Fractional order model of immune cells influenced by cancer cells

In this paper, we study the mathematical model of interaction cancer cells and immune system cells presented Castiglione and Piccoli. As the interaction between cancer cells and the immune system is weak, when the immune system of the body begins to decrease, the cancer cells get stronger and increase rapidly. Helper CD4+ T and cytotoxic CD8+ T cells, cancer cells, dendritic cells and cytokine interleukin-2 (IL-2) cells are involved in the mathematical model of this competition in the living body. As can be seen in the literature, since the cancer cells have memory structure, fractional models describe the struggle between the cancer cells and immune system give more meaningful results than classical models as closer to the reality. The main motivation of the present work is to generalize the model in Castiglione and Piccoli [J. Theor. Biol. 247 (2007) 723–732] by using Caputo fractional derivative. The main aim is to analyze the behaviors of system cells by changing of the fractional parameter. In this sense, we study on the stability analysis of treatment free and the fixed points of the prescribed model. To get the numerical solutions, we apply the Adam-Bashforth-Moulton (ABM) algorithm and also illustrate the results by the graphics held by Matlab program. We have reached the excellent result that cancer cells decrease as θ diminishes in this process.