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] 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 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 : 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 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): 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 * * ( , , , , ) ( ) 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: 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 * * ( , , , , ) 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. 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. ; 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. 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 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. Comparison of the coronavirus pandemic dynamics in Ukraine and neighboring countries Long-term predictions for COVID-19 pandemic dynamics in Ukraine, Austria and Italy Як довго українці сидітимуть на карантині? How long will the Ukrainians stay in quarantine? SIR-simulation of Corona pandemic dynamics in Europe Динаміка COVID-19 епідемії в Україні та Києві після покращання тестування. COVID-19 epidemic dynamics in Ukraine and Kyiv after testing has improved Corona pandemic global stabilization? GLOBAL STABILIZATION TRENDS OF COVID-19 PANDEMIC Simulations and predictions of COVID-19 pandemic with the use of SIR model Hidden periods, duration and final size of COVID-19 pandemic Статистика пандемії COVID-19 в Україні та світі COVID-19 pandemic statistics in Ukraine and world Coronasummer in Ukraine and Austria Waves of COVID-19 pandemic. Detection and SIR simulations Study of the COVID-19 epidemic dynamics in Ukraine and its regions by methods of probabilistic and statistical analysis of evolutionary models New waves of COVID-19 pandemic in Ukraine. (In Ukrainian) COVID19 pandemic dynamics -Mathematical simulations Coronavirus disease (COVID-2019) situation reports A Contribution to the mathematical theory of epidemics Mathematical Biology I/II Comparison of mathematical models for the dynamics of the Chernivtsi children disease Statistics based models for the dynamics of Chernivtsi children disease Statistics-based predictions of coronavirus epidemic spreading in mainland China Applied regression analysis Borysenko and Oleksii Rodionov for their support and help in collecting and processing data.