On network-based epidemiological models: Analysis, simulations, and continuum limit

This work focuses on network epidemic models that emphasise the role of social contact networks within a given population, coupled with biological processes governing the spread of infectious diseases. In particular, we address the modelling and analysis of disease dynamics on large networks. As a foundational epidemiological framework, we consider the SEIR (Susceptible-Exposed-Infectious-Removed) model, which describes the transmission of infectious diseases among individuals partitioned into subpopulations. We investigate the long-term behaviour of this model while accounting for heterogeneity in infections and social interactions. Applying the theory of graphons, we explore the natural question of the large-population limit and analyze the model's behaviour as the network size approaches infinity. After establishing the existence and uniqueness of solutions to the selected models, we derive analytical results concerning the spectral properties of epidemiological networks. We also present preliminary numerical experiments. The proposed framework can be readily extended to other compartmental models in epidemiology. Given its ability to incorporate heterogeneous interconnections among individuals, it provides a natural approach for studying epidemic phenomena. Moreover, this framework can be further adapted to dynamic networks, including cases where biological parameters vary over time.