Nisar Kottakkaran Sooppy, Farman Muhammad, Jamil Khadija, Hincal Evren, Sambas Aceng
Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam bin Abdulaziz University, 11942, Al Kharj, Saudi Arabia.
Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan.
Sci Rep. 2025 Jul 17;15(1):25953. doi: 10.1038/s41598-025-09944-6.
In this paper, fractional calculus has proven to be invaluable in disease transmission dynamics and the creation of control systems, among other real-world problems. To investigate vaccine and treatment dynamics for disease control, this work focuses on Kawasaki illness and uses a unique fractional operator called the modified Atangana-Baleanu-Caputo derivative. The stability analysis, positivity, boundedness, existence, and uniqueness, are treated for the proposed model with novel fractional operators. Additionally, it investigates the effects of different parameters on the reproductive number. It verifies the existence and uniqueness of the solutions to the suggested model using Banach fixed point and the Leray-Schauder nonlinear alternative theorem. Employs Lyapunov functions to determine the model equilibria analysis global stability. The numerical simulation and results utilized the two-step Lagrange interpolation approach at various fractional order values. The results are contrasted with those obtained using the widely recognized ABC method and comparisons are also made to show the effects of the proposed method for the epidemic system. This model advances beyond existing Kawasaki disease models by incorporating fractional-order dynamics, which capture memory effects and long-range dependencies in biological systems, offering more accurate representations of disease progression. The inclusion of chaos stability control provides novel insights into managing complex, nonlinear behaviors, enhancing both theoretical understanding and potential clinical applications.
在本文中，分数阶微积分已被证明在疾病传播动力学以及控制系统的创建等其他实际问题中具有极高的价值。为了研究用于疾病控制的疫苗和治疗动力学，这项工作聚焦于川崎病，并使用了一种名为修正的阿坦加纳 - 巴莱努 - 卡普托导数的独特分数阶算子。针对所提出的具有新型分数阶算子的模型，进行了稳定性分析、正性、有界性、存在性和唯一性的研究。此外，它还研究了不同参数对繁殖数的影响。利用巴拿赫不动点定理和勒雷 - 绍德尔非线性择一性定理验证了所建议模型解的存在性和唯一性。采用李雅普诺夫函数来确定模型平衡点并分析全局稳定性。数值模拟和结果在不同分数阶值下采用了两步拉格朗日插值法。将结果与使用广泛认可的ABC方法获得的结果进行对比，并进行比较以展示所提出方法对流行病系统的影响。该模型通过纳入分数阶动力学超越了现有的川崎病模型，分数阶动力学能够捕捉生物系统中的记忆效应和长程依赖性，从而更准确地描述疾病进展。混沌稳定性控制的纳入为管理复杂的非线性行为提供了新的见解，增强了理论理解和潜在的临床应用。
