Early epidemic spread, percolation and Covid-19
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Journal of Mathematical Biology (2020) 81:1143–1168 https://doi.org/10.1007/s00285-020-01539-1
Early epidemic spread, percolation and Covid-19 Gonçalo Oliveira1 Received: 18 June 2020 / Revised: 18 June 2020 / Published online: 18 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Human to human transmissible infectious diseases spread in a population using human interactions as its transmission vector. The early stages of such an outbreak can be modeled by a graph whose edges encode these interactions between individuals, the vertices. This article attempts to account for the case when each individual entails in different kinds of interactions which have therefore different probabilities of transmitting the disease. The majority of these results can be also stated in the language of percolation theory. The main contributions of the article are: (1) Extend to this setting some results which were previously known in the case when each individual has only one kind of interactions. (2) Find an explicit formula for the basic reproduction number R0 which depends only on the probabilities of transmitting the disease along the different edges and the first two moments of the degree distributions of the associated graphs. (3) Motivated by the recent Covid-19 pandemic, we use the framework developed to compute the R0 of a model disease spreading in populations whose trees and degree distributions are adjusted to several different countries. In this setting, we shall also compute the probability that the outbreak will not lead to an epidemic. In all cases we find such probability to be very low if no interventions are put in place.
Contents 1 Introduction . . . . . . . . . . . . . 2 Generating functions . . . . . . . . . 2.1 From the generating vertex . . . 2.2 By following a random infection 3 The basic reproduction number . . . 4 Containing an outbreak . . . . . . . 4.1 The case when R0 < 1 . . . . . 4.2 The case when R0 > 1 . . . . . 4.3 The case when R0 = 1 . . . . . 4.4 Lower bounds on P . . . . . . .
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Gonçalo Oliveira [email protected] Universidade Federal Fluminense, Rio de Janeiro, Brazil
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1144 5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Delta and Poisson . . . . . . . . . . . . . . . . . . . . . . . 5.2 Polynomial and Poisson I . . . . . . . . . . .
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