Bayesian inference of epidemics on networks via Belief Propagation (1307.6786v2)

Abstract: We study several bayesian inference problems for irreversible stochastic epidemic models on networks from a statistical physics viewpoint. We derive equations which allow to accurately compute the posterior distribution of the time evolution of the state of each node given some observations. At difference with most existing methods, we allow very general observation models, including unobserved nodes, state observations made at different or unknown times, and observations of infection times, possibly mixed together. Our method, which is based on the Belief Propagation algorithm, is efficient, naturally distributed, and exact on trees. As a particular case, we consider the problem of finding the "zero patient" of a SIR or SI epidemic given a snapshot of the state of the network at a later unknown time. Numerical simulations show that our method outperforms previous ones on both synthetic and real networks, often by a very large margin.

• The paper develops novel BP equations to approximate infection and recovery times even with partial observations.
• It demonstrates superior performance over traditional methods through simulations on both synthetic and real-world networks.
• The approach integrates diverse observation models, offering a robust tool for epidemic tracing and public health decision-making.

Bayesian Inference of Epidemics on Networks via Belief Propagation

In the paper of epidemic modeling on networks, inferring the origins and dynamics of contagion spreads is crucial for both public health decision-making and theoretical understanding. The paper "Bayesian inference of epidemics on networks via Belief Propagation" introduces a novel approach using Bayesian inference and graphical models to tackle this problem. Specifically, it leverages Belief Propagation (BP), a powerful algorithm from statistical physics, to compute the posterior distribution of the epidemic state at each node given partial observations.

Overview of the Approach

The research focuses on stochastic epidemic models, particularly the Susceptible-Infected-Recovered (SIR) and Susceptible-Infected (SI) models, defined over networks. These models allow the paper of the time evolution of epidemics, with nodes transitioning through states—susceptible, infected, and recovered—based on probabilistic processes. The authors develop a method to infer the initial state distribution (including identifying the "zero patient") by integrating BP into the Bayesian inference framework, accommodating very general observation models including variable or incomplete observations.

The main contribution is the derivation of BP equations that accurately approximate the probability of infection and recovery times conditioned on observed states, allowing the determination of the epidemic's origin and parameters even in scenarios with unobserved nodes or unknown timing.

Technical Details

The paper formulates the inference problem by expressing the posterior probability distribution of the initial state given the final observed state using Bayes' theorem. The infection and recovery times across nodes are modeled as a graphical factor model, where factors represent probabilistic transmissions and transitions between states. BP is applied to this factor graph to estimate marginals efficiently.

What distinguishes this method from prior approaches is its ability to incorporate diverse forms of observation data, such as state observations at unknown times or observations with mixed data types. Moreover, while exact on tree structures, BP provides a robust approximation for general graphs, showing superior performance compared to previous methods like Dynamic Message Passing (DMP) and centrality measures.

Numerical Simulations and Results

Extensive numerical simulations on both synthetic and real-world network data demonstrate the efficiency and accuracy of the BP-based inference. The method outperforms traditional approaches in identifying epidemic seeds, especially in complex network topologies, achieving higher probabilities of correct inference and providing lower ranks for true epidemic origins.

The paper also addresses scenarios with incomplete information—critical in practical applications—such as unobserved nodes and unknown epidemic age. BP remains effective under these conditions, even when traditional methods falter, proving the robustness of this approach in realistic and noisy environments.

Implications and Future Directions

The implications of this research are twofold. Practically, it offers a potent tool for public health authorities and epidemiologists to trace epidemic origins, estimate the spread dynamics, and evaluate intervention strategies. Theoretically, it enhances understanding of epidemic processes on complex networks, contributing to the fields of computational epidemiology and network science.

Future work may explore extensions to more complex models, including multi-stage diseases or adaptive networks with changing topologies—a scenario increasingly relevant in digital contact tracing and dynamic social behaviors. Additionally, further exploration into parameter estimation within heterogeneous networks could enrich the practicality of this method.

In summary, this paper presents a rigorous and computationally efficient approach to epidemic inference on networks, expanding the toolkit available for tackling these pressing problems in both theoretical and applied contexts.