The University of Southampton
Courses

# MATH3052 Mathematical Biology

## Module Overview

Biology is undergoing a quantitative revolution, generating vast quantities of data that are analysed using bioinformatics techniques and modelled using mathematics to give insight into the underlying biological processes. This module aims to give a flavour of how mathematical modelling can be used in different areas of biology. Typically the models that are used in biology cannot be solved analytically. Nonetheless they give very useful information about the behaviour of the system. We will start by studying what we can say about differential equations that we cannot solve. For example, we cannot solve the equation of a simple pendulum analytically, but we can still say under what conditions it has periodic solutions. For biological oscillators this is usually what matters: it is important that your heart beats regularly, but whether your pulse rate is 68 or 71 beats per minute is less critical. Having introduced the mathematical tools needed to study ordinary differential equations, we will apply them to simple models of population dynamics, epidemics and biochemical reaction networks. One of the pre-requisites for MATH6149

### Aims and Objectives

#### Learning Outcomes

##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Develop the ability to explain mathematical results in language understandable by biologists.
##### Disciplinary Specific Learning Outcomes

Having successfully completed this module you will be able to:

• Understand and apply the concept of stability of a fixed point solution of a system of ordinary differential equations.
• Solve mathematically and interpret biologically simple problems involving one- and two-species ecosystems, epidemics and biochemical reactions.

### Syllabus

1. Fixed points of ordinary differential equations • Phase space of an ordinary differential equation • Linear stability analysis of fixed points 2. Bifurcations • Saddle-node, transcritical, pitchfork and Hopf bifurcations 3. Population dynamics • One- and two-species ecosystems: how species reproduce, interact and die 4. Infectious diseases • The SIR model: a simple model of an epidemic 5. Biochemical reaction networks • Enzyme kinetics and biochemical switches

### Learning and Teaching

#### Teaching and learning methods

Lectures, lecture notes, web support materials, private study.

TypeHours
Teaching60
Independent Study90
Total study time150

Jones D.S. & Sleeman B.D. (2010). Differential Equations and Mathematical Biology.

Nicholas Britton. Essential Mathematical Biology.

Strogatz, S.H,. Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering.

Alon, U. (2007). An introduction to systems biology: Design principles of biological circuits.

Leah Edelstein-Keshet (2005). Mathematical Models in Biology.

Murray J.D.. Mathematical Biology.

Glendinning, P. (1995). Stability, Instability and Chaos.

De Vries G., Hillen G., Lewis M., Müller J. and Schonfisch B. (2006). A Course in Mathematical Biology:Quantitative Modeling with Mathematical & Computational Methods.

### Assessment

#### Summative

MethodPercentage contribution
Exam  (2 hours) 60%
Online test 10%
Online test 10%
Weekly quizzes and puzzles 20%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Pre-requisites: MATH1059 AND MATH1060

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.