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

TL;DR

This paper models epidemic spread on large social contact networks using graphons, analyzing the long-term behavior, heterogeneity, and continuum limits, with theoretical results and preliminary simulations.

Contribution

It introduces a graphon-based framework for analyzing large-scale epidemic models, including existence, uniqueness, and continuum limit results.

Findings

Existence and uniqueness of solutions established.

Large population limit analyzed via graphons.

Preliminary numerical tests demonstrate model applicability.

Abstract

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.