Qualitative analysis and controllability of complex tumor model with different therapies with nonsingular kernel

In this paper, consider the immune response to avascular cancer under the effect of immunotherapy, chemotherapy, and their combinations, as well as vaccination regimens, is described using a fractional order model to observe the impact of different therapies for cancer treatment. The impact of vaccination therapy is viewed as a model parameter perturbation. The effect of the global derivative, the existence, and the boundedness of the suggested system are confirmed, which are the essential characteristics of epidemic problems. The proposed system is qualitatively examined as well to determine its stable points. The Lyapunov function is used to analyze global stability, and the equilibrium states of the second derivative test are quantitatively examined. To investigate the effects of the fractional operator on the suggested model, solutions are generated using the Mittag Leffler kernel, and numerical simulations are run to demonstrate the theoretical findings. Using MATLAB, the effects of cancer treatment with various drugs and parameter values are justified. The proposed system is also treated for controllability and observability for a linear control system to monitor the close-loop design with different therapies as an input and cancer cells as an output.