# Postgraduate opportunities in the Mathematics and Statistics Research Group (MSRG)

MSRG has many potential PhD projects in different areas of mathematics and statistics. See how our projects could make a difference.

We invite self-funded PhD students in the following areas of mathematics:

- Adaptive mesh methods
- Computer stochastic simulation
- Decision analysis
- Graph theory
- Mathematical pharmacology
- Molecular modelling
- Multi-objective optimisation
- Non-linear systems
- Numerical analysis
- Operational research
- Optimal transport
- Partial differential equations
- Stability analysis
- Transport networks

For further details, please get in contact with the members of MSRG.

## Examples of possible PhD projects

### Adaptive mesh methods for numerical weather and climate prediction

Represented by Dr Emily Walsh, Senior Lecturer in Mathematics.

Operational weather and climate models cannot yet resolve small-scale processes such as tropical cyclones or convective clouds. Adaptive mesh methods based on optimal transport theory have the potential to resolve these interactions efficiently. Such methods have successfully been applied to several geophysical models such as extra tropical cyclone prediction, ice sheet dynamics and tsunami prediction. However, operational use of adaptive methods is still considered some time away and the challenges associated with the efficient application of such methods are significant.

Example: Budd, C. J., Cullen, M. J., & Walsh, E. J. (2013) Monge-Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem. Journal of Computational Physics, 236(1), 247-270.

### Computer stochastic simulation

Represented by Dr Paul White, Associate Professor (Applied Statistics).

Dr Paul White and colleagues have supervised previous PhD programmes of study in the area of computer stochastic simulation used to evaluate statistical approaches, statistical tests, and in the development of new methodology. New bespoke PhD opportunities will be developed using this general approach based on the interest of applicants.

### Structured life course structural equation models

Represented by Dr Andrew Smith, Senior Lecturer in Statistics.

The structured life course modelling approach seeks to establish the most likely relationship between an exposure and outcome over the whole life course. Examples of relationships include critical or sensitive periods of development, or an accumulation of exposure model. The structured approach uses model selection methods such as the lasso and post-selective inference to establish the relationship most supported by data. Currently models are restricted to a single measurement of the outcome variable.

Structural equation models are used to express associations between several variables including models of exposures, covariates and outcomes that vary over time. It is possible that lasso methods can be applied to structural equation models in order to extend the structured life course modelling approach to multiple measurements of the outcome variable. Methods that are suitable for high-throughput applications (e.g. epigenetic studies) are desirable.

Example: Smith, A. Hardy, R. Heron, J. et al. (2016) A structured approach to hypotheses involving continuous exposures over the life course. International Journal of Epidemiology, 45 (4), pp. 1271-1279.

### Model reduction methods and operational models in mathematical pharmacology

Represented by Dr Lloyd Bridge, Senior Lecturer in Mathematics.

In the pursuit of drug discovery and therapeutics, pharmacological analysis is required to probe mechanisms of action of drug molecules (ligands) at cell surface receptors and to understand downstream signalling dynamics. Mathematical modelling is now providing new insights and tools for quantifying receptor behaviour and predicting signalling outcomes.

This project in the exciting and growing field of mathematical pharmacology will develop new minimal models for cellular responses triggered by ligand-receptor interactions (which will be validated using real data), using asymptotic and numerical analysis to bridge the gap between “systems biology” models and simplified operational models of agonism.

The student will contribute new results to a growing field, and will gain skills in scientific computing, mathematics and bio-modelling which will be applicable in a broad range of disciplines.

Example one: RJ, H.S., Quinn, A.M., Hughes, T.E. and Bridge, L.J. (2020) Analyzing kinetic signaling data for G-protein-coupled receptors. Scientific Reports (Nature Publisher Group), 10(1).

Example two: West, S., Bridge, L.J., White, M.R., Paszek, P. and Biktashev, V.N. (2015) A method of ‘speed coefficients’ for biochemical model reduction applied to the NF-kappaB system. Journal of Mathematical Biology, 70, p.591.

### Mathematical modelling of chronopharmacokinetics

Represented by Dr Lloyd Bridge, Senior Lecturer in Mathematics.

Pharmacokinetics (PK) is the study of drug distribution in the body, and is a topic which often draws on the results of ordinary differential equation (ODE) based models. Classical PK analysis often assumes that key parameters controlling drug absorption and elimination are constant. Many of these parameters may vary with time of day, meal timings and other time-dependent factors.

Mathematical modelling is a key contributor to modern chronopharmacometric analysis, towards the design and understanding of therapeutic treatments which take account of biological rhythms. Within many modelling frameworks, the approach to take to extend classical models to time-variant systems is clear. However, there is much left to explore, even for simple models.

This project considers chronoPK models as extensions of both specific and generalised PK compartmental models, exploring in particular:

- The effect on dosing regimen outputs of smoothed step changes in PK parameters, modelling disease onset for simple two-compartment models.
- The effect of circadian variation in PK parameters on effective and toxic drug levels in periodically forced multi-dosing scenarios.
- The effect of circadian variation in PK parameters on the analytical solutions to generalised drug-level models including a transit compartment train.
- The effect of phase differences in circadian parameters.
- Simple harmonic variation and generalised circadian rhythms represented by Fourier Series
- Issues of parameter identifiability and efficient evaluation of special functions which appear as solutions to the ODE models.

The student will contribute new results to a growing field, and will gain skills in scientific computing, mathematics and bio-modelling which will be applicable in a broad range of disciplines.

- Example one: Véronneau-Veilleux, F. and Bélair, J. (2017) Modeling circadian fluctuations of pharmacokinetic parameters. Mathematical Modelling of Natural Phenomena, 12(5), pp.146-161.

- Example two: Hof, F. and Bridge, L.J. (2021) Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments. Journal of Pharmacokinetics and Pharmacodynamics, 48(1), pp.99-131.

### Models of allosteric modulation in mathematical pharmacology

Represented by Dr Lloyd Bridge, Senior Lecturer in Mathematics.

Much pharmacological analysis seeks to understand the consequences of drug molecules (ligands) binding to cell surface receptors. Classical receptor theory has been built around assumptions of a single ligand binding to a single receptor to initiate a cellular response. Key outstanding research questions concern the implications of multiple ligands affecting each other’s binding and activation properties at monomeric (single) or oligomeric (“bound together”) receptors. Different ligand-receptor interactions offer a range of potential avenues for novel therapeutics. Mathematical (ordinary differential equation) models are required to develop receptor theory to account for dimerised receptors and allosteric modulators (ligands which bind to distinct sites on a receptor).

This project builds on recent mathematical results for dimerised receptors to

- generalise the existing models which capture a cooperativity effect (how the binding of one molecule affects effects driven by another drug), and
- extend the theory to simulate signalling dynamics influenced by allosteric modulators.

A new and novel output from this timely modelling project is to classify cell response dynamic behaviour in terms of ligand binding setup. As such, our analysis will translate to cutting-edge pharmacology research, which has potential impact in both academia and the pharmaceutical industry. The PhD student will gain skills in scientific computing, mathematics and bio-modelling which will be applicable in a vast range of fields.

- Example one: White, C. and Bridge, L.J. (2019) Ligand binding dynamics for pre-dimerised G protein-coupled receptor homodimers: linear models and analytical solutions. Bulletin of mathematical biology, 81(9), pp.3542-3574.

- Example two: Bridge, L.J., King, J.R., Hill, S.J. and Owen, M.R. (2010) Mathematical modelling of signalling in a two-ligand G-protein coupled receptor system: Agonist–antagonist competition. Mathematical biosciences, 223(2), pp.115-132.

### Non-local diffusion equations in evolutionary biology

Represented by Dr Robert Laister, Senior Lecturer.

These equations arise when local diffusion processes (such as those modelled via the Laplacian operator) are replaced with non-local ones (via an integral operator) in reaction-diffusion processes. This means that the classical unbounded diffusion operator is replaced by a non-local, bounded one.

Consequently, there is a lack of smoothing in the associated solution flow and this leads to interesting phenomena in the structure of the solution set. In particular, such equations are often more realistic as models for evolutionary biological processes. We are interested in studying the structure of equilibria, their stability and bifurcation in this biological setting.

Example: Grinfield, M., Hines, G., Hutson, V., Mischaikow, K. and Vickers, G. (2005) Nonlocal Dispersal. Diff. and Int. Eqns. 18, pp. 1299-132.

### Degenerate diffusion equations in population dynamics

Represented by Dr Robert Laister, Senior Lecturer.

In many biological systems, the movement of individuals is not governed purely by random walk process. Rather, individuals may move to avoid over-crowding or to form local aggregates to protect against predation. In these situations, density-dependent diffusion models (such as that given by the porous medium operator) are more realistic.

The absence of a strong maximum principle in these equations can lead to interesting solution behaviour such as finite speed of propagation,`dead cores’ and non-smooth (weak) solutions. We are particularly interested in models of this type in competitive reaction-diffusion systems.

Example: Suarez, A. (2004) Non-negative solutions for an heterogeneous degenerate competition model*.* ANZIAM Journal*.* 46, pp. 273-297.

### Semi-linear heat equations

Represented by Dr Robert Laister, Senior Lecturer.

Only recently has a complete classification been given for the local existence of non-negative solutions of semi-linear heat equations in Lebesgue spaces. However, there still remain many fundamental open problems such as uniqueness of solutions and the nature of blow-up.

Example: Laister, R., Robinson, J., Sierzega, M. and Vidal-Lopez, A. (2016) A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Annales de l'Institut Henri Poincaré - Analyse non linéaire*.* 33 (6), pp. 1519-1538.

### Delay-diffusion equations

Represented by Dr Robert Laister, Senior Lecturer.

Very little work has been done on degenerate reaction-diffusion equations with time delays. Even in the logistic case the asymptotic behaviour is not fully understood. There are many open problems regarding equilibria, their bifurcation and limit sets.

Example: Laister, R. (2013) Convergence to equilibrium in degenerate parabolic equations with delay*.* Nonlinear Analysis: Theory, Methods and Applications*.* 81, pp. 200-210.

### Preventing congestion in infrastructures

Represented by Dr Vadim Zverovich, Associate Professor.

The well-known Braess' paradox describes situations when adding a new link to a transport network might not reduce congestion in the network but instead increase it. Deeper insight into this paradox from the viewpoint of the structure and characteristics of networks may help transport planners to avoid the occurrence of Braess-like situations in real-life networks and thus prevent congestion in road networks. It is very likely that Braess' paradox may happen in a building during evacuation of people, thus increasing both congestion and evacuation time. This might have a considerable impact on the existing evacuation procedures.

Example: Zverovich, V. and Avineri, E. (2015) Braess' Paradox in a generalised traffic network. Journal of Advanced Transportation. 49 (1), pp. 114-138.

### Strongly perfect graphs

Represented by Dr Vadim Zverovich, Associate Professor.

In 1978 at a Monday Seminar in Paris, Berge introduced an important class of graphs called strongly perfect graphs. They form a subclass of perfect graphs. A graph is strongly perfect if every induced sub-graph contains an independent set that meets all maximal cliques. The known results about this class can be found in the survey paper by Ravindra (1999).

In contrast to perfect graphs, for strongly perfect graphs there is no conjecture similar to the Strong Perfect Graph Conjecture. This is probably due to the fact that for perfect graphs the number of forbidden sub-graphs of a fixed order does not exceed a constant, while strongly perfect graphs do not have this property. This means that it is much more difficult to formulate such a conjecture for strongly perfect graphs than for perfect graphs. The class of absorbantly perfect graphs was introduced by Hammer and Maffray in 1993.

A graph is absorbantly perfect if every induced sub-graph has a minimal dominating set meeting all maximal cliques. Absorbantly perfect graphs are important in this context because they form a superclass of strongly perfect graphs, while well-known bipartite graphs and Meyniel graphs are subclasses of strongly perfect graphs. In fact, we have the following chain of strict inclusions for these subclasses of perfect graphs:

{Bipartite} ⊂ {Meyniel} ⊂ {Strongly perfect} ⊂ {Absorbantly perfect} ⊂ {Perfect}.

The basic aims of the proposed project are as follows:

(a) formulate a characterisation conjecture on strongly perfect graphs;

(b) prove the conjecture for some important subclasses and then try to prove the entire conjecture.

Example: Ravindra, G. (1999) Some classes of strongly perfect graphs. Discrete Math. 206, pp. 197-203.

### Multi-criteria decision making

Represented by Dr Xiaodong Li, Lecturer.

Multi-Criteria Decision Making (MCDM) is the scientific process of aiding decision makers when their decisions involve multiple, conflicting criteria by which the goodness of a solution can be measured. There are a variety of techniques and methodologies within the field of MCDM involves techniques such as goal programming, where the distance between a solution that can be achieved and a set of decision maker set goals is minimised. We are interested in application of MCDM (particularly the distance-based) techniques with case studies in different industrial sectors.

Example one: Li, X., Beullens, P., Jones, D. and Tamiz, M. (2009) An integrated queuing and multi-objective bed allocation model with application to a hospital in China. Journal of the Operational Research Society*.* 60 (3), pp. 330-338, ISSN 0160-5682.

Example two: Mohammed, A., Wang, Q. and Li, X. (2016) A cost-effective decision-making algorithm for an RFID-enabled HMSC network design: A multi-objective approach. Industrial Management and Data Systems. ISSN 0263-5577.

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