where m = 2 is the number of parameters in the regression equation, [33]. The
corresponding experimental value F has to be compared with the critical value
F C ðk 1 ; k 2 Þ of the Fisher function at a desired significance or confidence level
(k 1 ¼ m À 1, k 2 ¼ n À m; see, e.g., [34]). This approach was successfully used in
[35] in order to find the duplication rate for the Chernivtsi children disease. Similar
statistics-based approach was applied in [36, 37] to link animal speeds with tail beat
amplitude, frequency and body length.
In [38], this method was used to calculate COVID-19 cases duplication time for
European region, USA, and the world. The corresponding accumulated numbers of
cases used for calculations and comparisons are shown in Table 2.1 and Fig. 2.1
(“circles”). Since in March 2020, the situations in mainland China and the Republic
of Korea were stable (the number of new cases was low), and the estimations of
global pandemic prospects have been done without taking into account the cases in
these two regions. The corresponding numbers V Gj are shown in the last column of
Table 2.1 and in Fig. 2.1 (magenta “circles”). The number of cases V Ej and V Uj is
also shown in Fig. 2.1 (brown and red “circles,” respectively). It can be seen that
V Ej , V Uj and V Gj numbers follow straight lines in the logarithmic scale. It means that
the epidemic dynamics in these regions was still exponential and far from stabilization (in comparison with Italy—“blue” circles).
To estimate the slopes of these lines, the linear regression for the values
logðV Yj Þ; Y ¼ E; U; G was used (see, (2.3)). The corresponding best fitting dashed
lines are shown in Fig. 2.1. To avoid influence of the initial epidemic outbreak
stages, calculations have been done only for V Yj [ 100; Y ¼ E; U; repeating values
were used only once. It can be seen that the most rapid was the epidemic in USA
(the number of cases duplicated every 2.31 days). The European and global
duplications rates were 2.91 and 3.65 days, respectively. Without stabilization by
mid-April 2020 (22 days after March 24), we could have in Europe 41 million
cases, 38 millions in USA and 211 millions globally. Fortunately, the tendencies
changed in April 2020, and these sad predictions became no more relevant.
To illustrate these stabilization tends, we will use some results from [39–41].
The accumulated number of cases confirmed in March and April 2020 is shown in
Figs. 2.2, 2.3, 2.4 and 2.5. We use the logarithmic scale to detect the periods of
exponential growth which is typical for initial stage of every epidemic. Figure 2.2
demonstrates the pandemic dynamics in the European region, USA and in the
world. It can be seen that in April 2020 the European and global dynamics (without
cases in mainland China and the Republic of Korea) and the number of cases in
USA showed deviation from the straight lines, i.e., the pandemic was stabilizing.
Many countries in Europe demonstrated clear stabilization. For example, we can
see it in Fig. 2.3, where the dynamics in Austria and its neighboring countries is
shown. Unfortunately, the number of cases in Italy, Spain, Germany, France, and
the UK was already higher than in China; in Turkey, Austria, and Switzerland it
was higher than in South Korea (see the horizontal lines in Fig. 2.2). In Slovenia,
Hungary, and Slovakia, the final number of cases is expected to be rather low. It
looks that quarantine measures and isolation of infected persons were very effective
2 Early Stages of Epidemics and Exponential Growth
9
corresponding experimental value F has to be compared with the critical value
F C ðk 1 ; k 2 Þ of the Fisher function at a desired significance or confidence level
(k 1 ¼ m À 1, k 2 ¼ n À m; see, e.g., [34]). This approach was successfully used in
[35] in order to find the duplication rate for the Chernivtsi children disease. Similar
statistics-based approach was applied in [36, 37] to link animal speeds with tail beat
amplitude, frequency and body length.
In [38], this method was used to calculate COVID-19 cases duplication time for
European region, USA, and the world. The corresponding accumulated numbers of
cases used for calculations and comparisons are shown in Table 2.1 and Fig. 2.1
(“circles”). Since in March 2020, the situations in mainland China and the Republic
of Korea were stable (the number of new cases was low), and the estimations of
global pandemic prospects have been done without taking into account the cases in
these two regions. The corresponding numbers V Gj are shown in the last column of
Table 2.1 and in Fig. 2.1 (magenta “circles”). The number of cases V Ej and V Uj is
also shown in Fig. 2.1 (brown and red “circles,” respectively). It can be seen that
V Ej , V Uj and V Gj numbers follow straight lines in the logarithmic scale. It means that
the epidemic dynamics in these regions was still exponential and far from stabilization (in comparison with Italy—“blue” circles).
To estimate the slopes of these lines, the linear regression for the values
logðV Yj Þ; Y ¼ E; U; G was used (see, (2.3)). The corresponding best fitting dashed
lines are shown in Fig. 2.1. To avoid influence of the initial epidemic outbreak
stages, calculations have been done only for V Yj [ 100; Y ¼ E; U; repeating values
were used only once. It can be seen that the most rapid was the epidemic in USA
(the number of cases duplicated every 2.31 days). The European and global
duplications rates were 2.91 and 3.65 days, respectively. Without stabilization by
mid-April 2020 (22 days after March 24), we could have in Europe 41 million
cases, 38 millions in USA and 211 millions globally. Fortunately, the tendencies
changed in April 2020, and these sad predictions became no more relevant.
To illustrate these stabilization tends, we will use some results from [39–41].
The accumulated number of cases confirmed in March and April 2020 is shown in
Figs. 2.2, 2.3, 2.4 and 2.5. We use the logarithmic scale to detect the periods of
exponential growth which is typical for initial stage of every epidemic. Figure 2.2
demonstrates the pandemic dynamics in the European region, USA and in the
world. It can be seen that in April 2020 the European and global dynamics (without
cases in mainland China and the Republic of Korea) and the number of cases in
USA showed deviation from the straight lines, i.e., the pandemic was stabilizing.
Many countries in Europe demonstrated clear stabilization. For example, we can
see it in Fig. 2.3, where the dynamics in Austria and its neighboring countries is
shown. Unfortunately, the number of cases in Italy, Spain, Germany, France, and
the UK was already higher than in China; in Turkey, Austria, and Switzerland it
was higher than in South Korea (see the horizontal lines in Fig. 2.2). In Slovenia,
Hungary, and Slovakia, the final number of cases is expected to be rather low. It
looks that quarantine measures and isolation of infected persons were very effective
2 Early Stages of Epidemics and Exponential Growth
9