Malaria is a complex disease with many factors influencing the transmission dynamics, including age. This research analyzes the transmission dynamics of malaria by developing an age-structured mathematical model using the classical integer order and Atangana–Baleanu–Caputo fractional operators. The analysis of the model focused on several important aspects. The existence and uniqueness of solutions of fractional order were explored based on some fixed-point theorems,such as Banach and Krasnoselski. The Positivity and boundedness of the solutions were also investigated. Furthermore, through mathematical analysis techniques, we analyzed different types of stability results, and the results showed that the disease-free equilibrium point of the model is proved to be both locally and globally asymptotically stable if the basic reproduction number is less than one, whereas the endemic equilibrium point of the model is both locally and globally asymptotically stable if the basic reproduction number is greater than one. The findings from the sensitivity analysis revealed that the most sensitive parameters, essential for controlling or eliminating malaria are mosquito biting rate, density-dependent natural mortality rate, clinical recovery rate, and recruitment rate for mosquitoes. Numerical simulations are also performed to examine the behavior of the model for different values of the fractional-order alpha,and the result revealed that as the value α reduces from 1, the spread of the endemic grows slower. By incorporating these findings, this research helps to clarify the dynamics of malaria and provides information on how to create efficient control measures.

中文翻译：

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通过经典和 ABC 分数算子分析疟疾传播动力学的年龄结构数学模型

疟疾是一种复杂的疾病，影响传播动态的因素有很多，包括年龄。本研究通过使用经典整数阶和 Atangana-Baleanu-Caputo 分数算子开发年龄结构的数学模型来分析疟疾的传播动态。模型的分析主要集中在几个重要方面。基于Banach和Krasnoselski等不动点定理探讨了分数阶解的存在性和唯一性。还研究了解的正性和有界性。此外，通过数学分析技术，我们分析了不同类型的稳定性结果，结果表明，当基本再生数小于1时，模型的无病平衡点被证明是局部和全局渐近稳定的，而如果基本再生数大于 1，则模型的地方性平衡点局部和全局渐近稳定。敏感性分析结果表明，控制或消除疟疾所必需的最敏感参数是蚊子叮咬率、密度依赖性自然死亡率、临床治愈率和蚊子招募率。还进行了数值模拟来检查模型在不同分数阶 α 值下的行为，结果表明，随着α值从 1 减小，地方病的传播速度减慢。通过整合这些发现，这项研究有助于阐明疟疾的动态，并提供有关如何制定有效控制措施的信息。