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

Abstract

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.

Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems

“…It is needed to mention the aim of using the piecewise fractional order derivative. In order to study multiple behaviors of a given dynamic system, which can not be possible otherwise, researchers in the field are empowered to use different concepts of derivative and integral operators at the same time by using the idea of piecewise derivatives and integrals (see details [37]). Because real world problems exhibit multi behaviors cannot be captured by using classical derivatives, stochastic, fractals-fractional, Mittag-Leffler type, etc.…”

Numerical Simulations for Deterministic-Stochastic Chaotic Cancer Model

“…Tumors are perilous disorders that can affect any tissue in the body and induce uncontrolled cell diffusion in that tissue. This disease is one of the main causes of mortality around the world; as a result, several mathematical models of tumor growth have been built, and countless publications have been written, in an effort to gain a better understanding of the behavior of this disease [2,8,17,18]. In this investigation, we adapt the cancer immune model presented in [17] by utilizing the idea of a piecewise differential model, in which the first is deterministic and the second part is stochastic.…”