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# Optimal and Memristor-Based Control of A Nonlinear Fractional Tumor-Immune Model

Amr M. S. Mahdy1,2,*, Mahmoud Higazy1,3, Mohamed S. Mohamed1,4

1 Department of Mathematics, College of Science, Taif University, Taif, 21944, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt
3 Department of Physics and Engineering Mathematics, Faculty of Electronic engineering, Menoufia University, Menouf, 32952, Egypt
4 Department of Mathematics, Faculty of Science, Al-Azher University, Nasr City, 11884, Egypt

* Corresponding Author: Amr M. S. Mahdy. Email:

(This article belongs to this Special Issue: Role of Computer in Modelling & Solving Real-World Problems)

Computers, Materials & Continua 2021, 67(3), 3463-3486. https://doi.org/10.32604/cmc.2021.015161

## Abstract

In this article, the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune. The fractional derivatives are described in the Caputo sense. The solutions derived using this method are easy and very accurate. The model is given by its signal flow diagram. Moreover, a simulation of the system by the Simulink of MATLAB is given. The disease-free equilibrium and stability of the equilibrium point are calculated. Formulation of a fractional optimal control for the cancer model is calculated. In addition, to control the system, we propose a novel modification of its model. This modification is based on converting the model to a memristive one, which is a first time in the literature that such idea is used to control this type of diseases. Also, we study the system’s stability via the Lyapunov exponents and Poincare maps before and after control. Fractional order differential equations (FDEs) are commonly utilized to model systems that have memory, and exist in several physical phenomena, models in thermoelasticity field, and biological paradigms. FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change. FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

## Keywords

A. M. S. Mahdy, M. Higazy and M. S. Mohamed, "Optimal and memristor-based control of a nonlinear fractional tumor-immune model," Computers, Materials & Continua, vol. 67, no.3, pp. 3463–3486, 2021.

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