Network Structures, Manifold Learning and Disease Management

Infectious diseases continue to pose a significant global health challenge. Mathematical modeling has emerged as a critical tool for understanding disease dynamics and informing public health interventions. This paper provides a comprehensive overview of modelling infectious disease epidemics on networks, progressing from the simple Erdős-Rényi random graph to more complex structures. We delve into the intricate relationship between network topology and disease transmission in the setting of graph structures embedded on a manifold. A particular emphasis is placed on the connection between random graph theory, percolation theory, and dynamical systems, providing a robust theoretical framework for analyzing disease spread. Furthermore, the paper addresses the complexities introduced by networks with a high density of short closed loops, which can significantly impact disease dynamics. By examining these factors, we aim to contribute to the development of more accurate and effective models for predicting and controlling infectious disease outbreaks. We see a potential model to make informed decisions regarding targeted public health interventions, optimize resource allocation, and ultimately mitigate the impact of future epidemics.