A fractional model of cancer-immune system with Caputo and Caputo-Fabrizio derivatives

Recently, it is important to try to understand diseases with large mortality rates worldwide, such as infectious disease and cancer. For this reason, mathematical modeling can be used to comment on diseases that adversely affect all people. So, this paper discuss mathematical model presented for the first time that examines the interaction between immune system and cancer cells by adding IL-12 cytokine and anti-PD-L1 inhibitor. The proposed ordinary differential new mathematical model is studied by considering in term of Caputo and Caputo-Fabrizio (CF) derivative. Stability analysis, existence, and uniqueness of the solution is examined for Caputo fractional derivative. Then numerical simulations of ordinary and fractional differential new mathematical model are given. It is obtained that a reduction (20%-80%) of the number of cancer cells for Caputo derivative and $(100\%)$ of the number of cancer cells for CF derivative. The reduction is one of the most important aspects of the new fractional model for the order discussed especially obtained for CF derivative.