Contributions to Stochastic Modelling of the Immune System

During the past century, mathematical modelling of biological processes rapidly developed into a field of high scientific interest. The diversity of mathematical approaches and biological applications is large. Some key objectives in this context are a better understanding of biological processes and, based on that, the prediction of processes which have not yet been investigated experimentally. The purpose of this thesis is to contribute to this area at the interface of stochastics and immunology. The thesis consists of three parts. In the first part, we investigate the question how certain immune cells (so-called T-cells) recognise structures which are potentially harmful to an organism. This is studied by means of a stochastic model. Mathematically, this problem can be described as the task to distinguish particular signals from a noisy background. A signal is represented by a sum of real-valued random variables and we are interested in the probability of the event that this sum becomes extremely large. To analyse this probability we use techniques from the theory of large deviations. We prove that sharp estimates on the probabilities of large deviations hold, also in conditional setups. The estimates are interpreted in the biological context of T-cell activation. In the second part, we analyse related problems in a setup which is independent of the biological application; as a consequence the results apply to a broader class of random variables. We establish strong large deviation results for certain conditional probability distributions. Moreover, we show that the behaviour of the random rate function can be characterised by an invariance principle. In the third part, we introduce a stochastic, individual-based model from population dynamics that describes the evolution of cancer. This model offers the possibility to include the effects of particular immunotherapies. It allows to survey the development of heterogeneous tumour cell populations under the influence of certain immune cells and specific aspects of an inflammatory environment. The relevance of particular stochastic phenomena for this context is studied and illustrated by examples. The models we investigate are examples of an effective interaction of mathematics and immunology. On the one hand, they show that biological concepts and questions can be stated more precisely with the help of mathematical models, and that the resulting models may be useful to predict the behaviour of a biological system. On the other hand, during the analysis of biological questions new mathematical problems and models arise, which are mathematically relevant, independent of the primary questions