A Graph-Based Modelling of Epidemics: Properties, Simulation, and Continuum Limit

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 that considers the heterogeneity of the infections and the social network. By relying on the theory of graphons we explore the natural question of the large population limit and investigate the behaviour of the model as the size of the network tends to infinity. After establishing the existence and uniqueness of solutions to the models that we will consider, we discuss the possibility of using the graphon-based limit model as a generative model for a network with particular statistical properties relating to the distribution of connections. We also provide some preliminary numerical tests.