The effective reproduction number R ti (t) shows the average number of people
infected by one person during the ith epidemic wave. Similar to (4.32), it can be
estimated as:
R ti ðtÞ ¼
Iðt þ 0:5s i Þ À Iðt À 0:5s i Þ
I
¼
1
I
Z
t þ 0:5s i
tÀ0:5s i
dIðt
Ã
Þ
dt à dt
Ã
%
s i
I
dIðtÞ
dt
ð9:27Þ
where s i is the average time of spreading the infection and can be estimated with the
use of
s i %
1
q i
ð9:28Þ
With the use of (9.2), (9.7) and (9.28), Eq. (9.27) can be rewritten as follows:
R ti ðtÞ %
S
m i
À 1
ð9:29Þ
The basic reproduction number R 0i is used for the initial periods of an epidemic
outbreak or for initial periods of consequent epidemic waves when the exponential
growth of the number of cases occurs. This characteristic can be estimated with the
use of equation similar to (4.33):
R 0i ¼ exp
s i ln 2
s d
ð9:30Þ
where s d is the doubling time (the period during which the number of cases
duplicates).
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9 General SIR Model and Its Exact Solution
infected by one person during the ith epidemic wave. Similar to (4.32), it can be
estimated as:
R ti ðtÞ ¼
Iðt þ 0:5s i Þ À Iðt À 0:5s i Þ
I
¼
1
I
Z
t þ 0:5s i
tÀ0:5s i
dIðt
Ã
Þ
dt à dt
Ã
%
s i
I
dIðtÞ
dt
ð9:27Þ
where s i is the average time of spreading the infection and can be estimated with the
use of
s i %
1
q i
ð9:28Þ
With the use of (9.2), (9.7) and (9.28), Eq. (9.27) can be rewritten as follows:
R ti ðtÞ %
S
m i
À 1
ð9:29Þ
The basic reproduction number R 0i is used for the initial periods of an epidemic
outbreak or for initial periods of consequent epidemic waves when the exponential
growth of the number of cases occurs. This characteristic can be estimated with the
use of equation similar to (4.33):
R 0i ¼ exp
s i ln 2
s d
ð9:30Þ
where s d is the doubling time (the period during which the number of cases
duplicates).
132
9 General SIR Model and Its Exact Solution