F10AN - Mathematical Biology B

Course leader(s):

Cathal Cummins (Edinburgh, January)

Aims

The course will teach the application of mathematical models to a variety of problems in biology and medicine. It will show the application of ordinary differential equations to simple biological and medical problems, the use of mathematical modelling in biochemical reactions, the application of partial differential equations in describing spatial processes such as cancer growth and pattern formation in embryonic development, and the use of delay-differential equations in physiological processes.

Syllabus

1. ODE models in biology and medicine (1.1 Introduction to mathematical modelling using ODE models; bacterial growth; growth in a chemostat; tumour-immune system dynamics; neural modelling and the Fitzhugh-Nagumo equations; revision of phase plane methods and non-dimensionalisation techniques.)

2. Reaction kinetics (2.1 the Law of Mass Action; modelling enzymatic reactions, including co-operative behaviour and substrate inhibition; analysis of a simple enzymatic reaction; pseudo-steady state hypothesis; matched asymptotics and singular perturbation theory; biological oscillators and demonstration of limit cycles in a simple model using the Poincare-Bendixson theorem; enzyme production.)

3. Biological movement and pattern formation (3.1 modelling cell movement; examples of patterning in biology, for example animal coat markings and bacteria patterns; the Turing mechanism as a model for pattern formation and the conditions for diffusion driven instability; patterns on one and two dimensional finite domains and applications to animal pigmentation; chemotaxis as a model for pattern formation.)

4. Travelling waves (4.1 reaction diffusion equations and their applications to wound healing, cancer growth, epidemiology; the Fisher equation - travelling waves and derivation of the wave speed; cubic kinetics; travelling waves for multiple populations and applications to epidemiology.)

5. Delay differential equations (5.1 Introduction to delay differential equations in modelling; derivation of a critical delay for stability in a single DDE. Construction of periodic solutions for piecewise constant negative feedback. Applications to modelling in physiological processes, for example Cheynes-Stokes breathing, hematopoietic regulation, testosterone secretion.)

Learning outcomes

By the end of the course, students should be able to do the following:

Interpret the biological significance of terms in mathematical models of biology and medicine.
Determine steady states, their stability and produce phase plane portraits.
Demonstrate an understanding of the Law of Mass Action and apply it to modelling simple chemical reactions.
Demonstrate oscillatory behaviour in simple biochemical models.
Demonstrate an understanding of the role and modelling of spatial movement terms in biological models.
Demonstrate an understanding of the key differences between travelling wave solutions of scalar reaction-diffusion equations with quadratic or cubic kinetics.
Demonstrate an understanding of the concept of a delay differential equation and the use of the delay as a bifurcation parameter, including the calculation of the critical delay for stability.

Further details

Curriculum explorer: Click here

SCQF Level: 10

Credits: 15