The Covid-19 pandemic has provided many modeling challenges to investigate, evaluate, and understand various novel unknown aspects of epidemic processes and public health intervention strategies. This paper develops a model for the disease infection rate that can describe the spatio-temporal variations of the disease dynamic when dealing with small areal units. Such a model must be flexible, realistic, and general enough to describe jointly the multiple areal processes in a time of rapid interventions and irregular government policies. We develop a joint Poisson Auto-Regression model that incorporates both temporal and spatial dependence to characterize the individual dynamics while borrowing information among adjacent areas. The dependence is captured by two sets of space-time random effects governing the process growth rate and baseline, but the specification is general enough to include the effect of covariates to explain changes in both terms. This provides a framework for evaluating local policy changes over the whole spatial and temporal domain of the study. Adopted in a fully Bayesian framework and implemented through a novel sparse-matrix representation in Stan, the model has been validated through a substantial simulation study. We apply the model on the weekly Covid-19 cases observed in the different local authority regions in England between May 2020 and March 2021. We consider two alternative sets of covariates: the level of local restrictions in place and the value of the \textit{Google Mobility Indices}. The model detects substantial spatial and temporal heterogeneity in the disease reproduction rate, possibly due to policy changes or other factors. The paper also formalizes various novel model based investigation methods for assessing aspects of disease epidemiology.

翻译：COVID-19疫情为研究、评估和理解流行病过程中各种未知方面及公共卫生干预策略带来了诸多建模挑战。本文开发了一个能够描述疾病感染率的模型，可应对小区域单元中疾病动态的时空变化。该模型需具备灵活性、现实性和通用性，以在快速干预和不规则政府政策时期联合描述多个区域过程。我们构建了一个联合泊松自回归模型，该模型融合了时间和空间依赖性，以表征个体动态，同时借用相邻区域的信息。依赖性通过两组控制过程增长率和基线水平的时空随机效应来捕捉，但模型规格足够通用，可包含协变量效应以解释这两个项的变化。这为在整个研究时空域评估局部政策变化提供了一个框架。模型采用完全贝叶斯框架，通过斯坦（Stan）中一种新颖的稀疏矩阵表示实现，并经过大量模拟研究验证。我们将该模型应用于2020年5月至2021年3月期间英格兰不同地方当局地区每周观察到的COVID-19病例数据。我们考虑了两组替代协变量：当地限制措施级别和谷歌移动指数（Google Mobility Indices）的值。模型检测到疾病繁殖率存在显著的时空异质性，这可能是由于政策变化或其他因素所致。本文还形式化了多种基于模型的新颖研究方法，用于评估疾病流行病学的各个方面。