Hybrid fractional-order optimal control problem for immuno-chemotherapy with gene therapy and time-delay: numerical treatments

This paper investigates the optimal control problem for the tumor-immune system with time-delay in the presence of gene therapy and immuno chemotherapy. Riemann-Liouville fractional order integrals and Caputo fractional order derivatives are combined to form a hybrid fractional order operator. For consistency with the physical model problem, a new parameter ϑ is presented. A stability and bifurcation analysis of the proposed model is performed. The positivity, boundedness, and existence of optimal control for the proposed model are discussed. Grünwald-Letnikov nonstandard finite difference method with the discretization of the hybrid fractional order operator is constructed to solve the proposed model. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.