This work presents a new mathematical model to depict the effect of obesity on cancerous tumor growth when chemotherapy as well as immunotherapy have been administered. We consider an optimal control problem to destroy the tumor population and minimi
ze the drug dose over a finite time interval. The constraint is a model including tumor cells, immune cells, fat cells, chemotherapeutic and immunotherapeutic drug concentrations with Caputo time fractional derivative. We investigate the existence and stability of the equilibrium points namely, tumor free equilibrium and coexisting equilibrium, analytically. We discretize the cancer-obesity model using L1-method. Simulation results of the proposed model are presented to compare three different treatment strategies: chemotherapy, immunotherapy and their combination. In addition, we investigate the effect of the differentiation order $alpha$ and the value of the decay rate of the amount of chemotherapeutic drug to the value of the cost functional. We find out the optimal treatment schedule in case of chemotherapy and immunotherapy applied.
In this paper we study the dichotomy of Poincarce map. We give a relation between the dichotomy of the Poincarce map and boundedness of solutions of certain periodic Cauchy problems
