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Department of Mathematical Sciences

# MATH3171 Mathematical Biology III

Mathematical Biology is one of the most rapidly growing and exciting areas of applied mathematics. Over the past decade research in biological sciences has evolved to a point that experimentalists are seeking the help of applied mathematicians to gain quantitative understanding of their data. Mathematicians can help biologists to understand very complex problems by developing models for biological situations and then providing a suitable solution. However, for a model to be realistic cross-talk between these two disciplines is absolutely essential.

Mathematics is now being applied in a wide array of biological and medical contexts and professionals in this field are reaping the benefits of research in these disciplines. Examples range from modelling physiological situations e.g. ECG readings of the heart, MRI brain scans, blood flow through arteries, tumor invasion and others. At the cellular level we can ask quantitative questions about the process by which DNA gets "transcribed" (copied) to RNA and then "translated" to proteins, the central dogma of molecular biology. Mathematical Biology also encompasses other interesting phenomena observed in nature, e.g. swimming behavior of microorganisms, spread of infectious diseases, and emergence of patterns in nature. In this course we shall examine some fundamental biological problems and see how to go about developing mathematical models that describe the biological situation with definite predictions that can then be tested to validate the model.

## Outline of Course

### Term 1

• Introduction to the ideas of applying mathematics to biological problems.
• Core applied modelling techniques such as stability analysis, weakly non-linear analysis, travelling wave solutions.
• ODE models in biology.
• Reaction diffusion equations and their applications in biology.
• Examples taken from the following: diffusion of insects and other species; the formation of spiral wave patterns in nature; hyperbolic models of insect dispersal and migration of a school of fish; glia aggregation in the human brain and possible connection with Alzheimer's disease; enzyme kinetics.

### Term 2

• The formation of patterns in nature: pattern formation mechanisms, morphogenesis. Questions such as how does a diffusion driven instability cause pattern formation (Turing instability).
• Examples taken from the following: the chemostat for bacteria production; branching growth of organisms; modelling the life cycle of the cellular slime mold Dictyostelium discoideum, and the phenomenon of chemotaxis; epidermal and dermal wound healing; epidemic models and the spatial spread of infectious diseases.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.