# Mathematics / Applied Mathematics MSc

## 5E: Mathematical Biology MATHS5072

**Academic Session:**2024-25**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 2**Available to Visiting Students:**No**Collaborative Online International Learning:**No

### Short Description

Mathematical biology is an exciting application of mathematics to biology. In this course students are introduced to the concepts and techniques involved in developing mathematical models of biological systems. Students will learn how to analyse the resulting models and interpret their results in the context of the biological questions being asked. In particular, topics considered include population dynamics and epidemiological processes in ecological systems, propagation of signals in nerve cells, biological waves, pattern formation and morphogenesis.

### Timetable

34 x 1 hr lectures and 12 x 1 hr tutorials and 5 lab hours in a semester.

### Excluded Courses

MATHS4106 - 4H: Mathematical Biology

### Assessment

Assessment

20% coursework and 80% Examination. Course work will involve the implementation of numerical solutions of differential equations and difference equations and the completion of related written exercises

Reassessment

Resit exam available to MSc students.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

### Course Aims

This course aims to teach the application of differential equations and difference equations to problems in biology. It will provide an understanding of the mathematical modelling methods used to develop such equations and it will provide training in the mathematical and numerical techniques used to analyse the solutions to these mathematical equations. Emphasis will also be placed on interpreting these results to answer the biological questions being asked.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

(a) Give a biological interpretation for the terms in difference equation and ordinary differential equation (ODEs) models for biological systems;

(b) Determine the steady states and stability of biological models;

(c) Use graphical methods (phase-plane analysis) to understand the dynamics of the nonlinear ODEs and difference equations;

(d) Calculate the stationary age distribution and long term population growth rate for Leslie matrix models;

(e) Derive infectious disease models and define the concepts of epidemic, endemic and disease-free states;

(f) Derive R0, the basic reproduction rate for a disease and relate it to vaccination strategies;

(g). State the FitzHugh-Nagumo equations and relate them to the Hodgkin-Huxley equations;

(h) Explain the concept of excitability and demonstrate it in the FitzHugh-Nagumo equations and other similar systems;

(i) State the Poincare-Bendixson Theorem and apply it to the FitzHugh-Nagumo equations and other similar systems;

(j) Derive partial differential equation (PDEs) descriptions of movement from a set of biological assumptions;

(k) Derive the persistence conditions for PDE movement models in bounded domains;

(l) Prove the existence of travelling waves for reaction diffusion equations with quadratic and cubic reaction

terms;

(m) Explain the concepts of morphogenesis and pre-patterning;

(n). Derive equations using the Law of Mass Balance and explain the behaviour of the corresponding solutions;

(o). Linearise reaction-diffusion equations and derive conditions for a Turing instability to occur;

(p) Calculate Turing instability patterns for a range of geometries and boundary conditions;

(q) Investigate the effect of parameters on the solution behaviour of systems of differential equations using appropriate numerical software and be able draw conclusions about the biological system that is studied;

(r) Numerically solve and investigate the behaviour of difference equations and interpret the results in the context of the biological problem being studied.

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.