Ukraine increased much more intensively (see blue “stars” in Fig. 6.16). Probably,
this indicates that the cancelation of the national lockdown was carried out too
early. May 10, 2020, corresponded to the maximum of non-isolated infected persons in Ukraine (see the blue dotted line in Fig. 6.16). The violation of quarantine
by Ukrainian authorities and population caused much more intensive increase in the
number of cases than in Austria.
To control the changes of epidemic parameters, we can use daily or weekly
numbers of new cases and their derivatives. Since these values are random, we need
some smoothing (especially for daily amounts, which are also characterized by
some weekly periodicity). For example, we can use the smoothed daily number of
accumulated cases:
V i ¼
1
7
X
j¼i þ 3
j¼iÀ3
V j ;
ð8:1Þ
or weekly number of cases w i (WHO reported only these values after August 15).
The first and second derivatives can be estimated with the use of following
formulas:
d
V
dt
t¼t i
%
1
2
V i þ 1 À
V iÀ1
ð
Þ
ð 8:2Þ
d
2
V
dt 2
t¼t i
%
V i þ 1 À 2
V i þ
V iÀ1
ð8:3Þ
dw
dt
t¼t i
%
1
14
w i þ 1 À w iÀ1
ð
Þ
ð 8:4Þ
d
2 w
dt 2
t¼t i
%
1
49
w i þ 1 À 2w i þ w iÀ1
ð
Þ
ð 8:5Þ
Coefficients 1/14 and 1/49 have appeared in formulas (8.4) and (8.5) due to the
fact that we will always measure time in days. For the first epidemic waves,
Eqs. (8.2) and (8.4) yield estimations of dV/dt (see (4.11) and (4.12)); Eqs. (8.3)
and (8.5) allow estimating d
2 V/dt
2 (see (4.26) and (4.27)). In particular, second
derivatives (8.3) and (8.5) have to change their signs from positive to negative at the
time moment t s , which can be calculated with the use of (4.31). Thus, the minimums of distributions (8.2) and (8.4) and zero points of (8.3) and (8.5) can be
treated as moments when the parameters of epidemic change. We will analyze the
distributions (8.2)–(8.5) for some countries and global statistics as well.
As usual, we will use the WHO official accumulated number of cases. Some data
sets are shown in many tables from previous chapters. In addition, some recent data
sets for daily (V j ) and weekly (w j ) accumulated number of cases in Ukraine and in
110
8 Identification of the New Waves of the COVID-19 Pandemic
this indicates that the cancelation of the national lockdown was carried out too
early. May 10, 2020, corresponded to the maximum of non-isolated infected persons in Ukraine (see the blue dotted line in Fig. 6.16). The violation of quarantine
by Ukrainian authorities and population caused much more intensive increase in the
number of cases than in Austria.
To control the changes of epidemic parameters, we can use daily or weekly
numbers of new cases and their derivatives. Since these values are random, we need
some smoothing (especially for daily amounts, which are also characterized by
some weekly periodicity). For example, we can use the smoothed daily number of
accumulated cases:
V i ¼
1
7
X
j¼i þ 3
j¼iÀ3
V j ;
ð8:1Þ
or weekly number of cases w i (WHO reported only these values after August 15).
The first and second derivatives can be estimated with the use of following
formulas:
d
V
dt
t¼t i
%
1
2
V i þ 1 À
V iÀ1
ð
Þ
ð 8:2Þ
d
2
V
dt 2
t¼t i
%
V i þ 1 À 2
V i þ
V iÀ1
ð8:3Þ
dw
dt
t¼t i
%
1
14
w i þ 1 À w iÀ1
ð
Þ
ð 8:4Þ
d
2 w
dt 2
t¼t i
%
1
49
w i þ 1 À 2w i þ w iÀ1
ð
Þ
ð 8:5Þ
Coefficients 1/14 and 1/49 have appeared in formulas (8.4) and (8.5) due to the
fact that we will always measure time in days. For the first epidemic waves,
Eqs. (8.2) and (8.4) yield estimations of dV/dt (see (4.11) and (4.12)); Eqs. (8.3)
and (8.5) allow estimating d
2 V/dt
2 (see (4.26) and (4.27)). In particular, second
derivatives (8.3) and (8.5) have to change their signs from positive to negative at the
time moment t s , which can be calculated with the use of (4.31). Thus, the minimums of distributions (8.2) and (8.4) and zero points of (8.3) and (8.5) can be
treated as moments when the parameters of epidemic change. We will analyze the
distributions (8.2)–(8.5) for some countries and global statistics as well.
As usual, we will use the WHO official accumulated number of cases. Some data
sets are shown in many tables from previous chapters. In addition, some recent data
sets for daily (V j ) and weekly (w j ) accumulated number of cases in Ukraine and in
110
8 Identification of the New Waves of the COVID-19 Pandemic