Small-coupling dynamic cavity: A Bayesian mean-field framework for epidemic inference

A novel generalized mean-field approximation, called the small-coupling dynamic cavity (SCDC) method, for epidemic inference and risk assessment is presented. The method is developed within a fully Bayesian framework and accounts for noncausal effects generated by the presence of observations. It is based on a graphical model representation of the epidemic stochastic process and utilizes dynamic cavity equations to derive a set of self-consistent equations for probability marginals defined on the edges of the contact graph. By performing a small-coupling expansion, a pair of time-dependent cavity messages is obtained, which capture the probability of individual infection and the conditioning power of observations. In its efficient formulation, the computational cost per iteration of the SCDC algorithm is linear in the duration of the epidemic dynamics. While the method is derived for the susceptible-infected (SI) model, it is straightforwardly applicable to many other Markovian epidemic processes, including recurrent ones. This linear complexity is particularly advantageous for recurrent epidemic processes, where inference methods are typically exponentially complex in the duration of the epidemic dynamics. The method exhibits high accuracy in assessing individual risk, as demonstrated by tests on the SI model applied to various classes of synthetic contact networks, where it performs on par with belief propagation techniques and generally exceeds the performance of individual-based mean-field methods. Additionally, the method was applied to recurrent epidemic models, where it showed interesting performance even for relatively large values of the infection probability, highlighting its versatility and effectiveness in challenging scenarios. Although convergence issues may arise due to long-range correlations in contact graphs, the estimated marginal probabilities remain sufficiently accurate for reliable risk estimation. Future work includes extending the method to non-Markovian recurrent epidemic models and investigating the role of second-order terms in the small-coupling expansion of the observation-reweighted dynamic cavity equations.