TL;DR
This paper reviews mathematical models of infectious disease spread on networks, exploring various network structures and their impact on epidemic dynamics, linking graph theory, percolation, and dynamical systems.
Contribution
It systematically analyzes epidemic modeling on networks, incorporating complex structures like correlations and heterogeneity, and discusses challenges with networks having many short loops.
Findings
Network structure significantly influences epidemic spread.
Correlations and heterogeneity affect transmission dynamics.
Networks with many short loops pose modeling challenges.
Abstract
Infectious disease remains, despite centuries of work to control and mitigate its effects, a major problem facing humanity. This paper reviews the mathematical modelling of infectious disease epidemics on networks, starting from the simplest Erdos-Renyi random graphs, and building up structure in the form of correlations, heterogeneity and preference, paying particular attention to the links between random graph theory, percolation and dynamical systems representing transmission. Finally, the problems posed by networks with a large number of short closed looks are discussed.
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