Nawaz Yasir, Arif Muhammad Shoaib, Ashraf Muhammad Usman
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000 Pakistan.
Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan.
Iran J Sci Technol Trans A Sci. 2021;45(6):2109-2119. doi: 10.1007/s40995-021-01214-0. Epub 2021 Sep 13.
In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh-Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model's epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.
在本论文中，提出了一种用于求解偏微分方程（PDEs）的一阶时间格式。对COVID-19疫情的数学模型进行了修正，其中还考虑了暴露个体的恢复率。通过求修正后的COVID-19模型的雅可比矩阵并对特征多项式应用劳斯 - 赫尔维茨准则，给出了其平衡态的线性稳定性。所提出的格式在时间上具有一阶精度，在空间上具有二阶精度。利用标准抛物型偏微分方程的冯·诺依曼稳定性准则给出了所提格式的稳定性。通过使用泰勒级数展开其中涉及的项，也给出了所提格式的一致性。该格式可用于获得正解的条件。通过选择所包含参数的合适值，可以扩大格式的稳定区域。最后，将所提格式与现有的非标准有限差分方法进行了比较。结果表明，非标准经典技术无法保留模型具有流行病学意义的解的独特特征，而所提方法能够做到这一点。如有需要，可为读者提供所提离散模型格式的计算代码，以便使用。
