## Final Activity Report Summary - M.M.M.A.T.D. (Mathematical Methods toward the Modelling and Analysis of Tumour Dynamics)

With reference to the originally planned project, the researcher developed his activity in three directions:

1. Modelling, Qualitative analysis and Simulation of mathematical models for the immune competition: The researcher proposed a new model to describe the competition between tumour and immune cells. Such competition is characterised by proliferation-destruction phenomena and the interacting entities are characterised by a microscopic state which is modified by interactions. The model is developed in the mathematical framework of the kinetic theory in order to include the description of the natural trend of immune cells to try to reach a healthy or sentinel level, even when they have been involved in the competition with the tumour cells. The qualitative analysis of the solution is performed showing some interesting features of the asymptotic behaviour of the solutions with reference both to the number densities of cells and the activation of the cells. This qualitative analysis is linked to a computational analysis which provides a relatively more detailed understanding of the asymptotic behaviour of the solutions, which can be followed by interesting biological interpretations.

In the same spirit, the researcher worked on the qualitative analysis and the development of a model of immune competition. The study concentrated on the very early stage of the competition, when interactions modify the microscopic state of cells, but not the size of the population (proliferating or destructive events are not yet relevant).

2. Development of computational schemes: In collaboration with the scientist in charge of the project, Prof. N. Bellomo, and two researchers from the Department of Hydraulics, Transport and Civil Infrastructures of the Politecnico di Torino, the researcher worked at a book devoted to the computational solution and related simulations of evolution problems from nonlinear models in applied sciences. The main object of this book is the development of computation solution, by the so-called generalized collocation method, devoted to the simulation of a large variety of nonlinear problems from various fields of applied sciences. In a more specific way, the researcher took care of the results concerning the collocation methods for integro-differential equations with particular emphasis to non-linear kinetic model from population dynamics and immune competition.

3. Conceptual aspects of kinetic theory: The researcher worked on various conceptual aspects of linear kinetic theory dealing with the well-posed-ness of kinetic equations with general boundary conditions. Recall that transport--like equations with conservative boundary conditions are arising naturally in various branches of kinetic theory (Boltzmann equation) and population dynamics.

More generally, the researcher gave a complete description of the conceptual aspects underlying initial and boundary value problems for first-order partial differential equations with general boundary conditions associated to general transport fields. The understanding of such (linear) first-order models is actually the key starting point of any kind of (non-linear) kinetic theory.

1. Modelling, Qualitative analysis and Simulation of mathematical models for the immune competition: The researcher proposed a new model to describe the competition between tumour and immune cells. Such competition is characterised by proliferation-destruction phenomena and the interacting entities are characterised by a microscopic state which is modified by interactions. The model is developed in the mathematical framework of the kinetic theory in order to include the description of the natural trend of immune cells to try to reach a healthy or sentinel level, even when they have been involved in the competition with the tumour cells. The qualitative analysis of the solution is performed showing some interesting features of the asymptotic behaviour of the solutions with reference both to the number densities of cells and the activation of the cells. This qualitative analysis is linked to a computational analysis which provides a relatively more detailed understanding of the asymptotic behaviour of the solutions, which can be followed by interesting biological interpretations.

In the same spirit, the researcher worked on the qualitative analysis and the development of a model of immune competition. The study concentrated on the very early stage of the competition, when interactions modify the microscopic state of cells, but not the size of the population (proliferating or destructive events are not yet relevant).

2. Development of computational schemes: In collaboration with the scientist in charge of the project, Prof. N. Bellomo, and two researchers from the Department of Hydraulics, Transport and Civil Infrastructures of the Politecnico di Torino, the researcher worked at a book devoted to the computational solution and related simulations of evolution problems from nonlinear models in applied sciences. The main object of this book is the development of computation solution, by the so-called generalized collocation method, devoted to the simulation of a large variety of nonlinear problems from various fields of applied sciences. In a more specific way, the researcher took care of the results concerning the collocation methods for integro-differential equations with particular emphasis to non-linear kinetic model from population dynamics and immune competition.

3. Conceptual aspects of kinetic theory: The researcher worked on various conceptual aspects of linear kinetic theory dealing with the well-posed-ness of kinetic equations with general boundary conditions. Recall that transport--like equations with conservative boundary conditions are arising naturally in various branches of kinetic theory (Boltzmann equation) and population dynamics.

More generally, the researcher gave a complete description of the conceptual aspects underlying initial and boundary value problems for first-order partial differential equations with general boundary conditions associated to general transport fields. The understanding of such (linear) first-order models is actually the key starting point of any kind of (non-linear) kinetic theory.