Exact equations for SIR epidemics on tree graphs

TL;DR

This paper derives exact equations for SIR epidemic models on tree graphs, providing a precise and computationally feasible way to predict infection dynamics in such networks.

Contribution

It proves that a specific pair-based moment closure accurately captures the expected infectious time series on acyclic networks, offering a new analytical tool.

Findings

Exact equations derived for SIR on tree graphs

Pair-based closure matches expected infectious time series

Method is straightforward to evaluate numerically

Abstract

We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this ``deterministic'' representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.