We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connexions between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a selection of numerical simulations for a selection of connected graphs.

中文翻译：

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连通图上流行病传播的动力学

我们提出了一个新模型，该模型描述了连通图上流行病传播的动力学。我们的模型包含一个PDE-ODE系统，其中在图的每个顶点处都有一个标准的SIR模型，并且顶点之间的连接由边上的热方程给出，并在顶点处用Robin边界条件进行了补充，在顶点处建模了入射边和顶点之间的交换。关联的顶点。我们描述了系统的主要特性，并得出了感染个体的最终总数。我们提出了一种基于空间有限差分的半隐式时间数值方案，该方案保留了连续模型的主要属性，例如解的唯一性和正性以及总种群的守恒。