Fractional derivative operators of non-integer order can be utilized as a powerful tool to model nonlinear fractional differential equations. In this paper, we propose numerical solutions for simulating fractional-order derivative operators with the power-law and exponential-law kernels. We construct the numerical schemes with the help the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes are applied to simulate the dynamical fractional-order model of the immune response (FMIR) to the uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Numerical results are then presented to show the applicability and efficiency on the FMIR.

中文翻译：

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人类抵抗IAV感染的分数阶系统的一些新数学模型

非整数的分数阶导数算子可以用作建模非线性分数阶微分方程的强大工具。在本文中，我们提出了用幂律和指数律内核模拟分数阶导数算子的数值解。我们借助分数演算的基本定理和Lagrange多项式插值来构造数值方案。这些方案适用于模拟对简单的甲型流感病毒（IAV）感染的免疫反应（FMIR）的动态分数阶模型，该模型重点在于通过先天和适应性免疫来控制感染。然后给出数值结果，以显示在FMIR上的适用性和效率。