A message passing approach for general epidemic models

TL;DR

This paper introduces a message passing framework for generalized epidemic models with arbitrary transmission and recovery time distributions, providing exact results on trees and bounds on complex networks.

Contribution

It develops a novel message passing approach for non-exponential epidemic models, extending analysis beyond traditional differential equation methods.

Findings

Exact on trees and locally tree-like networks

Provides bounds on outbreak sizes for non-tree networks

Matches well with numerical simulations

Abstract

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the SIR (susceptible-infected-recovered) model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally tree-like networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.