# PhD dissertation topics

### Improved lung cancer diagnosis: modelling ground glass opacities

Lung cancer is one of the cancer types causing the most number of deaths worldwide. Diagnostic tools have not improved substantially, and even if patients are diagnosed it is difficult to predict the tumour dynamics mainly because of the fractal like structure of the lung, which implies that diffusion models do not tend to provide the best possible approximation. Hence more accurate and advanced mathematical models are needed, which take into account the complex physical structure of the lung tissue.

### 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 very many important questions, relevant for the successful application 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.

### Modelling structured populations using partial differential equations

Structured population dynamics is a central part of mathematical biology. In any realistic mathematical model of populations it is natural to assume that individuals differ with respect to some characteristic. This may be chronological age, size, or any other structuring variable such as level of infectiousness. In this project the student will learn about a state of the art issue, which is of current interest to the biological/life science community, and then build structured population models and analyze them to help shed light on important underlying biological phenomena.

### Analysis of partial differential equations in structured population dynamics

Structured population models are often very challenging from the mathematical point of view. Even for a basic size-structured Gurtin-McCamy type of model, which takes the form of a quasilinear hyperbolic PDE with nonlinear and nonlocal boundary condition, the Principle of Linearised Stability has not been established to date. 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. Qualitative questions of PDEs, such as asymptotic behavior of solutions, will be addressed as central fixture.