Temporal and Spacial Studies of Infectious Diseases: Mathematical Models and Numerical Solvers

arXiv - QuanBio - Quantitative Methods Pub Date : 2024-09-05 DOI:arxiv-2409.10556

Md Abu Talha, Yongjia Xu, Shan Zhao, Weihua Geng

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Abstract

The SIR model is a classical model characterizing the spreading of infectious diseases. This model describes the time-dependent quantity changes among Susceptible, Infectious, and Recovered groups. By introducing space-depend effects such as diffusion and creation in addition to the SIR model, the Fisher's model is in fact a more advanced and comprehensive model. However, the Fisher's model is much less popular than the SIR model in simulating infectious disease numerically due to the difficulties from the parameter selection, the involvement of 2-d/3-d spacial effects, the configuration of the boundary conditions, etc. This paper aim to address these issues by providing numerical algorithms involving space and time finite difference schemes and iterative methods, and its open-source Python code for solving the Fisher's model. This 2-D Fisher's solver is second order in space and up to the second order in time, which is rigorously verified using test cases with analytical solutions. Numerical algorithms such as SOR, implicit Euler, Staggered Crank-Nicolson, and ADI are combined to improve the efficiency and accuracy of the solver. It can handle various boundary conditions subject to different physical descriptions. In addition, real-world data of Covid-19 are used by the model to demonstrate its practical usage in providing prediction and inferences.

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传染病的时空研究：数学模型和数值求解器

SIR 模型是描述传染病传播特征的经典模型。该模型描述了易感群体、感染群体和康复群体之间随时间变化的数量变化。费舍模型在 SIR 模型的基础上引入了扩散和创造等空间依赖效应，实际上是一个更先进、更全面的模型。然而，由于参数选择、2-d/3-d 空间效应的参与、边界条件的配置等方面的困难，Fisher 模型在数值模拟传染病方面的应用远不如 SIR 模型。本文旨在解决这些问题，提供了涉及空间和时间有限差分方案和迭代方法的数值算法，以及用于求解费希尔模型的开源 Python 代码。这个二维费雪求解器在空间上是二阶的，在时间上也达到了二阶，这一点通过分析求解的测试案例得到了可靠验证。为了提高求解器的效率和精度，我们将 SOR、隐式欧拉、交错曲柄-尼科尔森和 ADI 等数值算法结合在一起。它可以处理不同物理描述下的各种边界条件。此外，该模型还使用了 Covid-19 的实际数据，以证明其在提供预测和推论方面的实用性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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arXiv - QuanBio - Quantitative Methods

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