Modelling immune systems based on Atangana–Baleanu fractional derivative

Memory effects play a critical role in complex immune systems. In this paper, the recent efficient and realistic Atangana–Baleanu fractional order derivative, with non-local and non-singular kernel, was employed in two mathematical models for immune systems having multiple immune effectors. For each model, we derive the conditions under which a unique set of exact solutions exists. Stability analysis of equilibrium points of the two systems is carried out where the effects of model's parameters and fractional derivatives are examined. Furthermore, a recent numerical scheme is utilized to solve each model numerically and to compare theoretical results with those of numerical experiments. Results depict that memory influences induce stabilization of immune systems such that the solution trajectories of the model always converge to either a single immune effector or a persistent immune effector/antigen equilibrium states.