Abstract
In this study, we consider a three-dimensional discrete-time model to investigate the interaction between normal host cells with functional immune cells and tumor cells. Fixed point analysis is performed to study the stability of the discrete three-dimensional model and the sensitivity of the system analysis on the initial cell population. Necessary and sufficient conditions for optimal control of tumor cell growth have been created with the introduction of immune-chemotherapy drugs and the chaotic behavior of the system with branching having been demonstrated. The turbulence behavior of the system is shown by branching and Lyapunov power is performed for the integer-order discrete model and the turbulence effect is compared to the different fractional orders of the discrete model. Also, by numerical simulation, validating the theoretical results of the work and the effect of fractional derivative order on system chaos are investigated.