This article provides a very clear review of some of the individual-centric models used to describe the evolution of an epidemic, comparing the assumptions made in the case of the current COVID-19 epidemic. It provides numerous references of articles that have studied the different models presented here, and shows the robustness of the predictions made by the simple models, as well as the assumptions required to observe variations from these predictions.
A classic method for constructing an individual-centric model of epidemic transmission is a random graph, the peaks of which are the individuals and the edges of the links involving contamination. Many simple models propose to use a non-oriented graph as a connection graph, suggesting a symmetry of the contamination capacities of individuals. This is not necessarily the case for some types of contamination, which is why directed graph models have also been proposed recently.
Based on a review of existing results on epidemic dynamics on graphs, directed or not, the authors show that predictions of simple SIR-type models can be preserved or not by the hypothesis of edge direction. They also observe the different implications of contact tracing methods on these oriented graphs, noting a difference between upward tracing (identification of contaminants) and downward tracing (identification of contaminants).
The results obtained in this article are based on a model of an epidemic by an individual-centred model represented by a graph. Each node of the graph represents an individual, and each (directed) stop of the graph represents a possible contamination of one individual (the target) if the other (the source) is contaminated. Using standard Galton-Watson enumeration and process methods, the authors calculate the R0 parameter of each model, the probability of the epidemic spreading (probability of survival of the target tree of a typically contaminated node in the first generation), and the size of the epidemic if it spreads (probability of survival of the source tree). They then compare the results obtained with the values given by the SIR model.
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