The role of directionality, heterogeneity and correlations in epidemic risk and spread.

Modelization Infectiology
Allard A et al

Main result

The authors observe that Kermack-McKendrick's predictions about the relationship between the proportion of the population affected by an epidemic and the R0 parameter of that epidemic are not necessarily valid in a directed or undirected random graph model. A sufficient condition for the validity of this prediction is :
 - in the unsupervised case, that the number of neighbors per node is distributed according to a Poisson's law,
 - in the directed case, that the number of sources and targets of a node are independent, and that the number of sources follows a Poisson's law.
Finally, the authors note that classical structured SIR models correspond to non-directed random graph models, such that each node has a type, and the number of neighbors of a given node is distributed according to a Poisson distribution whose parameter depends on the type. In this case, the tree of origins becomes a multitype Galton-Watson process, and we find the Kermack-McKendrick predictions, by replacing the "mean" R0 parameter by the largest eigenvalue of the matrix associated with the process


It shows that the simple epidemic size predictions obtained from the SIR model remain valid only for some of the directed graph models.

Strength of evidence Weak

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.


This article compares the evolution of an epidemic for different classes of oriented random graphs.


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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