COURSE OBJECTIVES:
1. To introduce students to the application of mathematical modelling in the analysis of biomedical systems including populations of molecules, cells and organisms.
2. To show how mathematics, especially ordinary differential equations and computing can be used in an integrated way to analyse biomedical systems.
COURSE CONTENT:
1. Introduction to continuous models.
2. Population dynamics. Single-species populations. Malthus (exponential) model, Verhulst (logistic) model, Gompertz model. Mathematical models of tumour growth
3. Modelling loss of population (death, harvesting). Growth under restriction. Monod model. Chemostat model.
4. Parameter identification problem. Least squares method. Elements of numerical optimization.
5. Numerical solution of ODE.
6. Steady state solutions, stability, linearization. Systems of equations, phase-plane diagrams.
7. Population dynamics. Multiple species populations. Predator-prey systems, Lotka-Volterra model. Competition models.
8. Population biology of infectious diseases. SIR model.
9. Linear difference equations with applications. Qualitative behaviour. Cell division, an insect population.
10. Nonlinear difference equations with applications. Steady states, stability. Logistic difference equation. Density dependence, Nicholson-Bailey model.
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Mathematical models in biology. Reprint of the 1988 original. Classics in Applied Mathematics, L. Edelstein-Keshet, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
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Mathematical biology. I. An introduction, J.D. Murray, Springer-Verlag, New York, 2002.
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Mathematical biology. Modeling and analysis, A. Friedman, American Mathematical Society, Providence, 2018.
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Mathematical physiology. Vol. I: Cellular physiology. Second edition, J. Keener, J. Sneyd, Springer, New York, 2009.
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