基于分数阶反应扩散方程的传染病模型研究
Research on Epidemic Model Based on Fractional Reaction Diffusion Equation
DOI: 10.12677/PM.2021.117149, PDF,   
作者: 霍俊蓉：沈阳师范大学，数学与系统科学学院，辽宁 沈阳；张荣培：广东工业大学，应用数学学院，广东 广州
关键词: 分数阶反应扩散方程；有限差分法；传染病模型；图灵斑图；Fractional Reaction Diffusion Equation； Finite Difference Method； Integral Factor Method； Turing Pattern
摘要: 反应扩散方程在物理学、化学、医学及生物学等多个领域的不同模型中有广泛应用。分数阶反应扩散型方程是整数阶的推广，被广泛应用于研究传染病的传播过程、图像分析以及随机过程。本文采用分数阶反应扩散方程来模拟传染病模型的动态行为。首先，通过求解齐次Neumann边界条件下分数阶反应扩散方程的特征值，对其进行线性稳定性分析，得到图灵不稳定的条件。然后结合Kronecker积用有限差分法以及积分因子法在空间上以及时间上对反应扩散方程组进行离散并求解。最后，给出数值实验验证稳定性分析结果。本文的数值结果有助于预测传染病的流行趋势。
Abstract: Reaction diffusion equation is widely used in different models of physics, chemistry, medicine and biology. Fractional reaction diffusion equation is a generalization of integer order, which is widely used in the study of infectious disease propagation process, image analysis and stochastic process. In this paper, the fractional reaction-diffusion equation is used to simulate the dynamic behavior of the epidemic model. Firstly, by solving the eigenvalues of the fractional reaction-diffusion equation with homogeneous Neumann boundary conditions, the linear stability of the equation is analyzed, and the Turing instability condition is obtained. Then, combined with Kronecker product, the reac-tion-diffusion equations are discretized and solved by finite difference method and integration factor method in space and time. Finally, numerical experiments are given to verify the stability analysis results. The numerical results are helpful to predict the epidemic trend of infectious diseases.
文章引用：霍俊蓉, 张荣培. 基于分数阶反应扩散方程的传染病模型研究[J]. 理论数学, 2021, 11(7): 1326-1334. https://doi.org/10.12677/PM.2021.117149

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