maximum in daily number of new cases corresponds to zero value of the second
derivative d
2 V/dt
2 . Differentiation of (9.11) yields:
d
2 V
dt 2 ¼ a i S
dI
dt
þ I
dS
dt
ð9:21Þ
With the use of (9.1) and (9.2), Eq. (9.21) can be rewritten as follows:
d
2 V
dt 2 ¼ a
2
i SI S À I À m i
ð
Þ
ð 9:22Þ
According to (9.22), the maximum of the new daily cases corresponds to the value:
S s ¼ I s þ m i
ð9:23Þ
(provided it occurs between moments of time t
Ã
i and t ie ). Corresponding value of I s
may be calculated from (9.8):
I s ¼ m i ln S s À S s þ N i À R i À m i lnðN i À I i À R i Þ
ð 9:24Þ
Because function S decreases monotonically (see (9.1)), and the maximum of
I corresponds to S ¼ m i , Eq. (9.23) 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.
Let us estimate the time difference t max À t s (provided both time moments occur
between t
Ã
i and t ie ). Equations (9.23) and (9.24) yield the following nonlinear
equation for S s :
S s ¼
1
2
m i þ m i ln
S s
N i À R i À I i
þ N i À R i
ð9:25Þ
After solving (9.25), the point in time t s can be found by integration of (9.1)
t s ¼ t
Ã
i À
1
a
Z S s
N i ÀI i ÀR i
dU
U m i ln U À U þ N i À R i À m i lnðN i À I i À R i Þ
½
ð9:26Þ
In Eq. (9.26), the initial condition (9.5) and Eq. (9.8) were used. Formulas (9.20)
and (9.26) allow estimating the time difference t max À t s , but values of the SIR
parameters have to be calculated first.
9 General SIR Model and Its Exact Solution
131
derivative d
2 V/dt
2 . Differentiation of (9.11) yields:
d
2 V
dt 2 ¼ a i S
dI
dt
þ I
dS
dt
ð9:21Þ
With the use of (9.1) and (9.2), Eq. (9.21) can be rewritten as follows:
d
2 V
dt 2 ¼ a
2
i SI S À I À m i
ð
Þ
ð 9:22Þ
According to (9.22), the maximum of the new daily cases corresponds to the value:
S s ¼ I s þ m i
ð9:23Þ
(provided it occurs between moments of time t
Ã
i and t ie ). Corresponding value of I s
may be calculated from (9.8):
I s ¼ m i ln S s À S s þ N i À R i À m i lnðN i À I i À R i Þ
ð 9:24Þ
Because function S decreases monotonically (see (9.1)), and the maximum of
I corresponds to S ¼ m i , Eq. (9.23) 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.
Let us estimate the time difference t max À t s (provided both time moments occur
between t
Ã
i and t ie ). Equations (9.23) and (9.24) yield the following nonlinear
equation for S s :
S s ¼
1
2
m i þ m i ln
S s
N i À R i À I i
þ N i À R i
ð9:25Þ
After solving (9.25), the point in time t s can be found by integration of (9.1)
t s ¼ t
Ã
i À
1
a
Z S s
N i ÀI i ÀR i
dU
U m i ln U À U þ N i À R i À m i lnðN i À I i À R i Þ
½
ð9:26Þ
In Eq. (9.26), the initial condition (9.5) and Eq. (9.8) were used. Formulas (9.20)
and (9.26) allow estimating the time difference t max À t s , but values of the SIR
parameters have to be calculated first.
9 General SIR Model and Its Exact Solution
131