Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

Abstract

We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.

On the Dynamics of Immune-Tumor Conjugates in a Four-Dimensional Tumor Model

“…Various studies relating to the existence of chaotic dynamics of the dePillis-Radunskaya model [3] have been performed by Itik and Banks [4], Duarte et al [5], Letellier et al [6], Galindo et al [7], Abernethy and Gooding [8], Das et al [9]. Results of studies of a location of chaotic attractor and other compact invariant sets have been given by authors of this work in [10].…”

Exploring chronomodulated radiotherapy strategies in a chaotic population model

“…As stated in Section 1, several models describe the cell population dynamics related to cancer. Regardless of the type of treatment, the cancer models based on Gompertzian [16] or logistic growth, are the most common [29,31,32,34,38,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. Among the models on radiotherapy, we focus on those based on the action of the radiation on the cells as for instance [18,[30][31][32]34,56,[58][59][60].…”

Chaos synchronization and Nelder-Mead search for parameter estimation in nonlinear pharmacological systems: Estimating tumor antigenicity in a model of immunotherapy

“…In the course of these studies, currently and over the past two decades, significant attention has been paid to nonlinear dynamical chaotic systems, employing variations in logistic equations familiar to population models (e.g. Kuznetsov et al (1994) [12], Galindo et al (2015) [13]).…”