Exact solutions and bounds for network SIR and SEIR models using a rooted-tree approximation

TL;DR

This paper introduces an exact node-based model for network SIR and SEIR contagion dynamics on trees, providing explicit solutions and bounds, advancing understanding of epidemic spread on network structures.

Contribution

The paper proves the exactness of a new approximate model for SIR and SEIR on trees and extends analysis to complex SEIR models with multiple exposed classes.

Findings

Exact solutions for SIR on chains.

Upper bounds for node susceptibility probabilities.

Linear differential equations describing node-state evolution.

Abstract

In this paper, we develop a node-based approximate model for Markovian contagion dynamics on networks. We prove that our approximate model is exact for SIR (susceptible-infectious-recovered) and SEIR (susceptible-exposed-infectious-recovered) dynamics on tree graphs with a single source of infection and that the model otherwise gives upper bounds on the probabilities of each node being susceptible. Our analysis of SEIR contagion dynamics is generalised to SEIR models with arbitrarily many distinct classes of exposed state. In the case of trees with a single source of infection, our approach yields a system of partially-decoupled linear differential equations that exactly describes the evolution of node-state probabilities. We use this to state explicit closed-form solutions for SIR dynamics on a chain.