Inference of causality in epidemics on temporal contact networks.

Alfredo Braunstein,Alessandro Ingrosso

doi:10.1038/srep27538

Abstract

Investigating into the past history of an epidemic outbreak is a paramount problem in epidemiology. Based on observations about the state of individuals, on the knowledge of the network of contacts and on a mathematical model for the epidemic process, the problem consists in describing some features of the posterior distribution of unobserved past events, such as the source, potential transmissions, and undetected positive cases. Several methods have been proposed for the study of these inference problems on discrete-time, synchronous epidemic models on networks, including naive Bayes, centrality measures, accelerated Monte-Carlo approaches and Belief Propagation. However, most traced real networks consist of short-time contacts on continuous time. A possibility that has been adopted is to discretize time line into identical intervals, a method that becomes more and more precise as the length of the intervals vanishes. Unfortunately, the computational time of the inference methods increase with the number of intervals, turning a sufficiently precise inference procedure often impractical. We show here an extension of the Belief Propagation method that is able to deal with a model of continuous-time events, without resorting to time discretization. We also investigate the effect of time discretization on the quality of the inference.

Highlights

Investigating into the past history of an epidemic outbreak is a paramount problem in epidemiology
We simulated a large number of epidemic propagations, each one initiated from a unique random source
The nodes in the network are ranked in decreasing order of their estimated posterior probability of being the origin of the observed epidemics: the position of the true origin in the ranking provided by the algorithm is a good measure of the efficacy of the method

Summary

Introduction

Investigating into the past history of an epidemic outbreak is a paramount problem in epidemiology. This can be done by sub-dividing the time line into subintervals of length Δ (time-steps), aggregating all contacts falling in a given interval [tΔ,(t + 1)Δ] into a time-step dependent weight λijt equal a single quasi-instantaneous contact and kt the number of to1−(1−λ)kt, where λ is the probability of transmission in contacts the interval[8,6] Once this discrete time network has been constructed, the spread of infectious diseases on the community can be described through a discrete time SIR model, in which the transition probabilities between states defining each of these models depend on the time-step t. We will describe a very simple semi-continuous time stochastic model of infection dynamics that does not require coarsening or binning and is naturally equivalent to the

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Conclusion