Analysis of the Mathematical Model of Cancer Dynamics

Abstract

Qualitative analysis of dynamic systems is a powerful research tool used to solve a great deal of application problems. In mathematical models described by ordinary differential equations, the study of equilibrium positions has actually become the standard: their number and location depending on the parameters, as well as issues of local stability. In mathematical models in various fields of chemistry and biology, it is also important that the natural requirements for the possible values ​​of phase variables are not violated in the process of dynamics (as a rule, this is the non-negativity condition). More sophisticated study orientation is global sustainability.The need for a qualitative analysis of a dynamic system arises because, as a rule, differential equations are not integrated and it is impossible to define the general properties of the system trajectories by integration. However, even if the system of differential equations is integrated, the formulas can be so cumbersome that their use in the analysis of the system dynamics turns out to be a challenge.In this paper, we study an 8-dimensional system of ordinary differential equations that describes the dynamics of cancer during chemotherapy and immunotherapy. The phase space of the system is a non-negative orthant.The paper studies the equilibrium positions of the system. There is always one equilibrium in a system, but there may be two more equilibria. There are conditions for the system parameters found, under which the equilibrium position, being always present is asymptotically stable. Moreover, there is a provement that the asymptotic stability is global.