Karaagac Berat, Owolabi Kolade M, Pindza Edson
Faculty of Education, Department of Mathematics Education, Adiyaman University, Adiyaman, Turkey.
Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704 Akure, Ondo State Nigeria.
Int J Dyn Control. 2023 Mar 1:1-18. doi: 10.1007/s40435-023-01131-7.
The concept of a Caputo fractal-fractional derivative is a new class of non-integer order derivative with a power-law kernel that has many applications in real-life scenarios. This new derivative is applied newly to model the dynamics of diabetes mellitus disease because the operator can be applied to formulate some models which describe the dynamics with memory effects. Diabetes mellitus as one of the leading diseases of our century is a type of disease that is widely observed worldwide and takes the first place in the evolution of many fatal diseases. Diabetes is tagged as a chronic, metabolic disease signalized by elevated levels of blood glucose (or blood sugar), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the sense of fractional-fractal derivative. At first, the critical points of the diabetes mellitus model are investigated; then Picard's theorem idea is applied to investigate the existence and uniqueness of the solutions of the model under the fractional-fractal operator. The resulting discretized system of fractal-fractional differential equations is integrated in time with the MATLAB inbuilt Ode45 and Ode15s packages. A step-by-step and easy-to-adapt MATLAB algorithm is also provided for scholars to reproduce. Simulation experiments that revealed the dynamic behavior of the model for different instances of fractal-fractional parameters in the sense of the Caputo operator are displayed in the table and figures. It was observed in the numerical experiments that a decrease in both fractal dimensions and leads to an increase in the number of people living with diabetes mellitus.
卡普托分数阶-分形导数的概念是一类新的具有幂律核的非整数阶导数，在实际场景中有许多应用。这种新的导数最近被应用于糖尿病疾病动态的建模，因为该算子可用于构建一些描述具有记忆效应的动态模型。糖尿病作为本世纪的主要疾病之一，是一种在全球广泛观察到的疾病，在许多致命疾病的发展中位居首位。糖尿病被标记为一种慢性代谢疾病，其特征是血糖（或血糖）水平升高，随着时间的推移，会对身体的心脏、血管、眼睛、肾脏和神经造成严重损害。本研究致力于在分数阶-分形导数意义下对无遗传因素的糖尿病模型进行数学建模和分析。首先，研究糖尿病模型的临界点；然后应用皮卡德定理的思想来研究分数阶-分形算子下模型解的存在性和唯一性。所得的分形-分数阶微分方程离散系统通过MATLAB内置的Ode45和Ode15s软件包进行时间积分。还为学者们提供了一个逐步且易于适应的MATLAB算法以供重现。表和图中展示了模拟实验，这些实验揭示了在卡普托算子意义下模型对于不同分形-分数阶参数实例的动态行为。在数值实验中观察到，分形维数 和 的降低都会导致糖尿病患者人数增加。
