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Discrete SIR model on a homogeneous tree and its continuous limit

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. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.