A stochastic model for immunotherapy of cancer

In recent years, the use of mathematics in cancer research has caught on, with the rapid accumulation of data and applications of mathematical methodologies. The application of mathematics in cancer research is known as *mathematical oncology*. Mathematical oncology, starting from theoretical studies, tries to design clinical experiments with mathematical models. Mathematical models represent an useful tool for an interdisciplinary approach to cancer research. Indeed, various points of view, coming from several scientific areas, are fundamental to face the complexity of cancer evolution. In this setting, cancer ecology arises as promising quantitative approach. Cancer ecology looks at different groups of cells in an organism as interacting species of an ecosystem. From this perspective, cancer cells are a new species appearing in a stable ecosystem. Cancer cells represent a harmful and invasive species, abling to influence and change the interaction among the different types of healthy cells, that represent the pre-existing species. To promote their growth, cancer cells trigger a struggle for survival, which can lead to the extinction of certain types of cells and, in the worst cases, to the collapse of the entire ecosystem. The stochastic models (specially interacting particle systems) fit the noisy dynamics of cancer. One of the main problematic features of cancer is therapy resistance. Among therapies, immunotherapy represents a potential and effective treatment for patients with certain types of resistant cancer. Moreover a wide range of cancers are currently treated using immunotherapy; in particular melanoma has a good response to this kind of treatment. Therefore, in this thesis, we have chosen a stochastic model for immunotherapy of cancer. In particular the proposed model allows to simulate different treatment protocols, highlighting some counter-intuitive results: under some particular conditions therapy could work in favour of cancer resistance. Accordingly, the same type of cancer acts in very different ways with respect to the affected person and used therapy. Therefore, mathematical oncology could play a decisive role in the future of personalized medicine. In fact, patient-specific mathematical modelling, analysis and collection of clinical data could represent effective tools to develop patient-specific adaptive therapies and to face therapy resistance.