  Why are the epidemic patterns of COVID-19 so different among different cities
or countries which are similar in their populations, medical infrastructures,
and people's behavior? Why are forecasts or predictions made by so-called
experts often grossly wrong, concerning the numbers of people who get infected
or die? The purpose of this study is to better understand the stochastic nature
of an epidemic disease, and answer the above questions. Much of the work on
infectious diseases has been based on "SIR deterministic models," (Kermack and
McKendrick:1927.) We will explore stochastic models that can capture the
essence of the seemingly erratic behavior of an infectious disease. A
stochastic model, in its formulation, takes into account the random nature of
an infectious disease.
  The stochastic model we study here is based on the "birth-and-death process
with immigration" (BDI for short), which was proposed in the study of
population growth or extinction of some biological species. The BDI process
model ,however, has not been investigated by the epidemiology community. The
BDI process is one of a few birth-and-death processes, which we can solve
analytically. Its time-dependent probability distribution function is a
"negative binomial distribution" with its parameter $r$ less than $1$. The
"coefficient of variation" of the process is larger than $\sqrt{1/r} > 1$.
Furthermore, it has a long tail like the zeta distribution. These properties
explain why infection patterns exhibit enormously large variations. The number
of infected predicted by a deterministic model is much greater than the median
of the distribution. This explains why any forecast based on a deterministic
model will fail more often than not.
