The chaos and optimal control of cancer model with complete unknown parameters

Abstract In this paper, we study the chaos and optimal control of cancer model with completely unknown parameters. The stability analysis of the biologically feasible steady-states of this model will be discussed. It is proved that the system appears to exhibit periodic and quasi-periodic limit cycles and chaotic attractors for some ranges of the system parameters. The necessary optimal controllers input for the asymptotic stability of some positive equilibrium states are derived. Numerical analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out for various parameters values and different initial densities.

[1]  Y. Kuang,et al.  Biological stoichiometry of tumor dynamics: Mathematical models and analysis , 2003 .

[2]  A. Mitra,et al.  Synchronization among tumour-like cell aggregations coupled by quorum sensing: A theoretical study , 2008, Comput. Math. Appl..

[3]  Dmitriĭ Olegovich Logofet,et al.  Stability of Biological Communities , 1983 .

[4]  Ram Rup Sarkar,et al.  Cancer self remission and tumor stability-- a stochastic approach. , 2005, Mathematical biosciences.

[5]  Ami Radunskaya,et al.  The dynamics of an optimally controlled tumor model: A case study , 2003 .

[6]  Ami Radunskaya,et al.  A mathematical tumor model with immune resistance and drug therapy: an optimal control approach , 2001 .

[7]  Awad El-Gohary,et al.  Chaos and optimal control of cancer self-remission and tumor system steady states☆ , 2008 .

[8]  A. El-Gohary.  Optimal control of the genital herpes epidemic , 2001 .

[9]  Tomé,et al.  Stochastic lattice gas model for a predator-prey system. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  A. S. Al-Ruzaiza,et al.  Chaos and adaptive control in two prey, one predator system with nonlinear feedback , 2007 .

[11]  S. Menchón,et al.  Modeling subspecies and the tumor-immune system interaction: Steps toward understanding therapy , 2007 .

[12]  Katsuhiko Ogata,et al.  Modern Control Engineering , 1970 .

[13]  Masaki Katayama,et al.  Global asymptotic stability of a predator-prey system of Holling type , 1999 .

[14]  Awad I. El-Gohary,et al.  Optimal control of stochastic prey-predator models , 2003, Appl. Math. Comput..

[15]  Zeljko Bajzer,et al.  Mathematical modeling of cancer radiovirotherapy. , 2006, Mathematical biosciences.