Chaos and optimal control of equilibrium states of tumor system with drug. (English) Zbl 1198.37121
Summary: This article is devoted to study the chaos and optimal control problems of both tumor and tumor with drug systems. The stability and instability of the equilibrium states of these systems are investigated. This stability analysis indicates that these systems exhibit a chaotic behavior for some values of the system parameters. The optimal amount of drug and optimal dose for control of the equilibrium states that minimize the required Hamilton function are obtained. Analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
MSC:
| 37N25 | Dynamical systems in biology |
| 34H10 | Chaos control for problems involving ordinary differential equations |
| 92C50 | Medical applications (general) |
References:
| [1] | Dingli, D.; Cascino, D. M.; Josic, K.; Russell, J.; Bajzer, Z., Mathematical modeling of cancer radiovirotherapy, Math Biosci, 199, 55-78 (2006) · Zbl 1086.92024 |
| [2] | El-Gohary, A., Chaos and optimal control of cancer self-remission and tumor system steady states, Chaos, Solitons & Fractals, 37, 5, 1305-1316 (2008) · Zbl 1142.92327 |
| [3] | Feichtinger, G.; Forst, C.; Piccardi, C., A nonlinear dynamical model for the dynastic cycle, Chaos, Solitons & Fractals, 7, 2, 257-271 (1996) · Zbl 1080.91572 |
| [4] | Kuang, Y., Biological stoichiometry of tumor dynamics: mathematical models and analysis, Discret Contin Dyn S, 4, 3, 221-240 (2004) · Zbl 1056.34074 |
| [5] | El-Gohary, A., Optimal control of the genital herpes epidemic, Chaos, Solitons & Fractals, 12, 1817-1822 (2001) · Zbl 0979.92022 |
| [6] | DePillis, L.; Radunskaya, A., The dynamics of an optimally controlled tumor model: a case study, Math Comput Model, 37, 1221-1244 (2003) · Zbl 1043.92018 |
| [7] | Prehn, R., Stimulatory effects of immune-reactions upon the growths of untransplanted tumors, Cancer Res, 54, 4, 908-914 (1994) |
| [8] | Kuznetsov, V.; Makalkin, I.; Taylor, M.; Perelson, A., Nonlinear dynamics of immunogenic tumors parameter estimation and global bifurcation analysis, Bull Math Bio, 56, 6, 295-321 (1994) · Zbl 0789.92019 |
| [9] | Vaidya, V.; Alexandro, F., Evaluation of some mathematical models for tumor growth, Int J Biol Med Comput, 13, 19-35 (1982) |
| [10] | Hart, D.; Shochat, E.; Agur, Z., The growth law of primary breast cancer as inferred from mammography screening trials data, Br J Cancer, 78, 3, 382-387 (1998) |
| [11] | Kirschner, D.; Panetta, J., Modelling immunotherapy of tumor-immune interaction, J Math Biol, 37, 3, 235-252 (1998) · Zbl 0902.92012 |
| [12] | Lakshmanan, M.; Rajaseker, S., Nonlinear dynamics: integrability and chaos patterns (1965), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York |
| [13] | Ogata, K., Modern Control engineering (1970), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs (NJ) |
| [14] | El-Gohary, A.; Al-Ruzaiza, A. S., Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos, Solitons & Fractals, 34, 2, 443-453 (2007) · Zbl 1127.92040 |
| [15] | El-Gohary, A., Optimal control of stochastic lattice of preypredator models, Appl Math Computat, 160, 1, 15-28 (2005) · Zbl 1055.92041 |
| [16] | Angelis, Elena De; Lods, Bertrand, On the kinetic theory for active particles: a model for tumor immune system competition, Math Comput Model, 47, 1-2, 196-209 (2008) · Zbl 1134.92023 |
| [17] | Ramosand, C. A., Modeling subspecies and the tumor-immune system interaction: steps toward understanding therapy, Physica A: Statist Mech Appl, 386, 2, 713-719 (2007) |
| [18] | Misraand A, Mitra JC. Synchronization among tumour-like cell aggregations coupled by quorum sensing: a theoretical study. Comput Math Appl, in press.; Misraand A, Mitra JC. Synchronization among tumour-like cell aggregations coupled by quorum sensing: a theoretical study. Comput Math Appl, in press. · Zbl 1134.92015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
