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Discrete SIR model on a homogeneous tree and its continuous limit

Authors
  • Gairat, Alexander
  • Shcherbakov, Vadim
Type
Published Article
Journal
Journal of Physics A: Mathematical and Theoretical
Publisher
IOP Publishing
Publication Date
Oct 28, 2022
Volume
55
Issue
43
Identifiers
DOI: 10.1088/1751-8121/ac9655
Source
ioppublishing
Keywords
Disciplines
  • Fundamental Approaches Towards Predictive Epidemic Modelling
License
Unknown

Abstract

We study a discrete susceptible–infected–recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution.

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