Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.
Math Biosci Eng. 2022 Jul 28;19(11):10762-10808. doi: 10.3934/mbe.2022504.
In this paper, we apply the fractal-fractional derivative in the Atangana-Baleanu sense to a model of the human immunodeficiency virus infection of CD$ 4^{+} $ T-cells in the presence of a reverse transcriptase inhibitor, which occurs before the infected cell begins producing the virus. The existence and uniqueness results obtained by applying Banach-type and Leray-Schauder-type fixed-point theorems for the solution of the suggested model are established. Stability analysis in the context of Ulam's stability and its various types are investigated in order to ensure that a close exact solution exists. Additionally, the equilibrium points and their stability are analyzed by using the basic reproduction number. Three numerical algorithms are provided to illustrate the approximate solutions by using the Newton polynomial approach, the Adam-Bashforth method and the predictor-corrector technique, and a comparison between them is presented. Furthermore, we present the results of numerical simulations in the form of graphical figures corresponding to different fractal dimensions and fractional orders between zero and one. We analyze the behavior of the considered model for the provided values of input factors. As a result, the behavior of the system was predicted for various fractal dimensions and fractional orders, which revealed that slight changes in the fractal dimensions and fractional orders had no impact on the function's behavior in general but only occur in the numerical simulations.
在本文中，我们应用分数阶导数中的分数阶导数在 Atangana-Baleanu 意义上对一个人类免疫缺陷病毒感染 CD$ 4^{+} $ T 细胞的模型进行研究，该模型发生在感染细胞开始产生病毒之前。通过应用巴拿赫型和勒雷-斯卡尔型不动点定理，我们得到了所提出模型解的存在性和唯一性结果。为了确保存在紧密的精确解，我们对乌尔曼稳定性及其各种类型的稳定性分析进行了研究。此外，通过使用基本再生数，我们分析了平衡点及其稳定性。通过使用牛顿多项式逼近法、亚当斯-巴什福斯方法和预测校正技术，我们提供了三个数值算法来给出近似解，并对它们进行了比较。此外，我们以对应于零到一之间不同分形维数和分数阶的图形形式呈现数值模拟结果。我们分析了针对给定输入因子的所考虑模型的行为。结果，我们预测了各种分形维数和分数阶的系统行为，这表明分形维数和分数阶的微小变化通常不会影响函数的行为，而只会在数值模拟中出现。
