Mathematical Modeling in Cellular Dynamics: Applications to Cancer Research

Résumé

This thesis fits into the general framework of the study of population dynamics. The population particularly considered in this work is comprised of stem cells with both cases of healthy and cancerous cells being investigated. We propose and analyze three mathematical models describing stem cells dynamics. The first model is an age-structured partial differential model that we reduce to a delay differential system using the characteristics method. We investigate the existence and stability of the steady states of the reduced delay differential system. We, then, conduct some numerical simulations to illustrate the behavior of the steady states. In the second model, the duration of the cell cycle is considered to depend upon the total population of quiescent cells. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. We perform a detailed stability analysis of the resulting delay differential system. We confirm the analytical results by numerical simulations. The third and final model, proposed in this thesis, is an ordinary differential equations model describing healthy and cancerous stem cells dynamics and their interactions with immune system responses. Through this model, we show that the frequency, the duration of infections and their action (positive or negative) on immune responses may impact significantly tumor proliferation.

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