Hopf bifurcation, cascade of period-doubling, chaos, and the possibility of cure in a 3D cancer model. (English) Zbl 1353.92053

Summary: 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.


92C50 Medical applications (general)
37N25 Dynamical systems in biology
34C28 Complex behavior and chaotic systems of ordinary differential equations
Full Text: DOI


[1] Araujo, R. P.; McElwain, D. L., A history of the study of solid tumour growth: the contribution of mathematical modelling, Bulletin of Mathematical Biology, 66, 5, 1039-1091 (2004) · Zbl 1334.92187
[2] Chammas, R.; Silva, D.; Wainstein, A.; Abdallah, K., Imunologia clinica das neoplasias, Imunologia Clínica na Prática Médica, 447-460 (2009), São Paulo, Brazil: Atheneu, São Paulo, Brazil
[3] Chang, W.; Crowl, L.; Malm, E.; Todd-Brown, K.; Thomas, L.; Vrable, M., Analyzing Immunotherapy and Chemotherapy of Tumors Through Mathematical Modeling (2003), Claremont, Calif, USA: Department of Mathematics, Harvey-Mudd University, Claremont, Calif, USA
[4] Kirschner, D.; Panetta, J. C., Modeling immunotherapy of the tumor—immune interaction, Journal of Mathematical Biology, 37, 3, 235-252 (1998) · Zbl 0902.92012
[5] Galach, M., Dynamics of the tumor-immune system competition—the effect of time delay, International Journal of Applied Mathematics and Computer Science, 13, 3, 395-406 (2003) · Zbl 1035.92019
[6] Itik, M.; Banks, S. P., Chaos in a three-dimensional cancer model, International Journal of Bifurcation and Chaos, 20, 1, 71-79 (2010) · Zbl 1183.34064
[7] de Pillis, L. G.; Radunskaya, A., The dynamics of an optimally controlled tumor model: a case study, Mathematical and Computer Modelling, 37, 11, 1221-1244 (2003) · Zbl 1043.92018
[8] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (2004), New York, NY, USA: Springer, New York, NY, USA · Zbl 1082.37002
[9] Itik, M.; Banks, S. P., On the structure of periodic orbits on a simple branched manifold, International Journal of Bifurcation and Chaos, 20, 11, 3517-3528 (2010) · Zbl 1208.34037
[10] Pontryagin, L. S., Ordinary Differential Equations (1962), New York, NY, USA: Addison-Wesley, New York, NY, USA · Zbl 0112.05502
[11] Letellier, C.; Denis, F.; Aguirre, L. A., What can be learned from a chaotic cancer model?, Journal of Theoretical Biology, 322, 7-16 (2013) · Zbl 1406.92313
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