Dynamical analysis of a novel mathematical model of colorectal cancer

In the present work, a novel model of colorectal cancer is proposed, and studied both analytically and numerically. We discuss cancer pathology and its onset, along with its interactions with immune system and other cell populations. We also shortly describe the colon anatomical structure and point out some of the main features of colorectal cancer. Subsequently, we provide a qualitative introduction to the new colorectal cancer model, consisting of five interacting cell populations: stem, semi-differentiated, fully differentiated, tumor, and immune system. Then, we move on, writing down our original set of model equations, which consists of five coupled nonlinear ordinary differential equations (ODEs). This system of ODEs is investigated by means of Nonlinear Dynamical Systems Theory. For such a system, we investigate the properties of ODEs solutions, along with providing new and original proof of their existence, uniqueness, positivity, and boundedness. As next, the steady states of our dynamical system are determined, and their stability investigated. This is performed by adopting both linearization and center manifold reduction. In addition, we also provide a novel characterization of the ``tumor-free'' steady state, thus identifying its bifurcation properties. Thereafter, we also validate our original analytical results by performing extensive numerical simulations. This allows us a full exploration of the parameter space, and to observe the variety of dynamical behaviors of the corresponding solutions. Numerical simulations widely contribute to confirm the goodness of our analytical results, obtained both by linear approximation and by reduction to the center manifold. Next, we build up different bifurcation diagrams, which provide us with additional confirmation of the validity of our theoretical findings, and explore the effects produced by varying other parameters in the system. We conclude by summarizing the results obtained in the present work, along with suggesting future directions in this research line.