Quantifying structured populations using partial differential equations 

Structured population dynamics is a central part of mathematical biology. In any realistic mathematical model of populations individuals differ with respect to some characteristic. This characteristics may be chronological age, size, or any other structuring variable, such as level of infectiousness. In this project the candidate will learn about a state of the art topics, which is of current interest to the biological/life science community, build a structured population model and then analyse/quantify its basic properties. 

Improved lung cancer diagnosis: quantifying ground glass opacities 

Lung cancer is one of the deadliest type of cancer worldwide. Part of the reason is that diagnostic tools have not improved substantially, and even if patients are diagnosed in early stages (I-II) it is difficult to predict the tumour dynamics. This is partly  because of the many technical challenges around the visualisation and analysis of CT scan data. Mathematical models, such as reaction diffusion PDE models can help to quantify the temporal density distribution of cells, and to aid predictions for individual patients.

Wolbachia infection dynamics in arthropod species 

Wolbachia is one of the most common endosymbionts infecting around 70% of all insect species. In recent years it has been discovered that Wolbachia can be used potentially as a biological control tool to fight mosquito born diseases, such as malaria, dengue fever, West Nile virus, etc. There are number of important questions, relevant for the successful introduction of Wolbachia, which can be addressed by using mathematical models. These questions include: successful introduction of a genetically modified Wolbachia strain into a wild mosquito population, optimal release strategies, competition between resident and new strains, etc. 

Analysis of partial differential equations in structured population dynamics 

Structured population models are often very challenging from the mathematical point of view. A simple one-species size-structured Gurtin-McCamy model takes the form of a quasilinear hyperbolic PDE with nonlinear and nonlocal boundary conditions, which are known to be notoriously difficult to analyse both from the computational and analytical points of view. In this project the focus will be on gaining familiarity with some functional analytic techniques, such as the theory of semigroups of operators and the spectral theory of positive operators; and then applying them to address qualitative questions of PDEs, such as asymptotic behavior of solutions and existence of non-trivial steady states. 

Please feel free to contact me if you are interested in the projects above, or suggest a possible project of your own interest.