key: cord-0783879-0wvkuecp
authors: Nesteruk, I.
title: COVID-19 pandemic dynamics in Ukraine after September 1, 2020
date: 2020-12-22
journal: nan
DOI: 10.1101/2020.12.21.20248627
sha: b7f522cab3743594261589e04ad1200dbb521b8f
doc_id: 783879
cord_uid: 0wvkuecp

Background. The threats of the COVID-19 pandemic require the mobilization of scientists, including mathematicians. To understand how the number of cases increases versus time, various models based on direct observations of a random number of new cases and differential equations can be used. Complex mathematical models contain many unknown parameters, the values of which must be determined using a limited number of observations of the disease over time. Even long-term monitoring of the epidemic may not provide reliable estimates of its parameters due to the constant change of testing conditions, isolation of infected and quarantine. Therefore, simpler approaches should also be used, for example, some smoothing of the dependence of the number of cases on time and the known SIR (susceptible-infected-removed) model. These approaches allowed to detect the waves of pandemic in different countries and regions and to make adequate predictions of the duration, hidden periods, reproduction numbers, and final sizes of its waves. In particular, seven waves of the COVID-19 pandemic in Ukraine were investigated. Objective. We will detect new epidemic waves in Ukraine that occurred after September 1, 2020 and estimate the epidemic characteristics with the use of generalized SIR model. Some predictions of the epidemic dynamics will be presented. Methods. In this study we use the smoothing method for the dependence of the number of cases on time; the generalized SIR model for the dynamics of any epidemic wave, the exact solution of the linear differential equations and statistical approach developed before. Results. Seventh and eights epidemic waves in Ukraine were detected and the reasons of their appearance were discussed. The optimal values of the SIR model parameters were calculated. The prediction for the COVID-19 epidemic dynamics in Ukraine is not very optimistic: new cases will not stop appearing until June 2021. Only mass vaccination and social distancing can change this trend. Conclusions. New waves of COVID-19 pandemic can be detected, calculated and predicted with the use of rather simple mathematical simulations. The expected long duration of the pandemic forces us to be careful and in solidarity.The government and all Ukrainians must strictly adhere to quarantine measures in order to avoid fatal consequences.

The studies of the COVID-19 pandemic dynamics in Ukraine are presented in [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] and summarized in the book [15] . Different simulation and comparison methods together with the official WHO data sets, [16] were used. In particular the classical SIR model [17] [18] [19] , connecting the number of susceptible S , infected and spreading the infection I and removed R persons, was applied in [2] [3] [4] [5] [8] [9] [10] to simulate the first pandemic wave in Ukraine. The unknown parameters of this model were be estimated with the use of the cumulative number of cases V=I+R and the statistics-based method of parameter identification developed in [20, 21] .

The weakening of quarantine restrictions, changes in the social behavior and probably in the coronavirus activity causes change in SIR characteristics and the epidemic dynamics. To detect these changes, a simple method of numerical differentiations of accumulated number of cases was proposed in [11, 15] . To simulate these new pandemic waves, the SIR model was generalized in [12, 15] . In [12, 14, 15] the results of simulation of the first six epidemic waves in Ukraine are presented with the use of a procedure for sequentially determining the parameters of the model for each epidemic wave, starting with the first one.

This method requires considerable effort and time. The book [15] introduced a new algorithm for determining the optimal parameter values for a particular epidemic wave without calculating the dynamics of previous waves and presented calculations for the seventh epidemic wave in Ukraine.

In this paper, we will analyze the dynamics of the epidemic in Ukraine in the period from September 1 to December 20, 2020, calculate the parameters of the eighth wave and make some predictions.

The official information regarding the accumulated numbers of confirmed COVID-19 cases V j in Ukraine according to the official sources [22, 23] is shown in Table 1 . Unfortunately, WHO stopped to present the daily number of new cases in August 2020. The corresponding moments of time t j (measured in days, zero point is January 20, 2020) are also shown in this table. To calculate the SIR characteristics for the seventh epidemic wave in Ukraine, the data set for the period October 1-14 was used in [15] . We will use the period November 21 -December 4, 2020 to calculate the characteristics of the eighth epidemic wave. Other values presented in Table 1 1  225  125798  8  262  250538  14  299  535857   2  226  128228  9  263  256266  15  300  545689   3  227  130951  10  264  261034  16  301  557657   4  228  133787  11  265  265454  17  302  570153   5  229  135894  12  266  270587  18  303  583510   6  230  138068  13  267  276177  19  304  598085   7  231  140479  14  268  281239  20  305  612665   8  232  143030  15  269  287231  21  306  624744   9  233  145612  16  270  293641  22  307  635689  10  234  148756  17  271  298872  23  308  647976   11  235  151859  18  272  303638  24  309  661858   12  236  154335  19  273  309107  25  310  677189   13  237  156797  20  274  315826  26  311  693407   14  238  159702  21  275  322879  27  312  709701   15  239  162660  22  276  330396  28  313  722679  16  240  166244  23   277  337410  29   314  732625   17  241  169472  24  278  343498  30  315  745123  18  242  172712  25  279  348924  1  316  758264   19  243  175678  26  280  355601  2  317  772760   20  244  178353  27  281  363075  3  318  787891   21  245  181237  28  282  370417  4  319  801716   22  246  184734  29  283  378729  5  320  813306   23  247  188106  30  284  387481  6  321  821947   24   248  191671  31   285  395440   7  322  832758   25  249  195504  1  286  402194  8  323  845343   26  250  198634  2  287  411093  9  324  858714   27  251  201305  3  288  420617  10  325  872228   28  252  204932  4  289  430467  11  326  885039   29  253  208959  5  290  440188  12  327  894215   30  254  213028  6  291  450934  13  328  900666  1  255  217661  7  292  460331  14  329  909082   2  256  222322  8  293  469018  15  330  919704   3  257  226462  9  294  479197  16  331  931751   4  258  230236  10  295  489808  17  332  944381   5  259  234584  11  296  500865  18  333  956123   6  260  239337  12  297  512652  19  334  964448  7  261  244734  13  298  525176  20 335 970993 preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted December 22, 2020. ;

It must be noted that the data presented in Table 1 does not show all the COVID-19 cases in Ukraine. Many infected persons are not identified, since they have no symptoms. Many people know that they are ill, since they have similar symptoms as other members of families, but avoid to make tests. Unfortunately, one laboratory confirmed case can correspond to several other cases which are not confirmed and displayed in the official statistics. This fact reduces the accuracy of mathematical simulations.

To control the changes of epidemic parameters, we can use daily numbers of new cases and their derivatives. Since these values are random, we need some smoothing. For example, we can use the smoothed daily number of accumulated cases proposed in [11, 12, 14, 15] :

The first and second derivatives can be estimated with the use of following formulas:

The classical SIR model for an infectious disease [17] [18] [19] was generalized in [12, 15] 

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perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in 

must be constant for every wave and is not the volume of population.

To determine the initial conditions for the set of equations (4)-(6), let us suppose that at the beginning of every epidemic wave * i t :

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perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint It follows from (4) and (5) that

Integration of (9) with the initial conditions (8) yields:

It follows from (9) 

Formula (11) follows from (10) at I=0.

In [12, 15] the set of differential equations (4)- (7) was solved by introducing the function

corresponding to the number of victims or the cumulative confirmed number of cases. For many epidemics (including the COVID-9 pandemic) we cannot observe dependencies ( ), ( ) S t I t and ( ) R t but observations of the accumulated number of cases V j corresponding to the moments of time t j provide information for direct assessments of the dependence ( ) V t .

It follows from (5) and (6) that:

Eqs. (7), (10) and (13) yield:

Integration of (14) provides an analytical solution for the set of equations (4)- (6):

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perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint * * ( , , , , ) ( )

Thus, for every set of parameters , , , , (16) can be calculated and the corresponding moment of time can be determined from (15) . Then functions I(t)

and R(t) can be easily calculated with the use of formulas (10) and:

The final numbers of victims (final accumulated number of cases corresponding to the i-th epidemic wave) can be calculated from:

To estimate the final day of the i-th epidemic wave, we can use the condition:

which means that at if t t  less than one person still spreads the infection.

In the case of a new epidemic, the values of its parameters are unknown and must be identified with the use of limited data sets. For the first wave of an epidemic starting with one infected person, the number of unknown parameters is only four, since 1 1 I  and 1 0 R  . The corresponding statistical approach was proposed in [20, 21] and used [2] [3] [4] [5] [8] [9] [10] to simulate the first COVID-19 pandemic wave in Ukraine.

For the next epidemic waves (i > 1), the moments of time * Eq. (15) can be rewritten as follows:

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perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint * * ( , , , , ) 

using the standard formulas from, e.g., [24] . Values   and   can be treated as statistics-based estimations of parameters  and  from relationships (21) .

The reliability of the method can be checked by calculating the correlation coefficients r i (see e.g., [24] ) for every epidemic wave checking how close its value is to unity. We can use also the Ftest for the null hypothesis that says that the proposed linear relationship (20) fits the data set. The experimental values of the Fisher function can be calculated for every epidemic wave with the use of the formula: perpetuity.

preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in

The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint epidemic waves i = 1,2,3 ..., it is possible to avoid determining the four optimal unknown parameters , , , 17). This approach has been successfully used in [12, 14, 15] . In particular, six waves of the Covid-19 epidemic in Ukraine and four pandemic waves in the world were calculated.

Segmentation of epidemic waves and their sequential SIR simulations need a lot of efforts. To avoid this, a new method of obtaining the optimal values of SIR parameters was proposed in [15] .

First of all we can use the relationship

which follows from (12) . To estimate the value i V , we can use the smoothed accumulated number of cases (e.g., formula (1)). Then

where i corresponds to the moment of time * i t . To obtain one more relationship, let us use (7) and (13)

To estimate the average number of new cases dV/dt at the moment of time * i t , we can use (2). Thus we have only two independent parameters i N and i  . To calculate the value of parameter i  , some iterations can be used (see details in [15] ).

The COVID-19 pandemic characteristics for Ukraine in autumn 2020 are shown in Fig. 1 .

Differentiation of the smoothed number of accumulated cases (eq. (1), line) with the use of All rights reserved. No reuse allowed without permission.

perpetuity.

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The copyright holder for this this version posted December 22, 2020. ; formulas (2) ("triangles") and (3) (stars) allow us to detect the changes in epidemic dynamics. It can be seen that after November 15 the average daily number of new cases ("triangles") started to decrease. Similar short periods of the epidemic stabilization occurred in May, June and August, 2020 (see [11, 12, 14, 15] ). Let us hope that the Christmas and New Year celebrations will not significantly worsen the existing positive dynamics. Fig. 1 . Pandemic dynamics in Ukraine after September 1, 2020. Accumulated number of cases (V j -"cicrles", Table 1 ; smoothed values -line, eq. (1)). "Triangles" show the first derivative (eq. (2)) multiplied by 100, "stars" -the second derivative (eq. (3) ) multiplied by 1000. Table 1 .

The severe jumps in d 2 V/dt 2 values occurred also in October and November, 2020 (see "stars" in Fig. 1) . Probably, this is due to the local elections and a presidential poll, which were held throughout Ukraine on October 25, 2020 and involved hundreds of thousands of people to campaign and work in election commissions (their number was about 30 thousand). This obviously increased the number of contacts and the likelihood of additional infections. The corresponding All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 22, 2020. ; seventh epidemic wave in Ukraine was considered in [15] (the results for the first six waves can be found in [12, 14, 15] ). The optimal values of SIR parameters; predictions of the epidemic wave duration if t and its final size i V  are shown in Table 2 . The characteristics of the eights wave (occurred after November 21, 2020) were calculated and presented in Table 2 . It can be seen that the predictions for Ukraine are very pessimistic. The number of cases will exceed 1.07 million and new cases will appear even in June 2021.

Sixth wave, I=6, n=14, [14, 15] Seventh wave, i=7, n=14, [15] Eighth wave, i=8, n=14

Period taken for perpetuity.

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The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint accumulated number of cases V j taken for calculations are shown by circles; "triangles" and "stars" correspond the cases taken only for comparisons and verifications of the predictions. The absence of sharp jumps of the second derivative after November 21, 2020 (see "stars" in Fig. 1 ), allowed us to predict quite accurately the further epidemic dynamics (compare solid blue line and "stars" in Fig.2 ). Blue dashed line in Fig.2 show that the number of persons spreading the infection diminished in December 2020. Let us hope that the Christmas and New Year celebrations will not significantly worsen this positive dynamics. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 22, 2020. ; https://doi.org/10.1101/2020.12.21.20248627 doi: medRxiv preprint

Constant changes in the Covid-19 pandemic conditions (i.e., in the peculiarities of quarantine and its violation, in situations with testing and isolation of patients, in coronavirus activity due to its mutations etc.) lead to new pandemic waves and cause the changes in the values of parameters of the mathematical models. To identify these changes, a simple method was proposed based on the numerical differentiation of smoothed dependences of the accumulated number of cases. The results

show that smoothing dependence of the accumulated number of cases and its differentiation can provide fairly accurate and useful information about the course of the epidemic, identify important changes in its dynamics and provide timely recommendations for quarantine measures or control of social distancing.

To simulate different pandemic waves (periods with more or less constant values of its dynamics parameters) a new generalized SIR model and its exact solution have been proposed.

Procedure of its parameters identification were developed and successfully applied to calculate the characteristics of several pandemic waves in Ukraine.

The pandemic duration projections are unfortunately not optimistic. We hope that effective quarantine measures and successful vaccination will be able to reverse this trend. Very long duration of the pandemic requires correction of our behavior, we can not live as before it occurred. 3. If you (or others) have any suspicious symptoms, do your best to avoid the spread of the infection.

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Borysenko and Oleksii Rodionov for their support and help in collecting and processing data.