**Estado**Finalizada

**Trabajo fin de master**

**y**

**Tesis doctoral**

Although cancer is not the leading cause of death in developed countries, still it's the first as far as mortality rates are concerned. Population aging will result globally in significant increases in the rankings for most noncommunicable diseases, particularly cancers. Also increasing levels of tobacco smoking in many middle- and low-income countries will contribute to increase deaths from cardiovascular disease, chronic obstructive pulmonary disease and some cancers.

Despite of the enormous efforts taken to find a method which could solve the problem definitively or at least turn cancer into chronic not lethal disease, we still have not an answer. The main reason is complexity of the problem and that cancer can not be generalized, in fact this term that encompasses more than 200 types of malignancies, each of them with some special features, its causes, its evolution and its specific treatment. Mathematical modeling can be a great help on this field because it provides a powerful tool for simulations "in silico", which nowadays are an element of any investigation. There exist thousands of models of tumor development, simpler and more complicated ones, none of them is the perfect one. That is why the main objective should be usefulness of a model as far as applications are concerned, not its complexity. The same rule might be applied to different elements of mathematics involved in medical problems, for example statistical methods.

The thesis consists of three parts: analysis of three models of cancer-immune system competition, a stochastic model of lymphoma and statistical analysis of clinical data.

In the first chapter three models of competition between cancer and immune system are presented and analyzed. Each of them is a simple system of ODEs, and the motivation are mathematical models of competing species in an ecosystem. For each of the models their strong and week points are discussed in the terms of agreement with biological reality and stability analysis and numerical simulations are performed. The final model is a model of competition between an artificially induced tumor and the adaptive immune system. The aim of this work was to reproduce experimental, found two possible outcomes depending on the initial quantities of tumor and adaptive immune cells. We came to the conclusion that the hypothesis of an equilibrium state before the treatment possibly refers to a small solution that tends to zero, but it has not disappeared yet. So, the final model is a known model of two species competition with finite carrying capacities and it has two groups of solutions depending on the initial conditions. In the first one, the immune system wins against the tumor cells, so the cancer disappears (elimination). In the second one, cancer keeps on growing.