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
Linked modules
Pre-requisites: MATH1059 AND MATH1060
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:
- Solve mathematically and interpret biologically simple problems involving one- and two-species ecosystems, epidemics and biochemical reactions.
- Understand and apply the concept of stability of a fixed point solution of a system of ordinary differential equations.
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.
| Type | Hours |
|---|---|
| Teaching | 60 |
| Independent Study | 90 |
| Total study time | 150 |
Resources & Reading list
Textbooks
Glendinning, P. (1995). Stability, Instability and Chaos. Cambridge.
Alon, U. (2007). An introduction to systems biology: Design principles of biological circuits. Chapman & Hall.
Murray J.D.. Mathematical Biology. Springer-Verlag.
De Vries G., Hillen G., Lewis M., Müller J. and Schonfisch B. (2006). A Course in Mathematical Biology:Quantitative Modeling with Mathematical & Computational Methods. SIAM.
Jones D.S. & Sleeman B.D. (2010). Differential Equations and Mathematical Biology. Chapman & Hall.
Nicholas Britton. Essential Mathematical Biology. Springer.
Strogatz, S.H,. Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering. Addison-Wesley.
Leah Edelstein-Keshet (2005). Mathematical Models in Biology. SIAM.
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
| Method | Percentage contribution |
|---|---|
| Final Assessment | 60% |
| Coursework | 40% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
| Method | Percentage contribution |
|---|---|
| Exam | 100% |
Repeat Information
Repeat type: Internal & External