Epidemic Inference in Metapopulation Models: optimization algorithm through forward-backward propagation

The epidemic inference allows us to make predictions on the evolution of an epidemic to develop containment measures or infer the infection channels and the origin of the epidemic. In this thesis work, we discuss a metapopulation structure approach to stochastic inference, where the total population is partitioned in many subpopulations which interact with each other. We assume to receive daily information about the aggregate number of infected individuals at the single population level. The information from such time-scattered observations together with some prior information can be used to develop a Bayesian framework to address different epidemic inference problems such as to predict the future evolution of the outbreak (epidemic forecast), to infer the current state of the epidemics in unobserved populations (risk assessment), or to infer the past state of the epidemics (causal paths, patient zero). To address these problems, we provide a common Bayesian framework to compute the joint probability of the overall history 𝓗 of the metapopulation system given a set of observations 𝓞. We derive the prior distribution P[𝓗] associated with the stochastic metapopulation system, as well as the likelihood of the data P[𝓞|𝓗], which is derived from a simple Gaussian sampling process for the observations. The prior distribution is formulated using a Path Integral approach to the underlying stochastic process. By applying a saddle-point method, we derive a set of deterministic equations for the epidemic state variables and their associated conjugate fields. These equations are integrated using a forward-backward algorithm, which efficiently identifies the most probable epidemic trajectory consistent with the observed data. The numerical results demonstrate a strong agreement between the inferred trajectories and ground truth simulations, validating the effectiveness of the proposed framework. Additionally, due to numerical challenges and the complexity of the underlying optimization landscape, we implement gradient-based optimization methods, which enhance convergence and allow for more stable and accurate inference.