d
2 V
dt 2 ¼ a
2 SI S À I À m
ð
Þ
ð 4:27Þ
According to (4.27) the maximum of daily cases increase corresponds to the value
S s ¼ I s þ m
ð4:28Þ
where corresponding value I s may be calculated from (4.8):
I s ¼ m ln S s À S s þ N À m ln N À 1
ð
Þ
ð4:29Þ
Because function S decreases monotonically (see (4.1)), and the maximum of
I corresponds to S ¼ m, Eq. (4.29) shows that the daily number of new cases starts
to decline before the maximum number of the infected (and spreading) persons is
achieved. This conclusion is very important, because watching a new epidemic, we
record the daily number of new cases or the accumulated number of cases. If the
daily amount begins to decline, it does not mean that the number of carriers of the
infection is also declining and quarantine can be relaxed. To estimate the corresponding time difference t max À t s , we continue the analysis of the system of
Eqs. (4.1)–(4.3).
Equations (4.28) and (4.29) yield the following nonlinear equation for S s :
S s ¼
1
2
m þ m ln
S s
N À 1
þ N
ð4:30Þ
After solving (4.30), the point in time t s can be found by integration (4.1)
t s ¼ t
Ã
1 À
1
a
Z S s
NÀ1
dU
U m ln U À U þ N À m ln N À 1
ð
Þ
½
ð4:31Þ
Equations (4.24), (4.25), (4.30), (4.31) allow estimating the time difference
t max À t s , but values of the SIR parameters have to be calculated first.
Important epidemic characteristics—the basic and effective reproduction numbers—can be also estimated with the use of SIR model. The effective reproduction
number R t (t) shows the average number of people infected by one person [79]. In
terms of SIR model, it can be estimated as follows:
R t t
ð Þ ¼
I t þ 0:5s
ð
ÞÀI t À 0:5s
ð
Þ
I
¼
1
I
Z
t þ 0:5s
tÀ0:5s
dI t
Ã
ð Þ
dt à dt
Ã
%
s
I
dI t
ð Þ
dt
ð4:32Þ
4 Classical SIR Model and the Exact …
31
2 V
dt 2 ¼ a
2 SI S À I À m
ð
Þ
ð 4:27Þ
According to (4.27) the maximum of daily cases increase corresponds to the value
S s ¼ I s þ m
ð4:28Þ
where corresponding value I s may be calculated from (4.8):
I s ¼ m ln S s À S s þ N À m ln N À 1
ð
Þ
ð4:29Þ
Because function S decreases monotonically (see (4.1)), and the maximum of
I corresponds to S ¼ m, Eq. (4.29) shows that the daily number of new cases starts
to decline before the maximum number of the infected (and spreading) persons is
achieved. This conclusion is very important, because watching a new epidemic, we
record the daily number of new cases or the accumulated number of cases. If the
daily amount begins to decline, it does not mean that the number of carriers of the
infection is also declining and quarantine can be relaxed. To estimate the corresponding time difference t max À t s , we continue the analysis of the system of
Eqs. (4.1)–(4.3).
Equations (4.28) and (4.29) yield the following nonlinear equation for S s :
S s ¼
1
2
m þ m ln
S s
N À 1
þ N
ð4:30Þ
After solving (4.30), the point in time t s can be found by integration (4.1)
t s ¼ t
Ã
1 À
1
a
Z S s
NÀ1
dU
U m ln U À U þ N À m ln N À 1
ð
Þ
½
ð4:31Þ
Equations (4.24), (4.25), (4.30), (4.31) allow estimating the time difference
t max À t s , but values of the SIR parameters have to be calculated first.
Important epidemic characteristics—the basic and effective reproduction numbers—can be also estimated with the use of SIR model. The effective reproduction
number R t (t) shows the average number of people infected by one person [79]. In
terms of SIR model, it can be estimated as follows:
R t t
ð Þ ¼
I t þ 0:5s
ð
ÞÀI t À 0:5s
ð
Þ
I
¼
1
I
Z
t þ 0:5s
tÀ0:5s
dI t
Ã
ð Þ
dt à dt
Ã
%
s
I
dI t
ð Þ
dt
ð4:32Þ
4 Classical SIR Model and the Exact …
31