This work is motivated by an original dataset of reported mumps cases across nine regions of England, and focuses on the modeling of temporal dynamics and time-varying dependency patterns between the observed time series. The goal is to discover the possible presence of latent routes of contagion that go beyond the geographical locations of the regions, and instead may be explained through other non directly observable socio-economic factors. We build upon the recent statistics literature and extend the existing count time series network models by adopting a time-varying latent distance network model. This approach can efficiently capture across-series and across-time dependencies, which are both not directly observed from the data. We adopt a Bayesian hierarchical framework and perform parameter estimation using L-BFGS optimization and Hamiltonian Monte Carlo. We demonstrate with several simulation experiments that the model parameters can be accurately estimated under a variety of realistic dependency settings. Our real data application on mumps cases leads to a detailed view of some possible contagion routes. A critical advantage of our methodology is that it permits clear and interpretable visualizations of the complex relations between the time series and how these relations may evolve over time. The geometric nature of the latent embedding provides useful model based summaries. In particular, we show how to extract a measure of contraction of the inferred latent space, which can be interpreted as an overall risk for the escalation of contagion, at each point in time. Ultimately, the results highlight some possible critical transmission pathways and the role of key regions in driving infection dynamics, offering valuable perspectives that may be considered when designing public health strategies.

翻译：本研究受英格兰九个地区报告的腮腺炎病例原始数据集启发，重点在于对观测时间序列之间的时间动态和时变依赖模式进行建模。目标是探索可能存在的、超越地区地理位置的潜在传染途径，这些途径或许可以通过其他非直接可观测的社会经济因素来解释。我们基于近期统计学文献，通过采用时变潜在距离网络模型，扩展了现有的计数时间序列网络模型。该方法能够有效捕捉序列间与时间上的依赖关系，这两者均无法直接从数据中观测到。我们采用贝叶斯分层框架，并利用L-BFGS优化算法和哈密顿蒙特卡洛方法进行参数估计。通过多项模拟实验，我们证明该模型参数能够在多种现实依赖设置下被准确估计。对腮腺炎病例实际数据的应用，揭示了一些潜在传染途径的详细图景。本方法的关键优势在于，它能够对时间序列间的复杂关系及其随时间演变的过程提供清晰且可解释的可视化呈现。潜在嵌入的几何特性提供了基于模型的有用摘要。特别地，我们展示了如何提取推断潜在空间的收缩度量，该度量可被解释为每个时间点上传染升级的总体风险。最终，结果突显了一些可能的关键传播路径以及核心区域在驱动感染动态中的作用，为公共卫生策略的设计提供了值得考量的宝贵视角。