This work is concerned with epidemiological models defined on networks, which highlight the prominent role of the social contact network of a given population in the spread of infectious diseases. In particular, we address the modelling and analysis of very large networks. As a basic epidemiological model, we focus on a SEIR (Susceptible-Exposed-Infective-Removed) model governing the behaviour of infectious disease among a population of individuals, which is partitioned into sub-populations. We study the long-time behaviour of the dynamic for this model, also taking into account the heterogeneity of the infections and the social network. By relying on the theory of graphons, we address the natural question of the large population limit and investigate the behaviour of the model as the size of the network tends to infinitely. After establishing the existence and uniqueness of solutions to the selected models, we discuss the use of the graphon-based limit model as a generative model for a network with particular statistical properties related to the distribution of connections. We also provide some preliminary numerical tests.

翻译：本文研究定义在网络上的流行病模型，这些模型凸显了给定人群的社会接触网络在传染病传播中的关键作用。我们特别关注超大规模网络的建模与分析。作为基础流行病模型，我们聚焦于描述个体间传染病行为的SEIR（易感-潜伏-感染-移除）模型，其中人群被划分为若干子群体。我们研究该模型动力系统的长期行为，同时考虑了感染异质性与社会网络的影响。借助图论理论，我们探讨了大规模人群极限这一自然问题，并分析了当网络规模趋于无穷大时模型的行为。在建立选定模型解的存在唯一性后，我们讨论了基于图极限模型作为生成模型的应用，该模型可生成具有特定连接分布统计特性的网络。最后，我们提供了初步数值测试结果。