COVID-19 outbreaks in China in late December 2019, then in the United States (US) in early 2020. In the initial wave of diffusion, the virus respectively took 14 and 33 days to spread across the provinces/states in the Chinese mainland and the coterminous US, during which there are 43％ and 70％ zero entries in the space-time series for China and US respectively, indicating a zero-inflated count process. A logistic growth curve as a function of the number of days since the first case appeared in each of these countries accurately portrays the national aggregate per capita rates of infection for both. This paper presents two space-time model specifications, one based upon the generalized linear mixed model, and the other upon Moran eigenvector space-time filtering, to describe the spread of COVID-19 in the initial 19 and 58 days across the Chinese mainland and the coterminous US, respectively. Results from these case studies show both models shed new light on the role of spatial structures in COVID-19 diffusion, models that can forecast new cases in subsequent days. A principal finding is that describing the spatio- temporal diffusion of COVID-19 benefits from including a hierarchical structural component to supplement the commonly employed contagion component.