and the time of cases duplication:
s d ¼
ln 2
c
ð2:4Þ
To estimate the values of parameters c and b, we can treat the values
y j ¼ logðV j Þ and corresponding time moments t j as random variables and use the
observations of the accumulated number of cases (e.g., presented in Table 1.1)
and the linear regression [33] in order to calculate the coefficients c
_ and b
_
of the
regression line
y
_ ¼ c
_ t þ b
_
ð2:5Þ
using the standard formulas (see, e.g., [33]):
c
_ ¼
n
P n
j¼1 y j t j À
P n
j¼1 y j
P n
j¼1 t j
n
P n
j¼1 t 2
j À
P n
j¼1 t j
2
;
ð2:6Þ
b
_ ¼
P n
j¼1 y j À c
_ P n
j¼1 t j
n
ð2:7Þ
Here n is the number of observations.
Values c
_ and b
_
can be treated as statistics-based estimations of parameters c and
b from relationships (2.1) or (2.3). The reliability of this estimation can be checked
by calculating the correlation coefficient r:
r ¼
n
P n
j¼1 y j t j À
P n
j¼1 y j
P n
j¼1 t j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n
P n
j¼1 y 2
j À
P n
j¼1 y j
2
n
P n
j¼1 t 2
j À
P n
j¼1 t j
2
s
ð2:8Þ
If the values of r
j j are close to unit, there is a strong linear relationship between
variables y and t.
We can use also the F-test for the null hypothesis that says that the proposed
linear relationship (2.3) fits the data set. The experimental value of the Fisher
function can be calculated with the use of the formula:
F ¼
r
2
ðn À mÞ
ð1 À r 2 Þðm À 1Þ
ð2:9Þ
8
2 Early Stages of Epidemics and Exponential Growth