key: cord-0861157-7zo2l8wf authors: Nesteruk, Igor title: SIR-simulation of Corona pandemic dynamics in Europe date: 2020-04-25 journal: nan DOI: 10.1101/2020.04.22.20075135 sha: 3d5ccbf00fe55daa84c75763f3c6df76128664db doc_id: 861157 cord_uid: 7zo2l8wf The SIR (susceptible-infected-removed) model, statistical approach to the parameter identification and the official WHO daily data about the confirmed cumulative number of cases were used to estimate the characteristics of COVID-19 pandemic in Italy, Spain, Germany, France, Austria and Moldova. The final sizes and durations of epidemic outbreaks in these countries are calculated. 7697 199 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The official data about the accumulated numbers of confirmed COVID-19 cases V j in Italy, Spain , Germany, France, Austria and in the republic of Moldova from WHO daily situation reports (numbers 33-87), [1] are presented in Tables 1 and 2 . The corresponding moments of time t j ( measured in days) are also shown in these tables. The data sets presented in Table 1 were used only for comparison with corresponding SIR curves. Table 2 29 36 97689 78797 57298 39642 8813 263 30 37 101739 85195 61913 43977 9618 298 31 38 105792 94417 67366 51477 10182 353 1 39 110574 102136 73522 56261 10711 423 2 40 115242 110238 79696 58327 11129 591 3 41 119827 117710 85778 63536 11525 591 4 42 124632 124736 91714 67757 11766 752 5 43 128948 130759 95391 69607 11983 864 6 44 132547 135032 99225 73488 12297 965 7 45 135586 140510 103228 77226 12640 1056 8 46 139422 146690 108202 81095 12969 1174 9 47 143626 152446 113525 85351 13248 1289 10 48 147577 157022 117658 89683 13560 The SIR model for an infectious disease [2] [3] [4] [5] relates the number of susceptible persons S (persons who are sensitive to the pathogen and not protected); the number of infected is I (persons All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075135 doi: medRxiv preprint who are sick and spread the infection; please don't confuse with the number of still ill persons, so known active cases) and the number of removed R (persons who no longer spread the infection; this number is the sum of isolated, recovered, dead, and infected people who left the region);  and  are constants. To determine the initial conditions for the set of equations (1-3), let us suppose that at the moment of the epidemic outbreak 0 t , [5, 6] : The analytical solution for the set of equations (1-3) was obtained by introducing the function , corresponding to the number of victims or cumulative confirmed number of cases, [5, 6] : Thus, for every set of parameters N,  , , 0 t and a fixed value of V the integral (6) can be calculated and the corresponding moment of time can be determined from (5) . and R(t) can be easily calculated with the of formulas, [5, 6] . Function I has a maximum at S   and tends to zero at infinity, see [2, 3] . In comparison, the number of susceptible persons at infinity 0 S   , and can be calculated from the non-linear equation, [5, 6] : The final number of victims (final accumulated number of cases) can be calculated from: To estimate the duration of an epidemic outbreak, we can use the condition: All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. which means that at final t t  less than one person still spread the infection. In the case of a new epidemic, the values of this independent four parameters are unknown and must be identified with the use of limited data sets. A statistical approach was developed in [5] and used in [6] [7] [8] [9] [10] [11] [12] to estimate the values of unknown parameters. The registered points for the number of victims V j corresponding to the moments of time t j can be used in order to calculate for every fixed values N and  with the use of (6) and then to check how the registered points fit the straight line (5) . For this purpose the linear regression can be used, e.g., [13] , and the optimal straight line, minimizing the sum of squared distances between registered and theoretical points, can be defined. Thus we can find the optimal values of  , 0 t and calculate the correlation coefficient r . Then the F-test may be applied to check how the null hypothesis that says that the proposed linear relationship (5) fits the data set. The experimental value of the Fisher function can be calculated with the use of the formula: where n is the number of observations, m=2 is the number of parameters in the regression equation, [13] . The corresponding experimental value F has to be compared with the critical value Usually the number of cases during the initial period of an epidemic outbreak is not reliable. To avoid their influence on the results, only V j values for the period March 28 -April 10, 2020 ( 35 48 j t   ; n=14; (1, 2) C F n =18.6; see Table 2 ) were used to calculate the epidemic characteristics in every country. Since during the quarantine, the international people exchange is quite limited, we can apply the SIR model for every country assuming its parameters to be constant All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. Table 5 . Italy, prediction 5 Spain, prediction 1 Germany Optimal values of parameters, final sizes and durations (last two rows). All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. . The SIR curves and markers representing the V j values taken for calculations ("circles"); for comparisons ("triangles") and verifications of predictions ("stars") are shown in Figs. 1-6. For Italy, Spain, France and Moldova, the second sets of optimal parameters (from Table 5 ) were used to calculate SIR curves. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. . It can be seen that the previous predictions for Italy and Austria [11] were more optimistic. Fresh data sets has showed that the final number of cases in Italy could reach 226,000 and their appearance can stop only after August 18, 2020 (see Table 5 , prediction 6). The epidemic stop in Austria is expected after May 21, 2020 (see Table 4 ). These estimations are valid only when the quarantine measures, isolation rate and the coronavisus activity will be same as for the periods taken for calculations. versus time. Numbers of infected I (green), removed R (black) and victims V=I+R (blue). [11] . In particular, according to the prediction in Italy the first COVID-19 cases could happen even after November 27, 2019. This results correlates with the All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. . information form Giuseppe Remuzzi, director of the Mario Negri Institute for Pharmacological Research that "virus was circulating before we were aware of the outbreak in China", [16]. Table 5 and Fig. 3 show that the epidemic outbreak in France could happen around January 25, 2020. This estimation correlates with the results of paper [17] , where the first COVID-19 cases with 5 Chinese tourists are described. They were from Wuhan and arrived in Europe on January 17-22, 2020. Table 3 -5) . By comparison, in South Korea this time was approximately 4.3 hours, [8] . Figs. 1-6 illustrate that close to the epidemic outbreaks, the information about number of cases is not complete, since it is difficult to detect all the infected persons, especially those with mild illness. As a result, SIR simulation using data sets from the initial epidemic period has limited All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 25, 2020. . https://doi.org/10.1101/2020.04.22.20075135 doi: medRxiv preprint accuracy, and predictions of final size and duration are too optimistic (see [10] [11] [12] ). The data quality may be very different and unpredictable. The only way to obtain the reliable results is to compare the V(t) curve with V j data obtained after day of calculations. If the discrepancy after some days of observation is too large, new calculations must be performed with the use of fresh data. "Stars" in Figs. 1-3, 5,6 illustrate that the accuracy of calculations is good enough. Probably, the prediction for Spain is too optimistic and can be updated later. Numbers of infected I (green), removed R (black) and the number of victims V=I+R (blue line). It looks, that SIR model can determine the real time of epidemic outbreak. Due to the data incompleteness, the "hidden" period could be very long. For example, the real epidemic outbreaks All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. in Italy and Spain probably happened in November, 2019 and January, 2020 respectively (see Table 5 and Figs. 1, 4) . There is no reason to think that the "hidden" period in China was shorter. According to [18] , the first laboratory confirmed case was recorded on December 8, 2019. Therefore the real "zero" patient could get infection in October-November, 2019 and probably had no connection with the Wuhan fish market. Chinese authorities notified WHO about the epidemic outbreak only on January 3, 2020 (see [18] ). This delay and obstacles to the dissemination of truthful information (an example is doctor Li Wenliang ) has led to tragic consequences in Europe and other parts of the world. SIR curves could be useful to estimate the number of persons who are still spreading the infection, so known "hidden" patients (see green lines in Figs. 1-6) . These information could be useful to plan the relaxation of quarantine measures. If medical and other services are able to quickly isolate persons who have contacted, for example, with 100 "hidden" patients, then there is no need to wait until the number of patients on the green curve reaches the value of unity. On the other hand, such attenuations may extend the time of occurrence of new cases and the number of deaths. A compromise between medicine and economic interests must be found here. The SIR (susceptible-infected-removed) model and statistical approach to the parameter are able to make some reliable estimations for the epidemic dynamics, e.g., the real time of the outbreak, final size and duration of the epidemic and the number of persons spreading the infection versus time. This information may be useful to regulate the quarantine activities and to predict the medical and economic consequences of the pandemic. Unfortunately, the number of patients in Europe already exceeds one million. The risk of catching the infection will persist until at least mid-August, 2020. Such fatal consequences could have been avoided if timely and truthful information came from China. Coronavirus disease (COVID-2019) situation reports A contribution to the mathematical theory of epidemics 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 Characteristics of coronavirus epidemic in mainland China estimated with the use of official data available after February 12, 2020 Estimations of the coronavirus epidemic dynamics in South Korea with the use of SIR model Comparison of the coronavirus epidemic dynamics in Italy and mainland China Stabilization of the coronavirus pandemic in Italy and global prospects 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? Applied Regression Analysis Maximal speed of underwater locomotion Clinical and virological data of the first cases of COVID-19 in Europe: a case series Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia. The new england journal of medicine I would like to express my sincere thanks to Gerhard Demelmair and Ihor Kudybyn for their help in collecting and processing data.