what the doctor wants to know quickly ishow effective the drug is: is it actually killing cancer cells and if so,how many is it killing and how fast? Unfortunately, relationships ofthis type can be extremely complex, particularly inside the humanbody. For cancer treatment in patients, it is not simply a case thatmore drug kills more cells. It is very unpredictable at present.

The situation is made more complicated by the aggressivenature of anti-cancer drugs, which often kill healthy cells as anunwanted side-effect. There is a delicate balancing act here,says Dr Davidson. The patient needs enough drug to kill thetumour and no more. Can we use our methods to help fine-tune doses so that the patient receives as little as is necessaryof the right drug?

The Dundee mathematicians are working closely with scientistsin the Cyclacel laboratories to produce models of how the doseof the drug impacts on cell death and how this is reflected inthe concentrations of biomarker. Dr Davidson says, Ultimatelythis kind of approach could lead to mathematical models useddirectly by clinicians who would be able to key in informationabout a particular patient and a particular drug. The modelwould tell the physician the most appropriate treatment regime and how to measure its effectiveness.

Contact: Dr F Davidson, email: fdavidso@maths.dundee.ac.uk

NEW INSIGHTS INTO ANTI-CANCER DRUGS

Cancer is a devastating disease that claims thousands of liveseach year. One of the big challenges faced by doctors whentreating certain cancers is to find out quickly whether aparticular treatment is effective or not. For cancer tumours thatare within the body, it can be very difficult to measure howquickly the tumour is shrinking in response to treatment orindeed if it is shrinking at all. It might take several weeks oreven months before any change becomes apparent on a scan,for example. If it is found that the drug being used has notbeen effective, then for patients with particularly aggressivecancers this delay can have serious consequences.

One possible way around this is to find other methods ofmeasuring whether cancer cells are being killed. When cells die, tell-tale biochemical markers are released by the cell intothe blood stream and it is possible that these biomarkerscould be used as a measure of the effectiveness of a drug.

At the University of Dundee, mathematicians Dr Fordyce Davidsonand Professor Mark Chaplain are working with a local drugdevelopment company, Cyclacel Ltd, to investigate how the dose of anti-cancer drugs relates to the release of biomarkers and how this in turn equates to the actual numbers of cells that have been killed.

At Cyclacel, scientists led by Dr Bob Jackson present cultures ofcells with a drug and study the type and amount of biomarker

The mathematical sciences have played a key

role in health studies for many years. Statistical

analysis of the incidence of disease has helped

pinpoint the causes of many illnesses such

as certain types of cancer. But as new

technologies begin to provide vast quantities

of information for biologists and medical

scientists, maths is playing a more important

part than ever. Biologists are increasingly

turning to mathematicians to help sift through

the masses of data produced by techniques such as genetic screening and new methods for

imaging processes that occur in cells. And in other ways too, mathematicians are uniquely

placed to unravel many of the complicated interactions that occur in living systems.

14 ENGAGING MATHS

A healthy outlook:

mathematics in medicine and biology

ENGAGING MATHS 15

GETTING A MATHEMATICAL HOLD ON LIFE

Biological systems are inherently complicated. Processes thatlook relatively straightforward, such as the division of a cell,involve hundreds of different individual mechanisms working in concert. Similarly, diseases are seldom the result of a singlefactor a faulty gene, for example. They usually arise from a range of interacting components, which could be genetic,physiological and environmental.

At the University of Warwick, Professor David Rand specialisesin applying mathematical techniques to biological problems.The approach helps to tease out the most important aspects of complicated biological systems, and often suggests newlines of enquiry for experimental biologists to take.

Often mathematics is key to any sensible approach to thesehighly complex systems, Professor Rand says. As technologyadvances, biologists are obtaining vast amounts of information from such things as the human genome project. Mathematicsand statistics are vital tools in sorting, storing and making senseof these data.

The complexity of living systems can be overwhelming atfirst glance. The organism has mechanisms for receiving and processing constantly changing information from its environment; there are issues of growth, reproduction and death. The more complex the organism, the morecomplex the network of these interactions.

Two examples of complex biologicalsystems that Professor Rand is working onare the circadian clock and how the braincontrols energy balance within the body. Circadian clocks are 24-hour biological oscillators, found in a wide variety of organisms including mammals, insects, fungi and plants.They regulate the daily rhythm of activity in all our cells anddirectly control the expression of about one-tenth of our genes.Correct regulation of these rhythms is key to health. We are all aware from jet-lag of how disrupting the circadian clock can upset our daily rhythms but the circadian clock can alsoplay an important role in the development of certain cancers.

One of the questions about circadian clocks is why the set ofinteractions between the genes involved has such a complexstructure. What are the design principles underlying this? To unravel this mystery, Professor Rand and his collaborators,including Professor Andrew Millar from the University ofEdinburgh, are developing advanced mathematical andstatistical tools to analyse images of cells from Professor Millarsexperiments to deduce the structure of this interaction and adevelop a theory that relates this to the function of the clock.

Controlling energy balance is about whether more food isneeded by the body or whether there is sufficient fuel onboard. Understanding how this works is important in diseasessuch as anorexia and obesity. Professor Rand and Dr Hugo vanden Berg are modelling this system mathematically, providingnew insights into the processes involved. A challenging aspectof this problem is the need to span several scales, to go fromneurons to networks and interconnected regions of the brainand from there to the feedback loops of the whole organismthat connect the brain signal to the key energy-storing andenergy-using organs and tissue of the body.

Contact: Professor D Rand,email: dar@maths.warwick.ac.uk

SHAPING UP TO VIRUSES

Viruses are responsible for some of the most devastatingdiseases known to man, from AIDS and hepatitis to polio and various cancers. Because of this, medical science isconstantly seeking new and better ways to combat viruses.

At the University of York, mathematician Dr Reidun Twarock is providing important new insights into one of the mostfascinating aspects of viruses their geometry. Perhapssurprisingly, most viruses have a well-defined three-dimensional, highly symmetrical structure. Essentially a virusconsists of an external shell, the capsid, which encloses thegenetic material that the virus needs in order to replicate.

The capsid is made from small clusters of protein molecules called capsomeres that join together in a specific geometricarrangement. For example there could be a mixture ofhexagonal and pentagonal capsomeres that are bondedtogether to form an overall shape that resembles a football.

The big question that we want to answer is how thecapsomeres link to one another to form these larger geometricbodies says Dr Twarock. If we understand this we are then in a position to disrupt the process of assembly in some wayand therefore destroy the virus.

The assembly of the proteins into thecapsid is not an arbitrary process. It is governed both by the shape of the protein clusters similar to the panels on a football and the nature of thechemical bonds between them.

Working closely with biologists, Dr Twarock has discovered thatconventional theories of how these clusters knit together arenot sufficient to explain all the observed patterns that occur innature, and that more complex ideas need to be incorporated.

Using this information, the researchers have created simulationsof how capsid proteins assemble, and are investigating how itmight be possible to use drugs to alter the shape of the proteinclusters so that they are unable to aggregate into the correctthree-dimensional structure. For example it can be shown inthe test-tube that by changing certain chemical conditions,capsomeres can lose their characteristic curvature that isnecessary for them to form an enclosed, roughly spherical,shell. Instead they aggregate into sheets or tubes.

The mathematical work is also providing key insights into the biochemical nature of the external and internal surfaces of the capsid. Various structures on these surfaces are vital for enabling the virus to dock onto the outside of a cell, forexample. By looking at the geometric constraints on theorganisation of the three-dimensional structure of a virus we can see the outer and inner location of various functionalelements of the protein molecules, which play important rolesin the way viruses infect cells and replicate, says Dr Twarock.

Contact: Dr R Twarock, email: rt507@york.ac.uk

Often mathematics is key

to any sensible approach to

highly complex systems

16 ENGAGING MATHS

COUGHS AND SNEEZES

We all know that coughs and sneezes spread diseases, but just exactly how does an infection move through a population?The question is important because the spread of a disease hasserious implications not only for the health of a population butalso for the authorities whose job it is to bring the epidemicunder control. Strategic decisions need to be made aboutwhere best to allocate resources and how much effort to spend on interventions.

At the University of Warwick, mathematical biologist Dr Ken Eames specialises in producing computer models of how an infection passes between individuals and groups withina population. The research is revealing some unusual patterns of transmission of disease and is providing clues about the bestway to pinpoint the carriers of the infection who are likely to be the most influential in spreading the disease.

Simple models of disease spread assume that there is randommixing of people within a population that any given personmight pass the infection to any other person. However, thesituation is more complicated: most of us move in fairly stablesocial circles, coming into repeated contact with relatively few people.

These closed circles tend to slow down the spread of anepidemic, says Dr Eames. While infection is likely to spreadquickly within the group, the opportunities for it to spreadbeyond the group are rarer. This adds an important layer of complexity to the dynamics of the epidemic.

By using information gathered by health workers mainlyrecords of the spread of sexually transmitted diseases, as theseare the best documented Dr Eames has built mathematicalsimulations of how diseases propagate. With simulations wecan look at various intervention strategies, he says. You canask: if I interfere with the spread of the disease at this or thatpoint, how much overall effect will it have? or is the extraeffort required to target specific groups worthwhile?.

Social interactions are key to the dynamics of the spread ofdisease and the closer these are examined the more complexthe picture becomes. For example in sexually transmitteddiseases there is a natural turnover of contacts, with not alllinks in the network active at the same time, says Dr Eames.Furthermore, most people tend to be monogamous theyhave only one partner at a time and not several partnersconcurrently. This has a huge impact on the way that thedisease moves through the network.

Ultimately, Dr Eames hopes that the types of mathematicalmodel that he is investigating will be useful in providing public health agencies with information that would allow them to predict the possible courses of a disease epidemic and to target resources accordingly.

Contact: Dr K Eames, email: k.eames@warwick.ac.uk

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