• For Entry: September
  • Duration: 12 months
  • School: Science & Engineering
  • Study Mode: Full Time

The focus of this course is using mathematics to solve biological and biomedical problems. Graduates from his course are well placed to make significant contributions to the rapidly expanding healthcare sector and major societal challenges including global sustainability.

TEF Gold - Teaching Excellence Framework

The focus of this course is using mathematics to solve biological and biomedical problems. Graduates from his course are well placed to make significant contributions to the rapidly expanding healthcare sector and major societal challenges including global sustainability.

The topics taught are directly linked to the research that we do, so you will be learning at the cutting edge of mathematical biology.

We are a relatively small division and operate with an excellent staff/student ratio. One advantage of this is that we can get to know each student personally, and so can offer a friendly and supportive learning experience. Staff are ready and willing to help at all levels, and in addition, our Student-Staff Committee meets regularly to discuss matters of importance to our students.

In this course we use a wide selection of mathematical software packages such as MATLAB, Maple and COMSOL.

We have a weekly seminar programme in the mathematics division, which features  a variety of mathematical biology topics.

Our current research

The University of Dundee has a long history of research in mathematical biology. Current mathematical biology research in Dundee involves the application of modern applied mathematics and computational modelling to a range of biological topics including cancer growth and treatment, ecology, plant dynamics and microbial biofilms. The mathematical biology group has close links with colleagues in Life Sciences and Medicine which ensure that research projects are driven by cutting-edge science.

Who should study this course?

This course suits graduates with a degree in mathematics or in a subject with strong mathematical components such as physics, who wish to develop their mathematical skills whilst focussing on solving biological and biomedical problems.

Teaching Excellence Framework (TEF)

The University of Dundee has been given a Gold award – the highest possible rating – in the 2017 Teaching Excellence Framework (TEF).

Read more about the Teaching Excellence Framework

TEF Gold - Teaching Excellence Framework

 

How you will be taught

You will learn by traditional methods such as lectures, tutorials, and workshops as well as via computer assisted learning. We teach the use of professional mathematical software packages in order to allow you to explore mathematics far beyond the limits of traditional teaching.

Individual reading and study takes a particularly important role in the Summer project. For the project, you will be guided to prepare your research project plan and to develop skills and competence in research including project management, critical thinking and problem solving, project reporting and presentation.

How you will be assessed

Assessment is via a mix of open book continual assessment and closed book examinations, with a substantial project completed over the Summer.

What you will study

This one year course involves taking four taught modules in semester 1 (September-December), followed by a further 4 taught modules in semester 2 (January-May), and undertaking a project over the Summer (May-August).

Modules

A typical selection of taught modules would be:

About the module

This module, aimed at the Level 5 student, takes an advanced look at dynamical systems. The time evolution of many biological, chemical, or physical processes, as well as systems considered in engineering or economics, can be described by difference or differential equations. Dynamical systems theory allows us to study these systems of equations and inver information about the behaviour of the corresponding biological, chemical or physical systems. It addresses questions like the existence and stability of solutions, how the behaviour of solutions changes depending on the system parameters, or determines the existence of strange attractors or chaos in the system.

This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.

Indicative Content

  • One-Dimensional Maps

    Definition, Cobweb Plot: Graphical Representation of an Orbit, Stability of Fixed Points, Periodic Points, Chaos: Lyapunov Exponents.

  • Ordinary Differential Equations

    Background, Examples of main Physical and Biological Processes described by Ordinary Differential Equations (ODEs), Existence and uniqueness of solutions of ODEs, Linearised Stability Analysis, Two-dimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

  • Partial Differential Equations

    Definitions, Background, Well-posedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for Reaction-Diffusion Equations.

Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via an exam (60%) and coursework (40%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

In this module the Level 5 student will learn to write their own code and to apply built-in "black box" solvers in MATLAB and COMSOL to mathematical modelling problems. This module is mandatory for Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology. This module may be taken in combination with another at Level 5 by students taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in each of the modules MA32005 and MA42003, or equivalent.

Indicative Content

  • MATLAB fundamentals

    Students will learn basic operations in MATLAB, and implement various finite difference schemes to solve ODEs (primarily initial value problems) originating in celestial mechanics, population dynamics, and cell biomechanics.

  • MATLAB ODE solvers for initial value problems

    Students will learn to use standard built-in solvers with MATLAB, particularly ode45 and ode23s, and possibly dde23. We will apply these solvers to initial value problems (and possibly delay differential equations) stemming from celestial mechanics, cell biomechanics, and population dynamics.

  • MATLAB random variables, stochastic processes, and SDEs

    After a brief introduction to stochastic differential equations (SDEs), students will learn MATLAB solution techniques, with applications to Brownian motion and related physical processes. We will also learn to simulate discrete and continuous stochastic processes, and generate samples from random variables with arbitrary distributions.

  • MATLAB ODE solvers for boundary value problems

    Students will implement a standard "shooting" method to solve a BVP from heat transfer. We will learn to use the standard built-in solvers, particularly bvp4c. We explore alternate solution techniques, such as by formulating the discretised equation as a linear algebraic system, and as the steady state solution to a PDE; these approaches help drive us towards PDE solution methods. The class will apply these solvers to boundary value problems stemming from heat transfer and fluid mechanics.

  • MATLAB for PDEs

    Students will implement explicit finite difference methods in MATLAB, with a focus on reaction-diffusion problems. The overall goal will be to solve coupled reaction-diffusion problems (with heterogeneous coefficients) and cell growth.

  • Weak formulations for partial differential equations; introduction to FEMs

    We repose PDEs using a weak formulation, using the context of function spaces. Using this framework, we develop an understanding of finite element methods (FEMs).

  • FEMs and COMSOL fundamentals

    Students will learn to solve reaction-diffusion equations using the built-in FEMs in COMSOL.

Delivery and Assessment

Delivery of this module will take a hands-on, interactive approach, where lectures are integrated with guided computer lab time. Assessment will be based on computational coursework (100%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This module, aimed at Level 5 students, gives a non-measure theoretic introduction to stochastic processes, considering the theory and some applications and going on to introduce stochastic differential equations and their solutions. This module may optionally be taken in combination with others by Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology or Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module would find it beneficial to have taken each of the modules MA32001 and MA51007, or equivalent.

Indicative Content

  • Probability fundamentals

    Elementary probability concepts such as random variables, expected value, moment generating and characteristic functions, conditional expections, probability inequalities and limit theorems, etc.

  • The Poisson process

    (Homogeneous) Poisson process and related examples such as interarrival and waiting time distributions and conditional distribution of the arrival times. Some practical examples such as the busy period of the M/G/1 queueing system. Introduction to the nonhomogeneous Poisson process.

  • Markov chains

    (Discrete-time) Markov chains and some related examples. Chapman-Kolmogorov equations and classification of states.

  • Continuous-time Markov chains

    Continuous-time Markov chains, birth and death processes, and the Kolmogorov differential equations.

  • Brownian motion and stochastic differential equations

    Basics of Brownian motion, Ito^ integral and Ito^ formula, and then stochastic differential equations (SDEs). A number of commonly used SDEs and their solutions will be discussed.

Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks and a presentation.

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This is a Level 5 course that offers a robust understanding of the inverse problems theoretical framework and methods suitable for medical and financial applications. The aim is to achieve comprehensive knowledge in the theoretical fundaments and general methodology for inverse problems in various heterogeneous media, including medical applications and finance. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.

Indicative Content

  • Examples of Inverse Problems

    Examples from medical applications and finance.

  • Inverse Methodology Preliminary Foundation

    Necessary Basic Definitions and Theorems in Measure Theory and Function Spaces

  • General Regularisation Theory

    Tikhonov′s regularization method. Landweber Iteration. The Discrepancy Principle of Morozov. Conjugate gradient method.

  • Galerkin Methods

    Galerkin General formulation. The Least Squares Method. The Dual Least Squares Method.

  • The Truncated Singular Value Decomposition Method

  • Stable inversion via the Mollification Method

  • Inverse Problems in General Heterogeneous Media and Medical Applications

    Backward heat conduction problem. Inverse problems in reaction-diffusion equations.

  • Inverse problems in finance

    Formulation of forward model: Black-Scholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time- and price-dependent volatilities.

Delivery and Assessment

The module is delivered in the form of lectures and assessed via coursework (100%) consisting of tests and homeworks.

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This module, aimed at the Level 5 student, covers the fundamentals of Measure Theory including Lebesgue Measure and Integration and Lebesgue Spaces as well as their pivotal implications in the modern analysis of partial differential equation and probability theory. This module may optionally be taken in combination with other modules at this level by Level 5 Students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001 or equivalent.

Indicative Content

  • Construction of a general measure space

    Definition and properties of a σ-algebra. The concept of measures as a non-negative σ-additive real valued set functions defined on a σ-algebra. Measurable sets, σ-finite measurable sets, the notion of algebra of generators of a σ- algebra of parts. Construction of the Lebesgue Measure. Charatheodory Extension Theorem: characterising the unique extension of a finitely additive non-negative real valued function on an algebra of generators to a measure on the generated σ-algebra.

  • Measurable functions

    Definition. Approximation by measurable simple/step functions as well as by continuous functions.

  • Construction of the Lebesgue Integral

    Construction of the integral for measurable indicators (characteristic) functions as well as for measurable simple/step functions. Construction of the integral for a general measurable function. Properties of Lebesgue Integrable functions (additivity, multiplication by scalars, positivity, monotonicity). Definition and basic vectorial properties of the space of Lebesgue Integrable Functions L1.

  • Limit theorems (concerning point-wise convergence, almost everywhere convergence, and convergence in measure)

    Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

  • Product Measures and the Fubini Theorem

  • Absolutely continuous measures and the Lebesgue-Radon-Nikodym Theorem

  • Definition and properties of general Lebesgue Spaces Lp and their embedding relations

If time permits then the following will also be covered:

  • The Sobolev Spaces H1 and their implications for the analysis of elliptic equations partial differential equations

  • Connection with probability theory, including random variables, cumulative distribution functions and martingales

Delivery and Assessment

The module is delivered in the form of lectures and seminars and assessed via an exam (60%) and coursework (40%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This Level 5 module will undertake analysis and computational simulation of mathematical models of cancer growth and treatment. In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.

Indicative Content

  • Overview of Cancer Growth and Spread

    The biology of cancer.

  • Avascular Solid Tumour Growth

    ODE models of solid tumour growth Greenspan′s model. Reaction-diffusion moving boundary models.

  • Tumour-induced Angiogenesis

    Continuum PDE models. Hybrid discrete-continuum models.

  • Cancer Invasion

    Continuum PDE models. Hybrid discrete-continuum models.

  • Immune Response to Cancer

    ODE models of the immune response to cancer. PDE models of the immune response to cancer.

  • Chemotherapy and Radiotherapy Treatment

    Chemotherapy drug scheduling models. The linear-quadratic radiobiological model.

Delivery and Assessment

The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

In this module we will undertake analysis and computational simulation of mathematical models of ecological and epidemiological systems including predator-prey systems, host-parasitoid systems, plant-herbivore systems, spread and transmission of disease (e.g. AIDS, SARS, measles, rabies). In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.

Indicative Content

  • Non-spatial models

    Difference equation models. Ordinary differential equation models. Delay difference/differential equation models.

  • Spatial models

    Integro-difference equation models. Partial differential equation models. Integro-differential equation models.

Delivery and Assessment

The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This module, aimed at Level 5 students, studies mathematical models of cellular physiology in metabolism, genetic networks, and electrophysiology. This module is mandatory for Level 5 students taking the MSci in Mathematical Biology and may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32009, or MA41002, or equivalent.

Indicative Content

  • Biochemical Reactions

    Enzyme kinetics. Glycolysis

  • Genetic Networks

    Central Dogma of biology, review of bifurcation theory. Feedback loops and Oscillation. Circadian rhythms. Cell cycle model.

  • Electrophysiology

    The Hodgkin-Huxley model. The FitzHugh-Nagumo equations. Small network dynamics of coupled neurons.

Delivery and Assessment

The module is delivered in the form of lectures and workshops/presentation classes and assessed via exam (70%) and coursework (30%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This module, aimed at the Level 5 student, specifically aims to develop students' knowledge, skills and understanding of the fundamentals of function spaces (infinite dimensional spaces) and linear operators and functionals defined on function spaces.

Functional analysis plays an important role in many areas of applied mathematics and is essential for the theory of Partial Differential Equations, Numerical Analysis, Probability theory and Theoretical Physics. Functional analysis originated from classical analysis and is formed by the study of infinite dimensional vector spaces and linear functions defined on these spaces. The theory of functional analysis was developed by some of the most famous mathematicians of the 20th century such as Hilbert, Schmidt, Riesz, Banach and von Neumann. Functional analysis can be characterised as a combination of infinite-dimensional linear algebra and classical analysis. Methods of functional analysis will allow us to analyse the properties of function spaces and to characterise solutions of integral and differential equations, arising in modelling of many biological and physical systems.

This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in MA32001, or equivalent.

Indicative Content

  • Function Spaces

    • Normed spaces and inner product spaces (Banach and Hilbert spaces)
    • Lebesgue integral and Lebesgue spaces
    • Orthogonality in Hilbert spaces
    • Weak derivatives and Sobolev spaces
    • Notion of weak convergence in Function Spaces
  • Linear Operators and Linear Functionals

    • Definition of linear operators and linear functionals
    • Definition of a norm of a bounded linear operator
    • Riesz and Lax-Milgram theorems
    • Definition, main properties and examples of compact operators
    • Spectral theory for compact operators
  • Well-posedness results for Integral and Partial Differential Equations

    • Application of Lax-Milgram theorem and theory of compact operators to prove the well-posedness results

Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

About the module

This module is a personal research project for students on the MSc in Applied Mathematics and Mathematical Biology, which runs over the summer. If you have questions about this module please contact your Advisor of Studies.

Prerequisites

Students taking this module must have achieved an average mark of at least 50% (C3) over the Mathematics MSc modules taken in Semesters 1 and 2 and have obtained at least 75 credits in these modules.

Indicative Content

  • Project

    Carry out a substantial project in an area of mathematics and document the work in a project report.

Delivery and Assessment

The project will require substantial independent work as well as regular meetings with the project supervisor(s). Assessment will be based on coursework (100%) consisting of the project report.

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 60 SCOTCAT credits or 30 ECTS credits.

Finally, all students will undertake a Personal Research Project under the supervision of a member of staff in the Mathematical Biology Research Group.

The Biomedical Sciences are now recognizing the need for quantitative, predictive approaches to their traditional qualitative subject areas. Healthcare and Biotechnology are still fast-growing industries in UK, Europe and Worldwide. New start-up companies and large-scale government investment are also opening up employment prospects in emerging economies such as Singapore, China and India.

Students graduating from this programme would be very well placed to take advantage of these global opportunities.

You should have, or expect to have, a 2.2 BSc (honours) or above, or a suitable alternative qualification, in a relevant mathematical discipline.

 EU and International qualifications


English Language Requirement

IELTS Overall 6.0
Listening 5.5
Reading 5.5
Writing 6.0
Speaking 5.5

 Equivalent grades from other test providers

 

English Language Programmes

We offer Pre-Sessional and Foundation Programme(s) throughout the year. These are designed to prepare you for university study in the UK when you have not yet met the language requirements for direct entry onto a degree programme.

 Discover our English Language Programmes

The fees you pay will depend on your fee status. Your fee status is determined by us using the information you provide on your application.

 Find out more about fee status

Fee statusFees for students starting 2017/18
Scottish and EU students £5,950 per year of study
See our scholarships for UK/EU applicants
Rest of UK students £5,950 per year of study
See our scholarships for UK/EU applicants
Overseas students (non-EU) £14,950 per year of study
See our scholarships for international applicants

You apply for this course via the UCAS Postgraduate (UKPASS) website which is free of charge. You can check the progress of your application online and you can also make multiple applications.

You'll need to upload relevant documents as part of your application. Please read the How to Apply page before you apply to find out about what you'll need.

  Degree Course Code
Apply NowMathematical Biology MScP044176

Course Contact

Dr Dumitru Trucu
Science and Engineering
pgt-maths@dundee.ac.uk
+44 (0)1382 384462

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