  This paper conducts a systematic statistical analysis of the characteristics
of the geographical empirical distributions for the numbers of both cumulative
and daily confirmed COVID-19 cases and deaths at county, city, and state levels
over a time span from January 2020 to June 2022. The mathematical heavy-tailed
distributions can be used for fitting the empirical distributions observed in
different temporal stages and geographical scales. The estimations of the shape
parameter of the tail distributions using the Generalized Pareto Distribution
also support the observations of the heavy-tailed distributions. According to
the characteristics of the heavy-tailed distributions, the evolution course of
the geographical empirical distributions can be divided into three distinct
phases, namely the power-law phase, the lognormal phase I, and the lognormal
phase II. These three phases could serve as an indicator of the severity degree
of the COVID-19 pandemic within an area. The empirical results suggest
important intrinsic dynamics of a human infectious virus spread in the human
interconnected physical complex network. The findings extend previous empirical
studies and could provide more strict constraints for current mathematical and
physical modeling studies, such as the SIR model and its variants based on the
theory of complex networks.
