Chaos in a Three-Dimensional Cancer Model

Abstract

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.

Dynamical Behavior Analysis and Control of a Fractional-order Discretized Tumor Model

“…[6]. Consider that the immune cells can die naturally and the main features of immune response involve memory, the natural death rate of immune cells is added to the second equation, and a fractional-order cancer model is given as follows:…”

“…Some authors discussed the problem of chaos and stability analysis of some biological models such as cancer and tumor model, stochastic lattice gas prey-predator model and many other models [5,6]. However, the current models are all restricted to integer-order ordinary differential equations.…”

Improved numerical solutions for chaotic-cancer-model

“…Recently third and fourth order of convergence techniques have been proposed and analyzed to get efficient results. [8][9][10][11][12][13][14][15][16][17] Frontini et. al.…”

Chaos Control in Three Dimensional Cancer Model by State Space Exact Linearization Based on Lie Algebra

“…Interactions of cancer cells with healthy host cells and immune system cells are the main components of these models and these interactions may yield different outcomes. Some important phenomena of cancer progression such as cancer dormancy, creeping through, and escape from immune surveillance have been investigated [1]. De Pillis and Radunskaya [2] included a normal tissue cell population in this model, performed phase space analysis, and investigated the effect of chemotherapy treatment by using optimal control theory whereas Kirschner and Panetta [3] examined the cancer cell growth in the presence of the effector immune cells and the cytokine IL-2 which has an essential role in the activation and stimulation of the immune system.…”