This report analyses the spread of COVID-19 in european countries, focusing especially on Italy
and Spain, for which short-term predictions are made for the cumulative number of hospitalizations,
ICUs, discharges and deaths. Taking into account the different magnitudes considered, for 1 day
predictions the mean probability of a right guess is of 0.99, 0.94 for 2 day predictions, 0.89 for 3 day
predictions and 0.83 for 4 day predictions2019/202

Pradas Rodriguez, Sergi

Rovira Ferrer, David

Simó Romaguera, Xavier

English

Consortium of Academic Libraries of Catalonia (CBUC)

Covid-19 analysis using the Gompertz Function

UPCommons. Portal del coneixement obert de la UPC

COVID-19 ANALYSIS USING THE GOMPERTZ FUNCTIONSergi Pradas Rodriguez,∗ David Rovira Ferrer,† and Xavier Simó Romaguera‡Projectes d’Enginyeria F́ısica II.Enginyeria F́ısica.(Dated: June 17, 2020)This report analyses the spread of COVID-19 in european countries, focusing especially on Italyand Spain, for which short-term predictions are made for the cumulative number of hospitalizations,ICUs, discharges and deaths. Taking into account the different magnitudes considered, for 1 daypredictions the mean probability of a right guess is of 0.99, 0.94 for 2 day predictions, 0.89 for 3 daypredictions and 0.83 for 4 day predictions.I. INTRODUCTIONCOVID-19 is an infectious disease caused by a viruscalled SARS-CoV-2. It usually causes tiredness, feverand breathing problems. It is not often serious, but itcan lead to severe problems, and ultimately death, forsome people.The objective of this project is to analyze, via the useof the Gompertz function, the evolution of the number ofcases, hospitalizations, ICUs, discharges and deaths dueto COVID-19, as well as make mid/long-term predictionsbased on current data. This analysis has been devel-oped with the help of and close collaboration with theBIOCOM-SC group from UPC, which has been workingon the study and evolution of COVID-19 since the firstdays of the epidemic. These results, among many moreregarding the spread of the disease, have been sent dailyto the European Commission for three months.This report is structured in the following way: in SectionII A the Gompertz function and its characteristics are in-troduced. In Section II B the process of systematizationof the information to be used in the predictions is ex-plained. This is followed by an account of the data anal-ysis of the significant magnitudes in Section II C. Then,in Section III A a study of the predicted final incidencefor different countries is carried out, and in Section III Bthe accuracy and reliability of the predictions is analysed.Finally, in Section IV the conclusions are drawn.II. METHODSA. GOMPERTZ FUNCTIONThe main objective of this project is to study the useof the Gompertz function to predict not only the futurenumber of cases of COVID-19, as it has already beenshown capable of doing [1], but to predict the spread ofthe disease in terms of hospitalized people, intensive caretreatment units occupied (referred as ICUs henceforth),discharges and deaths.∗ sergi.pradas@gmail.com† drovfer@gmail.com‡ xavi.simo.99@gmail.comThis empirical model is characterized by an initial expo-nential growth that, overtime, slows down. It is, thus,able to portray the evolution and, later, control of thedisease. It is described by the following equation:N(t) = Ke− ln(KN0)e−a(t−t0)(1)Where N(t) is the cumulated number of cases at time t(measured in days), or the magnitude studied, N0 is thevalue of such magnitude at time t0, a marks the slowingdown of the spreading’s rate, and K is the expected finalvalue of the variable under analysis at the end of the cri-sis. This model is fitted to the country or region beingFIG. 1. This figure shows the Gompertz function. As it canbe seen, the higher the a parameter is, the faster the spreadingis.analyzed in order to find K and a. A detailed explanationof this process will be carried later on in this report.It is, then, of major interest, to be able to predict whenthe maximum number of new cases in a day will be regis-tered. This is done by analyzing the first derivative of theGompertz function, as it represents the new cases eachday:dNdt= aKe− ln(KN0)e−a(t−t0)(ln(KN0)e−a(t−t0))(2)Therefore, by computing the second derivative and set-ting it to zero we get the day of maximum number of new2cases, marked as tpeak:tpeak = t0 +ln(ln(KN0))a(3)We can then analyze the error in such prediction by prop-agating the error:σ2tpeak =(δtpeakδa)2σ2a +(δtpeakδK)2σ2K (4)Where σa and σK are the error margins for a and K,respectively, as obtained by the fitting. Which, substi-tuting, yields:σ2tpeak =− ln(ln(KN0))a22 σ2a + 1aK ln(KN0)2 σ2K(5)Moreover, it is of crucial interest to be able to predictwhen the spreading of the disease reaches a critical pointregarding the final expected number of cases. We takethat critical point to be 90% of K, as it is then whengovernments can start to consider relaxing their measuresof control. In order to compute when that happens, weneed to consider N(t90) = 0.9K, with t90 the day whenthe 90% of the expected final number of cases is reached.Substituting into the Gompertz function yields:t90 = t0 −ln(− ln(0.9)ln ( KN0))a(6)The error in the prediction of this value is:σ290 =(∂t90∂K)2σ2K +(∂t90∂a)2σ2a (7)With:∂t90∂K= − 1aK ln (K/N0)&∂t90∂a= −ln(− ln (0.9)ln (K/N0))a2(8)B. RETRIEVAL OF DATA ANDSYSTEMATIZATIONGiven the main objective of this project is to study theextension of the Gompertz’s function fitting to other vari-ables apart from the number of cases, such as the numberof hospitalizations and ICUs, among others, we had towork on finding the information required for the analysisitself. In this sense we worked for several days on find-ing reliable and useful information about the magnitudesmentioned above for different countries in the EuropeanUnion. By useful information it is meant data in theform of csv or excel files that would make an systemati-zation of the actualization process viable. This proved tobe more difficult than first anticipated, as the majorityof official ministerial webpages showed the data in theform of graphical analysis as public release and were notmeant for use in further studies.In the end, it was possible to find such reliable infor-mation, with varying success, for three EU countries:Belgium[2] (including its regions), France[3], Sweden[4]and Switzerland[5]. The analysis of Sweden is of ma-jor interest, as it can show the effects of different controlmeasures, as Sweden has chosen to follow less restrictiveactions.Moreover, we were given the necessary data regard-ing Italy[6] and Spain[7] (and their respective regions andcommunities) by the BIOCOM-SC group from UPC. Tothis, one has to add the data regarding cases and deathscollected by the European Center for Disease Preventionand Control[8] (ECDC), that will make possible a supra-national analysis for European countries.However, given that every country organizes its data indifferent forms, be it by regions, days, sex or even age,an homogenization of the information was required. Thismeant that a Matlab code had to be developed, thatwould take into account those differences, and would gen-erate a file with the same format as those provided bythe BIOCOM-SC group. For Belgium this proved to bea complete success, as a retrieval of information regard-ing cases, hospitalizations, ICUs, discharges and deathsproved possible. The same can be said of France, ex-cept obtaining information of the separate regions provedmore difficult and could not be done, as opposed to Bel-gium. In the case of Sweden we were not able to find datain useful form regarding hospitalizations and discharges.On the other hand, some problems started to appear assoon as we started trying to make predictions:1. Belgium: ICUs are prevalence values. This meansthat we only have access to the current occupation,so we lack information about the cumulative values.2. France: Hospitalizations and ICUs are prevalencevalues, and not cumulative.3. Italy: Hospitalizations and ICUs are prevalencevalues.4. Spain: More general problems appeared due to in-consistencies and mistakes in the reported data.This meant that for the data presented in prevalence formno prediction was possible, as predictions are based oncumulative values.C. DATA ANALYSIS OF SIGNIFICANTMAGNITUDESIn this section we present the analysis of cases, hospi-talizations, ICUs, discharges and deaths due to COVID-19 carried out for different countries and regions withinthem. The main objective is to make short-term predic-tions that are useful for decision makers within govern-ments and other state institutions.The model is adjusted to make predictions for the cu-mulated values of the magnitudes indicated above when3certain conditions are met regarding the reported infor-mation:1. Cases: At least 3 days with more than 100 casesand 1 day with more than 200.2. Hospitalizations: At least 3 days with more than50 hospitalizations and 1 day with more than 70.3. ICUs: At least 3 days with more than 10 ICUs and1 day with more than 15.4. Discharges: At least 3 days with more than 10 dis-charges and 1 day with more than 15.5. Deaths: At least 3 days with more than 10 deathsand 1 day with more than 15 deaths.In case any of these criteria is not met, no predictionsare made for that variable. This fact is necessary to dif-ferentiate the phase in which the spreading of the diseaseis due to imported cases and the subsequent period inwhich new cases occur because of local transmission.The prediction process is based on a Non-Linear LeastSquares fitting of the parameters K and a of the Gom-pertz function using the Levenberg-Marquardt and theTrust-region algorithms. For this fitting only the daysfor which the criteria above are met are used in the pro-cess. In case there are more than 15 points that verify it,only the last 15 are used. In order to better capture thecurrent trend of the spreading of the disease, if there aremore than 9 points that meet the conditions, weights areapplied to the last 3 points in the fitting process. Then,if there are more than 6 points verifying the criteria, pre-dictions are made for the following 2 days; if there aremore than 9, for the following 3 days; if there are morethan 12, for the following 4 days and, if there are morethan 15, for the following 5 days.Moreover, we plot a variable we name ρ, that is related tothe reproduction number, that is, the number of new in-fections caused by a single case. It is evaluated as followsfor the day before the last reported information:ρ(t− 1) = Nnew(t) +Nnew(t− 1) +Nnew(t− 2)Nnew(t− 5) +Nnew(t− 6) +Nnew(t− 7)(9)Where Nnew(t) is the number of new confirmed cases atday t.As an example of the results obtained, in Figure (2) and(3) these are shown for Italy as a whole. The completereports for every country are shown in a shared Dropboxfolder [9].III. ANALYSIS AND RESULTSA. PREDICTION OF INCIDENCE ANDRELATED MAGNITUDESIn this section we show the results from the predictionsmade for the day of maximum number of new cases, theday t90, as defined above, and compare the current situa-tion to the final expected situation by using the incidence,FIG. 2. Total number of cases in Italy on April 24th. Innavy blue the reported number of cases, and in light blue thepredictions made. As it can be seen, the spread of the diseasewas not exponential anymore, and was starting to slow down.FIG. 3. ρ′s evolution in Italy as of April 24th. A black linemarks ρ = 1, that indicates when the spread slows down.that is, the number of cases for every 100000 citizens.This has been done for Italian and Spanish regions [9], aswell as for EU and EFTA (European Free Trade Associ-ation) countries, and the UK. This last analysis is shownin Figure (4).Moreover, a study of the convergence of tpeak, t90 andK, is done for Spain, Italy, France, Germany and the UK[9]. As an example, we show the results for Germany inFigure (5).An analysis of the convergence of K is also carried outfor Italian regions and Spanish autonomous communities[9]. For the case of EU-EFTA-UK as a whole the resultsare shown in Figure (6).4FIG. 4. Comparison of the current and final predicted situa-tion for EU-EFTA-UK countries as of June 2nd.FIG. 5. Convergence of tpeak and t90 for Germany as of May30th.FIG. 6. Convergence of K for EU-EFTA-UK as a whole as ofMay 30th.B. ACCURACY OF THE PREDICTIONSIn this section we present the analysis of the accuracy ofthe predictions shown in the previous sections for Bel-gium, France, Italy, Spain and Sweden, including notonly the countries, but also the predictions for Belgian,Italian and Spanish regions or autonomous communities.For the purpose of these calculations only the variables ineach country and region that are reported as cumulativevalues are used.In order to compute this we store the predictions madeeach day for every variable, as well as the corresponding99% confidence intervals. Then, that prediction is com-pared to the real values as reported by official sources. Incase the latter values fall into the predicted confidence in-tervals we consider the prediction to be right, in the caseit does not, we consider it to be wrong; this will be usedto determine the reliability of the predictions. The exactresults are shown in Table I.PROBABILITYVariables 1 day prediction 2 days 3 days 4 daysHospitalizations 0.98 0.94 0.88 0.83ICUs 0.99 0.95 0.91 0.88Discharges 0.99 0.92 0.85 0.78Deaths 0.99 0.96 0.90 0.84TABLE I. Reliability of the predictions made.Moreover, we compute the relative error as (reportedvalue - predicted)/(reported value). In order to showthese results we plot them via a boxplot for the relativeerror, and a bar graph for the probability of making acorrect prediction. The findings regarding hospitaliza-tions, ICUs, discharges and deaths are shown in Figure(7).For both the probability and the relative error the num-ber of samples used is shown in Table II. For hospitaliza-tions data from Belgium, Spain’s autonomous communi-ties and Switzerland is used due to other countries notproviding the corresponding cumulative numbers. Withrespect to ICUs, data from Spain and Sweden is used.Regarding discharges data from Belgium and its regions,Italy and its regions and France is used. Informationfrom Spain is not used due to problems with the histori-cal series. For deaths, data from Belgium, France, Italy,Spain, Sweden and Switzerland is considered.SAMPLES USEDVariables 1 day prediction 2 days 3 days 4 daysHospitalizations 1172 1152 1132 1112ICUs 261 257 253 249Discharges 756 744 732 720Deaths 1619 1594 1569 1544TABLE II. Samples used in the predictions.5(A)(B)FIG. 7. Errors of the predictions for hospitalizations, ICUs,discharges and deaths for up to 4-day predictions. N = 1refers to the prediction made for one day after the day it wasmade. The same goes for N = 2, 3 and 4. (A) Probabilitythat the predictions made fall in the prediction interval foreach day. (B) Relative errors of the predictions.We conclude that the relative errors for 1 and 2 day pre-dictions are well below 5%, indicating the reliability ofthe predictions. For 3 and 4 day predictions ICUs yieldmore than 5% relative error, and for 4 day predictionsdeaths also reach this threshold. As can be seen fromthe previous figures, discharges are an outlier, as theyshow significantly higher relative errors, even reaching10% for 4 day predictions. As will be explained in thefollowing section we suspect this is due to the ”weekendeffect”.IV. CONCLUSIONSThe main objective of this report was to analyze thespread of COVID-19 via the use of the Gompertz func-tion, and to make short-term predictions based on itsfitting to the data provided by the different institutionsinvolved in the monitoring of the disease. As the resultsprovided in the last section show, the mean relative er-ror for these short-term predictions is well below 10%,thus showing the potential of this function as a guide forpolicy makers. However, this approach suffers from someproblems.First of all, as is the general case for every predictionmodel, it relies entirely on the quality of the data thatis made public, thus suffering from inconsistencies andmistakes in the daily publication of the data. One suchhurdle is the effect known as ”weekend effect”, wherelower number of cases, hospitalizations and other magni-tudes are reported. This significantly modifies the day-to-day behaviour and influences the predictions that aremade based on the fitting to the reported values. Thishas proven to be specially significant in the case of re-ported discharges, where major differences appear be-tween workdays and weekends.Moreover, other problems arise due to inconsistencies inthe data reported by some governments and institutions.This is the case, for example, of Italy and Spain. Con-cerning the former, problems regarding discharges arespecially important. For the latter, multiple inconsis-tencies have appeared during the last months. Theseinclude changes in the criteria employed in the reportingof the data of different magnitudes, such as changes inwhat is considered a case of COVID-19 and the inabilityof correcting the historical series for some autonomouscommunities. An example of these problems is shown inFigure (8).FIG. 8. (A) Cumulative discharges in Friuli-Venezia-Giulia,Italy. (B) Hospitalizations in Madrid, Spain. Both show clearproblems in the historical series.To conclude, the Gompertz function has proven to be avery useful tool in analyzing the short-term the behaviourof the spread of COVID-19, provided the quality of thedata and the historical series allow for a good fitting ofthe function.V. FINAL COMMENTSWe would like to thank the BIOCOM-SC research groupfrom UPC, and especially Mart́ı Català and Clara Prats,for the constant help with the day-to-day problems re-garding the calculations and analyses.6REFERENCES[1] CATALA, Marti, et al. Empiric model for short-time pre-diction of COVID-19 spreading. medRxiv, 2020.[2] Sciensano, the Belgian institute for health. (2020). Epi-demiological follow-up of the COVID-19 disease in Bel-gium. Retrieved from https://epistat.wiv-isp.be/covid/[3] French Department of Public Health. (2020). Hospital-izations data regarding COVID-19 epidemic. Retrievedfrom https://www.data.gouv.fr/fr/datasets/donnees-hospitalieres-relatives-a-lepidemie-de-covid-19/#[4] Public Health Agency of Sweden. (2020). Historicdata on the outbreak of coronavirus disease. Retrievedfrom https://www.folkhalsomyndigheten.se/smittskydd-beredskap/utbrott/aktuella-utbrott/covid-19/bekraftade-fall-i-sverige/[5] Swiss Federal Office of Public Health. (2020).Data on confirmed cases of coronavirus infectionsand deaths due to the disease. Retrieved fromhttps://www.bag.admin.ch/bag/en/home/krankheiten/ausbrueche-epidemien-pandemien/aktuelle-ausbrueche-epidemien/novel-cov/situation-schweiz-und-international.html#1905396240[6] Italian Department of Civil Protection. (2020). Historicdata on the outbreak of coronavirus disease in Italy. Re-trieved from https://github.com/pcm-dpc/COVID-19[7] Spanish Health Department and Carlos III HealthInstitute. (2020). Historic data on the outbreakof coronavirus disease in Spain. Retrieved fromhttps://github.com/datadista/datasets/tree/master/COVID%2019[8] European Centre for Disease Prevention andControl. (2020). Number of new cases re-ported per day and per country. Retrievedfrom https://www.ecdc.europa.eu/en/publications-data/download-todays-data-geographic-distribution-covid-19-cases-worldwide[9] https: // www. dropbox. com/ sh/ vfok5ii8gawsrkp/AACi3xCxp-ibXFyvIzzlipwCa? dl= 0

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Covid-19 analysis using the Gompertz Function

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