For Germany, it is predicted that the first wave of the corona pandemic disease reaches its maximum of new infections on 11 April 2020 −3.4+5.4 days with 90% confidence. With a delay of about 7 days the maximum demand on breathing machines in hospitals occurs on 18 April 2020 −3.4+5.4 days. The first pandemic wave ends in Germany end of May 2020. The predictions are based on the assumption of a Gaussian time evolution well justified by the central limit theorem of statistics. The width and the maximum time and thus the duration of this Gaussian distribution are determined from a statistical χ2 -fit to the observed doubling times before 28 March 2020

Schlickeiser, Frank

Schlickeiser, Reinhard

English

MACAU: Open Access Repository of Kiel University

ArticleA Gaussian Model for the Time Development of theSars-Cov-2 Corona Pandemic Disease. Predictions forGermany Made on 30 March 2020Reinhard Schlickeiser 1,2,* and Frank Schlickeiser 11 Institut für Theoretische Physik, Lehrstuhl IV: Weltraum-und Astrophysik, Ruhr-Universität Bochum,D-44780 Bochum, Germany; schlickeiser@gmail.com2 Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstr. 15,D-24118 Kiel, Germany* Correspondence: rsch@tp4.rub.deReceived: 10 April 2020; Accepted: 10 May 2020; Published: 19 May 2020Abstract: For Germany, it is predicted that the first wave of the corona pandemic disease reachesits maximum of new infections on 11 April 2020 +5.4−3.4 days with 90% confidence. With a delay ofabout 7 days the maximum demand on breathing machines in hospitals occurs on 18 April 2020 +5.4−3.4days. The first pandemic wave ends in Germany end of May 2020. The predictions are based onthe assumption of a Gaussian time evolution well justified by the central limit theorem of statistics.The width and the maximum time and thus the duration of this Gaussian distribution are determinedfrom a statistical χ2-fit to the observed doubling times before 28 March 2020.Keywords: statistical methods in physics; Gaussian model; virus time evolution; health science1. IntroductionIn these days there is a very high interest in the societal, economical and political world tounderstand the time evolution of the first wave of infections of the population by the current Sars-Cov-2(corona) virus. The most important issues are the total duration and the peak time of the infectionevolution as well as the maximum number of daily infections. It would be most helpful for manypeople to have a reproducible, crude, but reliable estimate when this pandemic wave is over. It is thepurpose of this manuscript to provide such an estimate based on a simplified Gaussian model for thetime development of the pandemic outburst.The best justification for the Gaussian, or normal, distribution for the virus time evolution isgiven by the central limit theorem of statistics [1]. The central limit theorem states that in situations,when many (1) independent random variables are added, their properly normalized sum tendstoward a Gaussian distribution even if the original variables themselves are not normally distributed.The spread of the virus infection of populations with high number of persons certainly is such arandom process to which the central limit theorem is applicable. Each person in a given populationhas a probability distribution (normalized to unity) as a function of time of being infected: it is avery noncontinuous distribution being 1 at the day of infection and 0 on all other days. If one addsup these discrete distributions of persons living in villages and districts of towns of typical size ofabout 1000 persons one obtains quasi-continuous probability distributions for being infected whichcertainly will be different in hotspots of the disease and isolated rural areas. If we then add up a largenumber of these village probability distributions for all of Germany we obtain the daily infectionrate distribution which according to the central limit is close to a Gaussian distribution. Recently,the Gaussian model has also been used to predict the time evolution of the COVID-19 pandemic inPhysics 2020, 2, 164–170; doi:10.3390/physics2020010 www.mdpi.com/journal/physicsPhysics 2020, 2 165Italy [2] and China [3]. To the best of our knowledge this manuscript is the first application of theGaussian model to the COVID-19 pandemic in Germany. It differs from the earlier work [2,3] byproposing the use of monitored doubling times to infer the total duration and the peak time of the firstwave of infections.The analysis of Gaussian distribution function plays a central role in many problems of statisticalphysics and plasma physics e.g., in plasma kinetic theory they are referred to as drift-Maxwellian [4]or counterstreaming bi-Maxwellian [5,6] velocity distribution functions.Apart from consulting several reviews [7–9], as non-virologists we are not familiar with therecent relevant virology literature. Nevertheless, it is our hope that in these hard times an estimate byunbiased non-experts might be welcomed by specialists as well as the broad population, especiallyif some positive information and outlook is provided. We base our parameter estimates on publiclyavailable information, especially by the podcast [10] and the recent sophisticated modeling study forGermany [9]2. Gaussian ModelNumerical simulations and the empirical data of earlier epidemics [10] indicate that the timeevolution of epidemic waves is characterized by an early exponential rise until a pronounced maximumis reached followed by a rapid decrease. As argued above, we adopt a simple Gaussian model for thetime evolution of infections and explore its consequences. If I(t) denotes the number of infections perday, we assume that its time evolution is given by the Gaussian functionI(t) = I0 e−(t−E∆ )2, (1)shown in Figure 1, where I0 denotes the maximum value at time E and ∆ denotes the width ofthe Gaussian.By monitoring the new daily infections, one easily derives the relative change in the infectionsper dayP(t) =dI(t)/dtI(t)=d ln I(t)dt= 2E− t∆2, (2)where we used the distribution Equation (1).Figure 1. Gaussian model prediction of the time evolution of the number of infections I(t)/I0(I0 denotes the maximum number of infections.) for Germany for the best fit parameters, peak timeE = 27.5 days and width ∆ = 14.4 days. The time t = 0 in I(t) corresponds with the date 14 March2020, the day monitoring of the doubling times and drastic political actions against the virus started.Physics 2020, 2 1662.1. Doubling TimeThe monitored data are often given in terms of the doubling time D of the correspondingexponential function at any timeIa(t) = J0 et ln 2D . (3)Using the distribution of Equation (3) in Equation (2) provides for the relative change in dailyinfection rateP(t) =d ln Ia(t)dt=ln 2D. (4)Equating the two results in Equations (2) and (4) leads to the time-dependent Gaussiandoubling timeD(t) =AE− t , A = (ln√2)∆2 = 0.35∆2. (5)Figure 2 shows the monitored doubling times for Germany [11] starting from 15 March 2020 until28 March 2020. We assume that every value has an error of 15%. It starts at D(t = 0) = 2.6 days andincreases to D(t = 14) = 4.8± 0.72 days on 28 March 2020.The Gaussian doubling time modeling then provides D(t = 0) = A/E = 2.6 days,corresponding to∆ =√2.6Eln√2=√7.5E. (6)Moreover, Equation (5) reduces toD(E, t) =2.61− tE. (7)2.2. Statistical FitWe determine the value of the only free parameter E in Equation (7) by performing a χ2-fit to thedata shown in Figure 2. If m(ti) denote the observed doubling times at days ti, δm(ti) = 0.15m(ti) itserror and D(E, ti) the theoretical doubling time for given values of E, we calculateχ2(E) =N∑i=0(m(ti)− D(E, ti)δm(ti))2=N∑i=0(m(ti)− D(E, ti)0.15m(ti))2. (8)Currently, on 28 March 2020 we have N = 14 measurements. With one free parameter the numberof degrees of freedom is N − 1 = 13.The best fit with the minimum value of χ2min = 9.51 is provided for E = 27.5 days. The χ2min,p.d.f =9.51/13 = 0.73 per degree of freedom (p.d.f.) is less than unity indicating that our model fits the datavery well. For 90% confidence all values of χ2 ≤ χ2min + 2.71 = 12.22 are acceptable [12] yielding with90% confidence thatE = 27.5+5.4−3.4 days, (9)corresponding to 11 April 2020 +5.4−3.4 days. Consequently, the best fit Gaussian doubling time forGermany is given byDbest =71.527.5− t days. (10)As an aside we note that the variation in Equation (10) becomes infinitely large as t → 27.5.Moreover, for times t > E the doubling times becomes a decay half-life approaching 0 for very largetimes t E.Physics 2020, 2 167Figure 2. Monitored doubling times in Germany from 15 March until 28 March 2020 in comparisonwith the best fit.Moreover, after inserting the values of Equation (9) into Equation (6) yields with 90% confidence∆ = 14.4+1.4−1.0 days. (11)In Figure 3 we show the prediction of the doubling times in Germany until day 25 correspondingto 8 April 2020.Figure 3. The same as in Figure 1 but now the predicted doubling times until day 25 corresponding to8 April 2020.3. Predictions for Germany3.1. Total Number of InfectionsIt is known that during the whole duration of the first wave of the virus evolution 70% of the totalpopulation are infected [10], if nothing is done to reduce the number of infections. Scaling the totalpopulation in units of 105N5 (N5 = N/105, where N denotes the number of persons), we estimate that0.7q105N5 are infected during the whole duration of the first virus wave, where the quarantining factorq accounts for the currently taken political actions such as quarantining of elder and infected people,social distancing actions in the society as well as the closure of schools and daycare facilities.Integrating the Gaussian in Equation (1) over all times we then obtainPhysics 2020, 2 1687 · 104qN5 = I0∫ ∞−∞dt e−(t−E∆ )2= I0∆∫ ∞−∞dx e−x2=√π I0∆, (12)where we used the integral∫ ∞−∞ dx e−x2 =√π. Equation (12) yields for the maximum valueI0 =7 · 104qN5√π∆=4 · 104qN5∆. (13)With the 90% confidence value for ∆ from Equation (11) we obtain with the same confidence levelfor the maximum value in Equation (13)I0 = (2.78+0.20−0.22) · 103qN5. (14)3.2. Manageable InfectionsWe assume as typical medical parameters those suggested by the recent modeling by theRobert-Koch-Institute [9] (see their Figure 1): only about 20% of the infected people are seriouslyinfected, 5% have to be hospitalized and α% (α is often adopted to be unity) need access to breathingapparati for typically 1 week, corresponding to 7 days. We refer to the latter as NSSPs standing fornew seriously sick persons per day. As these numbers are uncertain we keep their scalings with thetypically adopted numbers.In Germany there are at most 56,000 breathing apparati in total available, corresponding for apopulation of 80 million people to 70b70 (with b70 = b/70 where b is the number of breathing apparati)breathing apparati available per 105N5 people. As a consequence, every day the hospitals can handleIn NSSPs withIn =70b70N57w= 10b70N5w. (15)The infections are handable by hospitals for all times if at the maximum of the virus evolution0.01αI0, denoting the maximum number of seriously sick persons per day needing access to breathingapparati, is less or equal to In, i.e.,0.01αI0 ≤ 10b70N5w. (16)Inserting the value of Equation (14) for the maximum value I0, the quantity N5 cancels out andwe obtain with 90% confidence the condition(2.78+0.20−0.24)αq ≤b70w, (17)which is equivalent toαqwb70≤ 0.36+0.04−0.03. (18)In order to handle all serious infections in German hospitals the condition in Equation (18) hasto be fulfilled. It seems that German hospitals can only ensure the best treatment of all NSSPs at themaximum of the first wave if either (1) the number of available breathing apparati can be increasedby a factor of 3, corresponding to b = 210 per day, or (2) the quarantine factor q can be reducedto q = 0.36+0.03−0.02 ' 1/3. The first option is unrealistic on short time scales. The second option ofreducing the quarantine factor q to about 0.3 by the currently taken and enduring social distancingand quarantining activities seems to be realistically achievable in Germany. We therefore will adoptthis optimistic value q = 0.3q0.3 (with q0.3 = q/0.3) in our further predictions. However we note thatwith such a small quarantine factor only 21, 000N5 persons will be infected during the whole firstPhysics 2020, 2 169wave of the virus, so that secondary and additional waves of infections are more likely to occur in thenear future.It is important to notice that the outbreak of serious sickness syndromes of NSSPs is delayed tothe infection time by about τ = 7 days [9,10]. This delay time has to be added to the above derivedmaximum time scale E, so that E + τ = 34.5+5.4−3.4 days, corresponding to 18 April 2020+5.4−3.4 as thepredicted day in Germany when the maximum number of NSSPs has to be treated. At this predictedday also the death rate is highest.3.3. Duration of the First WaveThe number of infections are significantly reduced by a factor 103 compared to the maximum I0at the timeT = E + 2.63∆ = 65.4+9.1−6.1 days (19)with 90 percent confidence. Consequently, the first pandemic wave will be over in Germany not before65 days or about 2 months with the indicated uncertainty. This corresponds to 38± 4 days after thetime of maximum.According to our predictions, the first pandemic wave in Germany will reach its maximum by18 April 2020 when about (8.3± 0.78)q0.3N5 NSSPs have to be treated in the hospitals. The wave hasa broad distribution from April 6 to 30 April 2020 with more than (4.2± 0.39)q0.3N5 but less than(8.3± 0.78)q0.3N5 NSSPs per day. The number of NSSPs needed to be treated at hospitals will sharplydrop to less than (0.0083± 0.00078)q0.3N5 NSSPs by 26 May 2020.As Germany has a population of about 80 million persons we have N5 = 800. Therefore,in absolute numbers German hospitals will have to cope with (6640± 624)q0.3 NSSPs at the maximumof the outburst on 18 April 2020, more than (3360± 312)q0.3 but less than 6640± 624 NSSPs per daybetween 6 April and 30 April 2020, before the total number per day drops below (6.64± 0.63)q0.3NSSPs after the end of May 2020. All errors have 90 percent confidence.This Gaussian analysis has also been applied successfully to 25 other countries with monitoredinput data on the infection and death rates until 2 April 2020, for detail see ref. [13]. The Gaussiananalysis is constantly updated to the latest data on the site on COVID-19 real time statistics andextrapolation using the Gaussian model [14]. It is impressive to see how well the Gaussian modelreproduces the monitored data on the death rates in now more than 68 countries.Knowing the time of maximum rush E+ 7 days of NSSPs to their hospitals, the maximum numberof NSSPs and the full width half maximum (FWHM)√2∆ would help the government and medicalagencies in these countries to optimize the managing of the disease wave. Moreover, it will allowmutual help in managing the peak of the disease wave by sharing breathing apparati and trainedmedical personal for a brief predictable time in cases where the times of maximum differ significantlyenough (by more than one FWHM) in these countries. Enough airplane capacities are available thesedays. The crucial medical personal should be supported financially with whatever it takes to motivatefor and honor their important work. We are convinced that many thousands of lives can be savedthis way.3.4. Final Important RemarkWe end with a sentence of caution: although the central limit theorem provides us with a verygood justification of the adopted Gaussian time distribution function it is not guaranteed that theactual virus time evolution follows this behavior. We will only know for sure after the first pandemicwave is over. It is possible that our estimates and we are wrong. We take this risk because we areconvinced that many persons will welcome our optimistic estimate that the first wave is over by endof May 2020. There is light at the end of a long tunnel. Our estimate might also help decision makersdetermine when to lift the current societal and economical lockdown.Physics 2020, 2 1704. Note Added in Proof (13 May 2020)Despite its simplicity the predictions of this Gaussian model made on 30 March 2020 agree verywell with the monitored data in Germany. The peak of the death rate occurred on 16 April 2020 inexcellent agreement with our predicted date 18 April 2020 +5.4−3.4 days. Moreover, the quarantine factor qwas lower than 0.03, well below the adopted value of 0.3 in this manuscript (for details see [15]).Author Contributions: R.S. wrote most of the manuscript, while F.S. provided the illustrations and critical readingof the manuscript. All authors have read and agreed to the published version of the manuscript.Funding: This research received no external funding.Acknowledgments: We thank Clarisse Luap and Martin Kröger, ETH Zurich, for their interest in this work andinitiating a follow-up study, including Janik Schüttler.Conflicts of Interest: The authors declare no conflict of interest.References1. Feller, W. An Introduction to Probability Theory and Its Applications; Wiley: New York, NY, USA, 1971.2. Ciufolini, I.; Paolozzi, A. Mathematical prediction of the time evolution of the Covid-19 pandemic in Italy bya Gauss error function and Monte Carlo simulations. Eur. Phys. J. Plus 2020, 135, 355. [PubMed]3. Lixiang, L.; Yang, Z.; Dang, Z.; Meng, C.; Huang, J.; Meng, H.; Wang, D.; Chen, G.; Zhang, J.; Peng, H.; et al.Propagation analysis and prediction of the COVID-19. Infect. Disease Model. 2020, 5, 282.4. Jackson, E.A. Drift Instabilities in a Maxwellian plasma. Phys. Fluids 1960, 3, 786. [CrossRef]5. Hadi, F.; Bashir, M.F.; Qamar, A.; Yoon, P.H.; Schlickeiser, R. On the ordinary mode instability for low betaplasmas. Phys. Plasmas 2014, 21, 052111. [CrossRef]6. Vafin, S.; Lazar, M.; Schlickeiser, R. The instability condition of the aperiodic ordinary mode for new scalingsof the counterstreaming parameters. Phys. Plasmas 2015, 22, 022129. [CrossRef]7. Anderson, R.M.; Anderson, R.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford UniversityPress: Oxford, UK, 1992.8. Hethcote, H.W. The Mathematics of Infectious Diseases. SIAM Rev. 2000, 42, 599. [CrossRef]9. an der Heiden, M.; Buchholz, U. Modellierung von Beispielszenarien an der SARS-CoV-2 Epidemie 2020 inDeutschland; Robert Koch-Institut: Berlin, Germany, 2020. (In German)10. Drosten, C. Coronavirusupdate (ndr.de/coronaupdate, 2020). Available Online: https://www.ndr.de/nachrichten/info/Coronavirus-Update-Die-Podcast-Folgen-als-Skript,podcastcoronavirus102.html (accessed on 13 May 2020).(In German)11. Endt, C.; Witzenberger, B. Süddeutsche Zeitung Online, Coronavirus in Deutschland (2020). Available online:https://www.sueddeutsche.de/wissen/corona-zahlen-weltweit-news-1.4844448 (accessed on 28 March 2020).12. Lampton, M.; Morgan, B.; Bowyer, S. Parameter estimation in X-ray astronomy. Astrophys. J. 1976, 208, 177.[CrossRef]13. Schüttler, J.; Schlickeiser, R.; Schlickeiser, F.; Kröger, M. Covid-19 predictions using a Gauss model, based ondata from April 2. MedRxiv 2020. [CrossRef]14. Kröger, M. Covid-19 Real Time Statistics & Extrapolation Using the Gauss Model (GM). Available online:http://www.complexfluids.ethz/corona (accessed on 15 May 2020).15. Schlickeiser, R.; Kröger, M. Clues from the First Covid-19 Wave and Recommendations for Social Measuresin the Future. Preprints 2020, 2020040379. [CrossRef]c© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

A Gaussian Model for the Time Development of the Sars-Cov-2 Corona Pandemic Disease. Predictions for Germany Made on 30 March 2020

For Germany, it is predicted that the first wave of the corona pandemic disease reaches its maximum of new infections on 11 April 2020      &minus; 3.4   + 5.4      days with 90% confidence. With a delay of about 7 days the maximum demand on breathing machines in hospitals occurs on 18 April 2020      &minus; 3.4   + 5.4      days. The first pandemic wave ends in Germany end of May 2020. The predictions are based on the assumption of a Gaussian time evolution well justified by the central limit theorem of statistics. The width and the maximum time and thus the duration of this Gaussian distribution are determined from a statistical     &chi; 2    -fit to the observed doubling times before 28 March 2020

Reinhard Schlickeiser

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A Gaussian Model for the Time Development of the Sars-Cov-2 Corona Pandemic Disease. Predictions for Germany Made on 30 March 2020

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