In this paper, we take a three-dimension discrete time model of the interaction of host cells with immune cells and tumor cells fixed point analysis was performed to analyze the stability of the system. the necessary conditions have been created to control the growth of cancer cells the introduction of chemotherapy and the disordered behavior was diagnosed the system by finding exponent and dimension of Lyapunov in order, and the numerical simulation of the system was done using the iterative fixed point method, as well as study the dissipative and Neimark-Sacker bifurcation of system. Finally, the tumor cells of the system and its disorder were controlled using the adaptive control technique, and a stable and regular system was obtained.
Cite this paper
Aziz, M. M. and Mohammed, S. A. (2023). Stability Analysis and Chaos Diagnosis of a Tumor-Host Immune Cell Interaction Model with Neimark Sacker Bifurcation. Open Access Library Journal, 10, e9577. doi: http://dx.doi.org/10.4236/oalib.1109577.
Ravichandran, C., Logeswari, K., Panda, S.K. and Nisar, K.S. (2020) On New Approach of Fractional Derivative by Mittag-Leffler Kernel to Neutralintegro-Differential Systems with Impulsive Conditions. Chaos, Solitons Fractals, 139, Article ID: 110012.
https://doi.org/10.1016/j.chaos.2020.110012
Panda, S.K., Karapinar, E. and Atangana, A. (2020) A Numerical Schemes and Comparisons for Fixed Point Results with Applications to the Solutions of Volterra Integral Equations in Dislocated Extended B-Metric Space. Alexandria Engineering Journal, 59, 815-827.
Panda, S.K., Abdeljawad, T. and Ravichandran, C. (2020) Novel Fixed Point Approach to Atangana-Baleanu Fractional and Lp-Fredholm Integral Equations. Alexandria Engineering Journal, 59, 1959-1970. https://doi.org/10.1016/j.aej.2019.12.027
Tuan, N.H., Mohammadi, H. and Rezapour, S. (2020) A Mathematical Model for COVID-19 Transmission by Using the Caputo Fractional Derivative. Chaos, Solitons Fractals, 140, Article ID: 110107. https://doi.org/10.1016/j.chaos.2020.110107
Rezapour, S., Mohammadi, H. and Jajarmi, A. (2020) A New Mathematical Model for Zika Virus Transmission. Advances in Difference Equations, 2020, Article No. 589. https://doi.org/10.1186/s13662-020-03044-7
Wodarz, D. (2005) Mathematical Models of Immune Effector Responses to Viral Infections Virus Control versus the Development of Pathology. Journal of Computational and Applied Mathematics, 184, 301-319.
https://doi.org/10.1016/j.cam.2004.08.016
Wodarz, D., May, R.M. and Nowak, M.A. (2007) The Role of Antigen Independent Persistence of Memory Cytotoxic T Lymphocytes. International Immunology, 12, 467-477. https://doi.org/10.1093/intimm/12.4.467
Khajanchi, S. and Nieto, J.J. (2019) Mathematical Modeling of Tumor-Immune Competitive System, Considering the Role of Time Delay. Applied Mathematics and Computation, 340, 180-205. https://doi.org/10.1016/j.amc.2018.08.018
Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1997) Chaos: An Introduction to Dynamical System. Springer-Verlag, New York.
https://doi.org/10.1007/978-3-642-59281-2
Alzabut, J., Selvam, A.G.M., Dhakshinamoorthy, V., Mohammadi, H. and Rezapour, S. (2022) On Chaos of Discrete Time Fractional Order Host-Immune-Tumor Cells Interaction Model. Journal of Applied Mathematics and Computing, 68, 4795-4820.
https://doi.org/10.1007/s12190-022-01715-0
Aziz, M.M. and Jihad, O.M. (2021) Stability, Chaos Tests with Adaptive and Feedback Control Methodsfor 3D Discrete-Time Dynamical System. International Journal of Electronics Communication and Computer Engineering, 12, 31-42.
Aziz, M.M. and Jihad, O.M. (2021) Stability & Chaos Tests of 2D Discrete Time Dynamical System with Hidden Attractors. Open Access Library Journal, 8, e7501.
https://doi.org/10.4236/oalib.1107501
Aziz, M.M. and Merie, D.M. (2020) Stability and Adaptive Control with Synchronization of 3-D Dynamical System. Open Access Library Journal, 7, Article No. e6075.
https://doi.org/10.4236/oalib.1106075
Aziz, M.M. and Merie, D.M. (2020) Stability and Chaos with Mathematical Control of 4-D Dynamical System. Indonesian Journal of Electrical Engineering and Computer Science, 20, Article 1242.
Liu, X. and Xiao, D. (2007) Complex Dynamic Behaviors of a Discrete-Time Predator-Prey System. Chaos, Solitons Fractals, 32, 80-94.
https://doi.org/10.1016/j.chaos.2005.10.081
Xin, B., Chen, T. and Ma, J.H. (2010) Neimark-Sacker Bifurcation in a Discrete-Time Financial System. Discrete Dynamics in Nature and Society, 2010, Article ID: 405639. https://doi.org/10.1155/2010/405639
