Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 1825 SW Broadway, Portland, OR, 97201, USA.
Natural Science Division, Pepperdine University, 24255 E Pacific Coast Highway, Malibu, CA, 90263, USA.
Bull Math Biol. 2024 Oct 1;86(11):132. doi: 10.1007/s11538-024-01364-3.
There is extensive evidence that network structure (e.g., air transport, rivers, or roads) may significantly enhance the spread of epidemics into the surrounding geographical area. A new compartmental modeling framework is proposed which couples well-mixed (ODE in time) population centers at the vertices, 1D travel routes on the graph's edges, and a 2D continuum containing the rest of the population to simulate how an infection spreads through a population. The edge equations are coupled to the vertex ODEs through junction conditions, while the domain equations are coupled to the edges through boundary conditions. A numerical method based on spatial finite differences for the edges and finite elements in the 2D domain is described to approximate the model, and numerical verification of the method is provided. The model is illustrated on two simple and one complex example geometries, and a parameter study example is performed. The observed solutions exhibit exponential decay after a certain time has passed, and the cumulative infected population over the vertices, edges, and domain tends to a constant in time but varying in space, i.e., a steady state solution.
有大量证据表明，网络结构（如航空运输、河流或道路）可能显著促进传染病在周围地理区域的传播。提出了一种新的房室模型框架，该框架将混合良好的（时间上的 ODE）人口中心（位于顶点处）、图的边缘上的 1D 旅行路线以及包含其余人口的 2D 连续统耦合起来，以模拟传染病如何在人群中传播。通过连接条件将边缘方程与顶点 ODE 耦合，而通过边界条件将域方程与边缘耦合。描述了一种基于边缘空间有限差分和 2D 域有限元的数值方法来逼近模型，并提供了方法的数值验证。该模型在两个简单示例和一个复杂示例几何图形上进行了说明，并进行了参数研究示例。观察到的解在经过一定时间后呈指数衰减，顶点、边缘和域上的累积感染人数随时间趋于常数，但在空间上变化，即稳态解。
