New observations on optimal cancer treatments for a fractional tumor growth model with and without singular kernel

The aim of this study is to examine a fractional optimal control problem (FOCP) governed by a cancer-obesity model with and without singular kernel, separately. We propose a new model including the population of immune cells, tumor cells, normal cells, fat cells, chemotherapeutic and immunotherapeutic drug concentrations. Existence and stability of the tumor free equilibrium point and coexisting equilibrium point are investigated analytically. We obtain the numerical solution of the fractional cancer-obesity model using L1 formula. The aim behind the FOCP is to find the optimal doses of chemotherapeutic and immunotherapeutic drugs which minimize the difference between the number of tumor cells and normal cells. To do so, we insert some weight constants as the coefficients of tumor and normal cells in the cost functional so that normal cell population is larger compared to tumor burden. On the other hand, we investigate the effect of obesity to the choice and schedules of treatment strategies in case of low and high caloric diets. Moreover, we discuss the choice of the differentiation operator, namely operators with and without singular kernel. Lastly, some illustrative examples are shown to examine the impact of the fractional derivatives of different orders on cancer-obesity model and we observe the contribution of the cost functional to eradicate tumor burden and regenerate normal cell population. Our model predicts the negative effect of obesity on the health of patient and we show that the most efficient treatment choice to eradicate the tumor is to apply combined therapy together with low caloric diet.