Discrete SIR model on a homogeneous tree and its continuous limit

Alexander Gairat, Vadim Shcherbakov

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This paper concerns a stochastic Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree. The paper consists of two parts. First, we study the distribution of the time it takes for a susceptible vertex to get infected. Specifically, we derive an exact analytical expression for this distribution in terms of a solution of a non-linear integral equation. This result is obtained under general assumptions on both the infection rate and recovery time. Namely, the infection rate can be time-dependent, and recovery time can be given by a random variable with an arbitrary distribution. Then we study the model in the limit when the vertex degree of the tree tends to infinity, and the 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. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.
Original languageEnglish
Article number434004
Number of pages20
JournalJournal of Physics A: Mathematical and Theoretical
Issue number43
Early online date29 Sept 2022
Publication statusPublished - 2 Nov 2022


  • SIR model, Bernoulli equation, homogeneous tree, logistic curve, basic reproduction number

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