A fractional stochastic differential modeling of cancer cells with an application to the immune response of tumor dynamics

In this paper, we present a fractional stochastic model that examines the response of cancer cells to the immune system. The model combines the long-term memory dependence of fractional derivatives with the stochastic nature of cancer cell growth. The geometric Brownian motion is used to present the stochastic nature of this model. By applying the global derivative from different versions of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, and converting them into the fractional integral version, we demonstrate the memory property of the model by maintaining the initial conditions. We also prove the stability of the model analytically in the two states of the ordinary differential equation and the fractional differential equation by obtaining the equilibrium points of the model in the disease-free state and the disease state. Additionally, we use the numerical method based on Lagrange polynomials and Newton’s polynomials to examine and compare the approximate solution of the model in two different states: the disease-free state and the disease state. Finally, using numerical simulation, we examine the stability of the model in the fractional-random state. We show that using Newton’s polynomial will preserve the stability condition better than Lagrange’s polynomial. Further, we demonstrate that the solutions of the stochastic fractional model are positive and bounded, and we also prove their uniqueness and existence.

کلیدواژه ها:

Cancer model ، Immune system ، Fractional stochastic differential equations ، Fractional derivatives ، Numerical approximations