Module MATH40340: Biomathematics III

Department: Mathematical Sciences

MATH40340: Biomathematics III

Prerequisites

• None.

Corequisites

• None.

Excluded Combination of Modules

• None.

Aims

• To impart an understanding of some ideas and mathematical methods that can be used to solve problems in theoretical and mathematical biology.

Content

• Subject to the approval and with the advice of the course director, students choose two courses from the following:
• BM31: Even years: Bayesian Methods (38 lectures): Review and introduction: Bayesian paradigm, conditional independence and conjugacy. Manipulation of multivariate probability distributions. Methods for computation: conjugacy, quadrature, Monte Carlo. Conjugacy for the normal linear model. Interpretation of Monte Carlo output. Concept of Markov chain Monte Carlo. Hierarchical modelling: Motivation, latent variables, random effects, conjugacy and semi-conjugacy. Bayesian graphical modelling: Directed acyclic graphs, Bayesian networks, conditional independence, moral graph, separation theorem. Doodles in WinBUGS. Specification of prior beliefs. Random number generation: generation of uniform pseudo-random numbers. Distribution function method, rejection sampling, log-concavity, adaptive rejection sampling Markov Chain Monte Carlo: Markov chains, equilibrium distribution. Metropolis-Hastings: Metropolis random walk, independence sample, Gibbs sampling. Analysis and interpretation of MCMC output. WinBUGS. Model comparison: Bayes factors, deviance information criterion (DIC). Case studies. Odd years: Bayesian Statistics (38 lectures): Bayesian paradigm, conditional independence and conjugacy. Manipulation of multivariate probability distributions. Foundations: Rational basis for subjective probability and Bayesian statistics, exchangeability and de Finetti’s representation theorem, parametric modelling. Exponential families: regular exponential families, canonical representation, sufficiency, conjugacy, linearity of posterior expectation. Non-parametrics/robustness: Introduction to Bayesian non-parametric inference or robustness. Bayes linear methods: Expectation as a primitive; evaluation and interpretation of adjusted expectations, variances and covariances; resolution transform and canonical structure; observed adjustment; belief diagnostics; partial adjustment; Bayes linear influence diagrams; exploiting exchangeability. Case studies.
• BM32: Even years: Continuum Mechanics (38 lectures): Description of fluid flows: continuum hypothesis, velocity field, particle paths and streamlines,vorticity and compressibility. Euler equations: ideal (‘perfect’) fluid. Bernoulli’s equation: irrotational flows, water waves. Navier-Stokes equations; viscous fluids and boundary conditions; boundary layers. Examples: flows in a channel and a pipe, vortices, aerodynamic lift . Compressible flows: sound waves. Elastic media. Motion of microorganisms. Odd years: Solitons (38 lectures): Nonlinear wave equations: historical introduction; properties; dispersion and dissipation.Progressive wave solutions: D’Alembert, KdV and sine-Gordon. Topological lumps and the Bogomolnyi bound. Backlund Transformations for Sine-Gordon Equation. The Liouville equation. Generation of multisoliton solutions of sine-Gordon by Backlund transformations. Breathers. Discussion of scattering. Conservation Laws in Integrable Systems. Hirota's method for multisoliton solutions of the wave and KdV equations. The Nonlinear Schrodinger Equation and its connection with the Heisenberg ferromagnet. The Inverse Scattering Method. Lax pairs. Spectrum of Schrodinger's operator for potential; Bargmann potentials. Scattering theory; asymptotic states, reflection and transmission coefficients, bound states. Two-Component Equations: MKdV equation and Miura transformation. Two-component Lax pair formulation of inverse scattering theory. Toda equations: Conservation laws in integrable finite-dimensional models. The Toda molecule and the Toda chain. Conservation laws for the Toda molecule. Integrability: Hamiltonian structures for the KdV equation. Integrability. Hierarchies of equations determined by conservation laws. Davydov’s solitons and biological applications.
• BM33: Dynamical Systems (38 lectures): Introduction: Smooth direction fields in phase space. Existence, uniqueness and initial-value dependence of trajectories. Autonomous Systems: Orbits. Phase portraits. Equilibrium and periodic solutions. Orbital derivative, first integrals. Equilibrium Solutions: Linearisation and linear systems. Hartman-Grobman, stable-manifold theorems. Phase portraits for non-linear systems, computation. Stability of equilibrium, Lyapunov functions. Periodic Solutions: Flow, section, maps, fixed points. Brouwer’s Theorem, periodic forcing, planar cycles. Poincar´e-Bendixson and related theorems. Orbital stability. Bifurcations: Hopf and other local bifurcations from equilibrium. Uses of bifurcations.
• BM34: Mathematical Biology (38 lectures) : Aim: Study of non-linear differential equations in biological models. Content: Introduction to the Ideas of Applying Mathematics to Biological Problems. Reaction Diffusion Equations and their Applications in Biology: Reaction diffusion and chemotaxis mechanisms. Application of the classical diffusion equation to dispersal of insects. Realistic modeling of insect dispersal with a nonlinear diffusion equation. Application of a nonlinear diffusion equation to patterns formed by herrings and gulls. Linear and nonlinear stability for the diffusion equation. Chemotaxis and slime mould aggregation. ODE Models in Biology: Enzyme kinetics. A little ODE stability theory. The chemostat and bacteria production. The Formation of Patterns in Nature: Pattern formation mechanisms, morphogenesis. Questions such as how does a tiger get its stripes. Diffusion driven instability and pattern formation. The famous work of Turing dealing with how patterns occur in nature. Conditions under which Turing instabilities will not form. Epidemic Models and the Spread of Infectious Diseases: Epidemic models. Spread of infectious diseases. Simple ODE model. Spatial spread of diseases. Spatial spread of rabies among foxes. Stability in epidemics and predator - prey models.
• BM35: Partial Differential Equations (38 lectures) : First-order equations and characteristics. Conservation laws. Systems of first-order equations, conservation laws and Riemann invariants. Hyperbolic systems and discontinuous derivatives. Acceleration waves. Classification of general second order quasi-linear equations and reduction to standard form for each type (elliptic, parabolic and hyperbolic). Energy methods for parabolic equations. Well-posed problems. Maximum principles for parabolic equations. Finite difference solution to parabolic and elliptic equations. Stability and convergence for solution to finite difference equations. Iterative methods of solving Ax = b.
• BM36: Statistical Methods (38 lectures): Statistical Computing: Introduction to statistical software for data analysis, and for reinforcing statistical concepts. Multivariate Analysis: Multivariate normal distribution, covariance matrix, Mahalanobis distance, principal component analysis. Likelihood Estimation: Likelihood- and scorefunction for multiparameter models, Fisher Information, confidence regions, method of support, LR tests, profile likelihood. Regression: General linear model: regression, analysis of variance, designed experiments, diagnostics, influence, outliers, transformations, variable selection, correlated errors, lack-of-fit tests. Generalised linear models: Framework, exponential families, likelihood and deviance, standard errors and confidence intervals, prediction, analysis of deviance, residuals, over-dispersion.
• BM37: Even years: Stochastic Processes (38 lectures): Probability Revision Conditional expectation, sigma fields. Examples of Markov Chains Branching processes, Gibbs sampler. Discrete Parameter Martingales. The upcrossing lemma, almost sure convergence, the backward martingale, the optional sampling theorem, examples and applications. Discrete Renewal Theory. The renewal equation and limit theorems in the lattice case, examples. General Renewal Theory. The renewal equation and limit theorems in the continuous case, excess life, applications. The renewal reward model. Poisson Processes Poisson process on the line, relation to exponential distribution, marked/compound Poisson processes, Cramer’s ruin problem, spatial Poisson processes. Continuous Time Markov Chains Kolmogorov equations, birth and death processes, simple queueing models. Topics Chosen From: stationary Gaussian processes, Brownian motion, percolation theory, contact process. Probability models for DNA sequence evolution. Odd years: Probability (38 lectures): Introductory examples: from finite to infinite spaces.Probability as Measure: s-fields. Measure and probability spaces. Main properties. Random Variables as measurable functions. Distributions. Generating and characteristic functions.Convergence almost surely, in probability, in L2. Borel-Cantelli lemmas. Kolmogorov 0-1 law. General theory: probability spaces and random variables. Integration: Expectation as integral. Inequalities. Monotone and dominated convergence. Limit results: Laws of large numbers. Weak convergence. The central limit theorem. Large deviations. Topics Chosen From: Random graphs. Percolation. Ergodic theorems. Stochastic integral. Probability models for DNA sequence evolution.

Learning Outcomes

• Students will, in each of the courses studied, have an understanding of the specialised mathematical theory together with mastery of a coherent body of knowledge demonstrated through one or more topics from the following: Statistics and Bayesian statistics, Continuum mechanics, Dynamical systems, Partial Differential Equations, Stochastic Processes.

• Students will develop specialised and advanced skills in the areas studied. They will be able to solve complex and specialised problems, draw conclusions and deploy abstract reasoning and mathematical intuition. They will develop their mathematical self-sufficiency and be able to read and understand advanced mathematics independently, in subjects relevant for applications in Biology.

• (1) Problem solving, written presentation of an argument.
• (2) The ability to learn actively and reflectively and to develop intuition, the ability to tackle material which is given both unfamiliar and complex.
• (3) Self-organisation, self-discipline and self-knowledge.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures indicate what is required to be learned and the application of the theory to examples.
• Formatively assessed assignments provide practice in the application of high level of rigour and sophisticated techniques of applied mathematics as well as feedback for the students and the lecturer on students' progress.
• The examinations assess the knowledge acquired and the ability to solve both standard and novel problems.
• The ability to solve problems will show that the key skills have been developed. (Group (1) is tested directly in the problem solving and group (2) either directly or indirectly by the testing of the knowledge acquired. For group (3), a student who has acquired the knowledge and skills to succeed in this module will necessarily have had to develop the ability to organise and execute a programme of work and will discover aspects of and limits to his/her ability.)

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

12 or more assignments (of problems) set and marked. (Other assignments set with solutions provided).

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University