On a Mathematical Model of Tumor-Immune Interaction with a Piecewise Differential and Integral Operator

The representation of mathematical models via piecewise differential and integral operators for dynamic systems has this potential to capture cross-over behaviors such as a passage from deterministic to randomness which can be exhibited by different systems. A 3D mathematical model, similar to the prey-predator system, of tumor-immune interaction with piecewise differential and integral operators is developed and analyzed. Three different scenarios, namely, cross-overs from deterministic to randomness, the Mittag-Leffler law to randomness, and a cross-over behavior from fading memory to the power-law and a random process, are considered. The existence, uniqueness, positivity, and boundedness of the solutions of the systems are proved via the linear growth and Lipschitz conditions. The numerical approximations by Toufik, Atangana, and Araz are used for approximation of solutions and simulation of the piecewise models in different scenarios. From the nondimensionalized version of the 3D model representation, it is shown that the parameter values have an impact on the growth of tumor cells, and activating the proliferation of the resting cells has negatively affected the development of tumor cells. Moreover, the dynamics of tumor-immune interaction exhibited a cross-over behavior, and this behavior is exposed by the piecewise modeling approach used for the representations.