A Monte Carlo approach to model COVID-19 deaths and infections using Gompertz functions
aa r X i v : . [ q - b i o . P E ] A ug A Monte Carlo approach to model COVID-19 deaths and infections using Gompertzfunctions
Tulio Rodrigues ∗ and Otaviano Helene Experimental Physics DepartmentPhysics Institute - University of So Paulo - Brazil (Dated: August 13, 2020)This study provides a phenomenological method to describe the exponential growth, saturationand decay of coronavirus disease 2019 (COVID-19) deaths and infections via a Monte Carlo ap-proach. The calculations connect Gompertz-type trial distributions of infected people per day withthe distribution of deaths adopting two gamma distributions to account for the elapsed time thatencompass the incubation and symptom onset to death periods. The analyses include death’s datafrom United States of America (USA), Brazil, Mexico, United Kingdom (UK), India and Russia,which comprise the four countries with the highest number of deaths and the four countries with thehighest number of confirmed cases, as of Aug 07, 2020, according to the World Health Organizationwebpage: https://covid19.who.int/table . The Gompertz functions were fitted to the data ofweekly averaged confirmed deaths per day by mapping the χ values. The uncertainties, variancesand covariances of the model parameters were calculated by propagation, taking into account thestandard deviations of the data for each epidemiological week. The fitted functions for the averagedeaths per day for USA and India have an upward trend, with the former having a higher growthrate and quite huge uncertainties. For Mexico, UK and Russia, the fits are consistent with a slopedown pattern. For Brazil we found a subtle trend down, but with significant uncertainties. TheUSA, UK and India data shown a first peak with a higher growth rate when compared to thesecond one (typically 2.7 to 3.7 times higher), demonstrating the benefits of non-pharmaceuticalinterventions of sanitary measures and social distance flattening the curves of the pandemic. Forthe case of USA, however, a third peak seems quite plausible, most likely related with the recentrelaxation policies. Brazil’s data are satisfactorily described by two highly overlapped Gompertzfunctions with similar growth rates, suggesting a two-steps process for the pandemic spreading. Forthe case of Mexico and Russia single peaks with smoother slopes fitted the data satisfactorily. The95% confidence intervals for the total number of deaths ( × ) predicted by the model for Aug31, 2020 are 160 to 220, 110 to 130, 59 to 62, 46.6 to 47.3, 54 to 63 and 16.0 to 16.7 for USA,Brazil, Mexico, UK, India and Russia, respectively. Our estimates for the prevalences of infectionsare in reasonable agreement with some preliminary reports from serological studies carried out inUSA and Brazil. The prevalences and 95% confidence intervals for Aug 1, 2020 were estimatedto be 8.3(5.7-10.9)%, 7.9(6.6-9.2)%, 6.7(6.0-7.5)%, 10.5(8.4-12.6)%, 0.6(0.5-0.7)% and 1.6(1.5-1.8)%for USA, Brazil, Mexico, UK, India and Russia, respectively. The method represents an effectiveframework to estimate the line-shape of the infection curves and the uncertainties of the relevantparameters based on the actual data, in contrast with more complex epidemiological models thatrequire a comprehensive knowledge of several parameters. The outbreak of the new coronavirus disease 2019(COVID-19) brought a challenging scenario worldwide[1–3], urging timely and effective responses from theauthorities regarding the availability of intensive careunits [4, 5], as well as the implementation of non-pharmaceutical interventions of social distance and pro-tective sanitary measures [6–8]. Epidemiological models[9–18] and other statistical approaches [19–21] have beenvery useful to guide actions to manage this crisis and toshed light on how to safely and gradually resume eco-nomics and social activities [22]. On the other hand,quantitative analyses are strongly susceptible to severaluncertainties, such as under-reporting of confirmed casesand deaths [23–25], lack of massive tests in some coun-tries [26], changes in policies and methods for reportingconfirmed cases and deaths as time evolves during the ∗ [email protected] pandemic growth, very distinct socio-economic patternsand health facilities capabilities among different countriesand also among different focuses of the disease withinthe same country. Such complex and puzzling scenariodirectly influences the forecast capabilities of the calcu-lations, supporting the need of a multidisciplinary coop-eration of the scientific community and the developmentof mathematical models to provide plausible estimatesfor the uncertainties of the relevant parameters [27–30].Therefore, the present analysis provides an effective phe-nomenological method to estimate the magnitude and therelevant uncertainties of some important quantities as thepandemic evolves, such as: (1) the peak day(s), growthrate(s) and total number of infected people for the re-constructed infection curves, (2) forecast distribution ofdeaths per day, and (3) forecast total number of deathsuntil Aug 31, 2020. It is worth mentioning, however, thatthe predictions of the model strongly depend on the pre-vailing conditions already in place for the specific countrywhich data were analyzed, as any substantial change ingovernmental policies, either in the direction of loosen-ing or tightening social distance, will generate differentdynamics for the spread of the virus.The best-fit parameters are generally strongly correlated,but its uncertainties reflect the dispersion of the data ateach epidemiological week, making the present analysisa suitable quantitative method to describe the pandemicline-shape behavior with few parameters. Moreover, suchapproach could also be useful for the identification of up-ward trends and its correlations with relaxation policiesand procedures as global social and economics activitiesare gradually resumed [31].The calculations assume that the total number of infectedpeople increase according with a Gompertz-type func-tion, which is a sigmoid curve with a lower growth rateat the beginning and at the end, such that: I ( t ) = N e − e − λ ( t − t , (1)where N represents the asymptotic number of infectedpeople for t → ∞ , λ is the growth rate and t the peaktime of the derivative of I ( t ). In such model, the numberof infected people per time period can be written as: G ( t ) = ddt I ( t ) = N λe − e − λ ( t − t e − λ ( t − t ) . (2)For the specific case of COVID-19, the data of con-firmed cases depend very strongly on testing and report-ing policies for the related country. These policies mayalso vary along time, distorting the shape of the distribu-tions. This complicated scenario disfavor the use of con-firmed cases as a reliable source of information to describethe pandemic dynamics, which is crucial to guide govern-ment actions and decisions. In order to overcome thesedifficulties, we have adopted the data of deaths per day,as they should be more consistent with the actual spread-ing mechanism of the virus. The connection between thetrial function of the number of infected people per dayand the number of deaths per day takes into account theprobability distribution function for the elapsed time be-tween the infection and the death, which can be satisfac-torily described by the sum of two time periods, namely,the incubation period t inc , and the symptom onset todeath period t s − d . Both periods are generated in theMonte Carlo algorithm assuming that they are indepen-dent and gamma distributed with an average of 5.1 and17.8 days and coefficient of variation of 0.86 and 0.45, re-spectively, as proposed elsewhere [32] based on previousstudies of Wuhan data [6, 33][34]. So, the time of thedeath t d can be written as: t d = t inf + t inc + t s − d , with t inf representing the time of infection.The analysis of the deaths data [35] were done consid-ering the weekly averaged deaths per day and its corre-sponding standard deviation of the average for the re-spective epidemiological week, counted retrospectivelyfrom the end date of Aug 7, 2020. The time countingconsidered the day of the first confirmed case for eachcountry and the first epidemiological weeks were chosen in such a way that all days of the week had at least onedeath, meaning that the present calculations do considerthe early stages of the pandemic, including from 19 upto 22 weeks depending on the country (see Table I). Themean day of each epidemiological week was chosen witha time bin of ± χ , defined as: χ = n X iw =1 [ ˜ F iw − y iw ] σy iw , (3)where ˜ F iw is the trial function for deaths calculated attime t iw (the mean day of the corresponding week), y iw the weekly averaged deaths per day and σy iw its stan-dard deviation. In that sense, the average values withhigher standard deviations had lower weights in the fit-ting procedure. The best fit parameters of the Gom-pertz functions ( N, λ and t ) where obtained by sorting5000 random sets of parameters around plausible guess-ing values and calculating the respective trial functionand χ for each candidate, assuming a total of 10 mil-lion infections using a time bin of one day. The trialfunctions ˜ F iw were calculated for each set of parame-ters by the connection between the 10 million infectionevents with the respective death events using the proba-bility distribution function of t inc + t s − d , herein denoted P rob (∆ t ) = P rob ( t d − t inf ) (see the insert of Fig.2). Thisprocedure was done several times with progressively nar-row bins for each parameter’s increments until the re-spective χ converged to the minimal value (the conver-gence criteria required that the χ obtained for 5000 ran-dom sets of parameters is lower than the χ obtained for4000 runs and their difference is lower than 0.05 units).The calculation of the total number of infected people N (three Gompertz functions for USA; two for Brazil, UKand India and one for Mexico and Russia) was performedassuming an infection-fatality-ratio ( ifr ) of 0.66% [33]. Aleast square method [36] was applied for the calculationof the uncertainties of the best fit parameters, which co-variance matrix can be written as: V b = ( ˜ F ′ ⊤ V − ˜ F ′ ) − , (4)where ˜ F = ˜ F ′ iw,j stands for the partial derivative of ˜ F iw at any t iw in respect to the P j parameter of each Gom-pertz function ( N, λ and t ). The variance matrix of thedeath’s data is diagonal, such that: V iw,iw = ( σy iw ) − .Given the lower number of deaths per day for Indiaand Russia and the huge variation of the data in eachweek, we have included an additional uncertainty of 5%( σy iw → σy iw + 0 . y iw ) in order to achieve a success-ful fitting. For the calculation of ˜ F ′ iw,j we have used theresulting convolution between the reconstructed curve ofinfected people G ( t ) and the probability density function P rob (∆ t ), such that:˜ F ′ iw,j = ddP j [ F C ( t iw , N k , λ k , t k )], with (5) F C ( t ) = if r · Z t k max X k =1 [ G ( τ, N k , λ k , t k ) P rob ( t − τ ) dτ ] , (6)where k max = 1 for Mexico and Russia, 2 for Brazil, UKand India and 3 for USA. The propagation of the uncer-tainties of the best fit parameters to the reconstructedinfection curve took into account the full co-variance ma-trix V b , as similarly described in [37], with the vector G ′ m being defined as: G ′ m = G ′ m, ... G ′ m,j max , (7)where G ′ m, , ... G ′ m,j max are the partial derivatives of theinfection curve G ( t m , N k , λ k , t k ) in respect to the param-eter P , ... P j max calculated at each day t m ( j max = 3for Mexico and Russia, 6 for Brazil, UK and India and 9for USA). Consequently, the uncertainty of the infectioncurve at each point can be written as: σG m = q G ′ ⊤ m V bG ′ m . (8)The uncertainties in the convoluted functions can be cal-culated as: F ± C ( t ) = F C ( t ) ± if r · Z t [ σG ( τ ) P rob ( t − τ ) dτ ] , (9)where σG ( τ ) is obtained by the interpolation of σG m .Figure 1 shows the weekly averaged deaths per day dis-tributions for all six countries (data points) and its re-spective convoluted functions F c ( t ) (dashed-dotted graylines). The upper and lower estimates F ± C ( t ) [95% Con-fidence Intervals(CI)] are presented by the red and bluedashed-dotted lines, respectively. The total number ofdeaths and its 95% CI at any given time t f can be writ-ten as: N d ( t f ) = N d ( t i ) + Z t f t i +1 F C ( t ′ ) dt ′ , and (10) N ± d ( t f ) = N d ( t i ) + Z t f t i +1 F ± C ( t ′ ) dt ′ , (11)where N d ( t i ) corresponds to the actual data of accumu-lated deaths until the day ( t i ) for each country.Table I summarizes all the results, including the modelpredictions for the total number of deaths and its 95%CI for Aug 31, 2020. The countries with well defined firstpeaks (USA and UK) present the highest initial growthrates (0 . ± . d − and 0 . ± . d − ) and uncer-tainties in the peak days of 1.0 and 2.2 days, respectively.The second peaks in both cases have much lower growthrates, demonstrating the flattening of the infection curvedue to non-pharmacological interventions of social dis-tance and sanitary measures. Besides UK, which clearlyshows a well controlled scenario, Mexico and Russia alsohave a trend down with modest uncertainties, which isa consequence of the fitting being successfully performedwith a single Gompertz function. For the case of India,the first Gompertz function has a quite small contribu-tion ( ∼ N and λ (-0.989), N and λ (-0.972), λ and λ (-0.841) and N and N (-0.973), influencing for the largeuncertainties. A similar situation also play a role for thehuge uncertainties found in the parameters of the thirdGompertz function for USA, which is weakly constrainedwith few data points. The death’s peak days have an av-erage shift of 22.9 days from the corresponding infection’speaks, which is the average of Prob ( t ) (the sum of theaverages of the two gamma functions). The estimatesfor the prevalences of infections at the beginning of eachmonth are also shown in Table I for all six countries.For the case of USA, our estimated prevalence for April3-4, 2020 [2.6% (95% CI, 2.1 to 3.0%)] is in good agree-ment with a preliminary serological study carried out inSanta Clara County [38] 2.8% (95% CI, 1.3-4.7%). Forthe case of Brazil, we found a prevalence of 2.6% (95% CI2.3-2.9%) within May 15-22, 2020, which is higher thanthe overall prevalence found in a survey that included90 cities of Brazil 1.4% (95% CI, 1.3-1.6%)[39]. On theother hand, our result for May 14, 2020 [(2.3%(95% CI,2.1-2.6%)] is in good agreement with the figures reportedin a preliminary research performed in the Brazilian Stateof Espirito Santo [40] 2.1% (95% CI, 1.67-2.52%). Obvi-ously that these comparisons should be done with parsi-mony, given the fact that our results refer to an overallestimate for each country and are strictly related with aninfection-fatality-ratio of 0.66% [33].The upper panel of Figure 2 shows the model predic-tions for the accumulated number of infections (solidlines) and the lower panel shows the correspondingmodel estimates for the accumulated deaths, in com-parison with the available data (data points) at each5 day time interval. Once again it is verified a niceagreement between the data and the model, with somediscrepancies found in the early stages of the pandemic(accumulated number of deaths . Prob ( t ), andthe two gamma distributions (incubation and symptomonset to deaths periods) are presented in the insert ofthe lower panel of Figure 2. In conclusion, we have presented a few-parametermodel to describe the dynamics of a pandemic in terms ofGompertz functions using Monte Carlo techniques to de-termine the best fit parameters and a least square method- weighted by the dispersion of the death’s data in eachepidemiological week - to estimate the relevant uncertain-ties. [1] F. Wu, S. Zhao, B. Yu, Y.-M. Chen, W. Wang, Z.-G.Song, Y. Hu, Z.-W. Tao, J.-H. Tian, Y.-Y. Pei, et al. , Anew coronavirus associated with human respiratory dis-ease in China, Nature , 265 (2020).[2] J. Bedford, J. Farrar, C. Ihekweazu, G. Kang, M. Koop-mans, and J. Nkengasong, A new twenty-first centuryscience for effective epidemic response, Nature , 130(2019).[3] T. Lancet, Emerging understandings of 2019-ncov,Lancet (London, England) , 311 (2020).[4] J. Phua, L. Weng, L. Ling, M. Egi, C.-M. Lim, J. V.Divatia, B. R. Shrestha, Y. M. Arabi, J. Ng, C. D. Gom-ersall, et al. , Intensive care management of coronavirusdisease 2019 (COVID-19): challenges and recommenda-tions, The Lancet Respiratory Medicine (2020).[5] P. Christen, J. D’Aeth, A. Lochen, R. McCabe,D. Rizmie, N. Schmit, A. Nayagam, M. Miraldo,P. White, P. Aylin, et al. , Report 15: Strengthening hos-pital capacity for the COVID-19 pandemic , Tech. Rep.(2020).[6] N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai,K. Ainslie, M. Baguelin, S. Bhatia, A. Boonyasiri, Z. Cu-cunub´a, G. Cuomo-Dannenburg, et al. , Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand. 2020, DOI ,77482 (2020).[7] M. U. Kraemer, C.-H. Yang, B. Gutierrez, C.-H. Wu,B. Klein, D. M. Pigott, L. Du Plessis, N. R. Faria, R. Li,W. P. Hanage, et al. , The effect of human mobility andcontrol measures on the COVID-19 epidemic in China,Science , 493 (2020).[8] C. J. Wang, C. Y. Ng, and R. H. Brook, Response toCOVID-19 in Taiwan: big data analytics, new technol-ogy, and proactive testing, Jama , 1341 (2020).[9] A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu,J. Edmunds, S. Funk, R. M. Eggo, F. Sun, M. Jit, J. D.Munday, et al. , Early dynamics of transmission and con-trol of COVID-19: a mathematical modelling study, TheLancet Infectious Diseases (2020).[10] C. E. Overton, H. B. Stage, S. Ahmad, J. Curran-Sebastian, P. Dark, R. Das, E. Fearon, T. Felton,M. Fyles, N. Gent, et al. , Using statistics and math-ematical modelling to understand infectious diseaseoutbreaks: COVID-19 as an example, arXiv preprintarXiv:2005.04937 (2020).[11] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri,A. Di Filippo, A. Di Matteo, and M. Colaneri, Mod-elling the COVID-19 epidemic and implementation ofpopulation-wide interventions in Italy, Nature Medicine, 1 (2020).[12] K. Prem, Y. Liu, T. W. Russell, A. J. Kucharski, R. M. Eggo, N. Davies, S. Flasche, S. Clifford, C. A. Pearson,J. D. Munday, et al. , The effect of control strategies toreduce social mixing on outcomes of the COVID-19 epi-demic in Wuhan, China: a modelling study, The LancetPublic Health (2020).[13] G. Gaeta, A simple SIR model with a large set of asymp-tomatic infectives, arXiv preprint arXiv:2003.08720(2020).[14] L. F. Scabini, L. C. Ribas, M. B. Neiva, A. G. Junior,A. J. Farf´an, and O. M. Bruno, Social interaction layers incomplex networks for the dynamical epidemic modelingof COVID-19 in Brazil, arXiv preprint arXiv:2005.08125(2020).[15] K. Leung, J. T. Wu, D. Liu, and G. M. Leung, First-waveCOVID-19 transmissibility and severity in China outsideHubei after control measures, and second-wave scenarioplanning: a modelling impact assessment, The Lancet(2020).[16] D. S. Candido, I. M. Claro, J. G. de Jesus, W. M. Souza,F. R. Moreira, S. Dellicour, T. A. Mellan, L. du Plessis,R. H. Pereira, F. C. Sales, et al. , Evolution and epidemicspread of sars-cov-2 in Brazil, Science (2020).[17] Y. Fang, Y. Nie, and M. Penny, Transmission dynamics ofthe COVID-19 outbreak and effectiveness of governmentinterventions: A data-driven analysis, Journal of medicalvirology , 645 (2020).[18] E. Massad, M. Amaku, A. Wilder-Smith, P. C. C. dosSantos, C. J. Struchiner, and F. A. B. Coutinho, Twocomplementary model-based methods for calculating therisk of international spreading of anovel virus from theoutbreak epicentre. the case of COVID-19, Epidemiology& Infection , 1 (2020).[19] J. Sch¨uttler, R. Schlickeiser, F. Schlickeiser, andM. Kr¨oger, Covid-19 predictions using a Gauss model,based on data from april 2, Physics , e0236387(2020).[23] B. Paix˜ao, L. Baroni, R. Salles, L. Escobar, C. de Sousa,M. Pedroso, R. Saldanha, R. Coutinho, F. Porto,and E. Ogasawara, Estimation of COVID-19 under-reporting in Brazilian States through sari, arXiv preprintarXiv:2006.12759 (2020). [24] S. G. Krantz and A. S. S. Rao, Level of underreportingincluding underdiagnosis before the first peak of COVID-19 in various countries: Preliminary retrospective resultsbased on wavelets and deterministic modeling, InfectionControl & Hospital Epidemiology , 1 (2020).[25] K. Jagodnik, F. Ray, F. M. Giorgi, and A. Lach-mann, Correcting under-reported COVID-19 case num-bers: estimating the true scale of the pandemic, PreprintmedRvix (2020).[26] J. Cohen and K. Kupferschmidt, Countries test tacticsin waragainst COVID-19 (2020).[27] A. Capaldi, S. Behrend, B. Berman, J. Smith, J. Wright,and A. L. Lloyd, Parameter estimation and uncertaintyquantication for an epidemic model, Mathematical bio-sciences and engineering , 553 (2012).[28] R. H. Gaffey and C. Viboud, Application of the CDCEbola Response modeling tool to disease predictions,Epidemics , 22 (2018).[29] T. Kuniya, Prediction of the epidemic peak of coron-avirus disease in japan, 2020, Journal of clinical medicine , 789 (2020).[30] A. Smirnova, B. Sirb, and G. Chowell, On stable param-eter estimation and forecasting in epidemiology by thelevenberg–marquardt algorithm with broydens rank-oneupdates for the jacobian operator, Bulletin of mathemat-ical biology , 4210 (2019).[31] J. Peto, Covid-19 mass testing facilities could end theepidemic rapidly, Bmj (2020).[32] S. Flaxman, S. Mishra, A. Gandy, H. Unwin, H. Cou-pland, T. Mellan, H. Zhu, T. Berah, J. Eaton,P. Perez Guzman, et al. , Report 13: Estimating the num-ber of infections and the impact of non-pharmaceuticalinterventions on COVID-19 in 11 European countries ,Tech. Rep. (2020).[33] R. Verity, L. C. Okell, I. Dorigatti, P. Winskill, C. Whit-taker, N. Imai, G. Cuomo-Dannenburg, H. Thompson,P. G. Walker, H. Fu, et al.
Update of X ray andgamma ray decay data standards for detector calibrationand other applications. V. 2: Data selection, assessmentand evaluation procedures (2007).[37] O. Helene, L. Mariano, and Z. Guimaraes-Filho, Use-ful and little-known applications of the least squaremethod and some consequences of covariances, NuclearInstruments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and AssociatedEquipment , 82 (2016).[38] E. Bendavid, B. Mulaney, N. Sood, S. Shah, E. Ling,R. Bromley-Dulfano, C. Lai, Z. Weissberg, R. Saavedra,J. Tedrow, et al. , COVID-19 antibody seroprevalence inSanta Clara County, California, MedRxiv (2020).[39] P. Hallal, F. Hartwig, B. Horta, G. D. Victora, M. Sil-veira, C. Struchiner, L. P. Vidaletti, N. Neumann, L. C.Pellanda, O. A. Dellagostin, et al. , Remarkable vari-ability in sars-cov-2 antibodies across Brazilian regions:nationwide serological household survey in 27 states,medRxiv (2020).[40] C. C. Gomes, C. Cerutti, E. Zandonade, E. L. N. Maciel,F. E. C. de Alencar, G. L. Almada, O. A. Cardoso, P. M.Jabor, R. L. Zanotti, T. Q. Reuter, et al. , A population-based study of the prevalence of COVID-19 infection inEspirito Santo, Brazil: methodology and results of thefirst stage, medRxiv (2020). C (t) F C+ (t) F C- (t) USA d : Apr 29, 2020 (a) 0 50 100 150 200 250050010001500 Brazil d : Jun 04, 2020 (b)0 50 100 150 200 25005001000 United Kingdom d : May 10, 2020 (c) Mexico d : Jun 15,2020 (d)0 50 100 150 200 25005001000 Mean day of the epidemiological week (d)
India* d : May 9, 2020 (e) 0 50 100 150 200 2500100200 Mean day of the epidemiological week (d) W ee k l y a v e r a g e d d e a t h s p e r d a y W ee k l y a v e r a g e d d e a t h s p e r d a y W ee k l y a v e r a g e d d e a t h s p e r d a y Russia* d : May 10, 2020 (f) FIG. 1. Weekly averaged deaths per day for USA (a), Brazil (b), Mexico (c), United Kingdom (d), India (e) and Russia (f)(data points) and the respective model estimations given by F C ( t ) (Gray dotted-dashed lines) and its 95% CI upper and lowerlimits (red and blue dashed dotted lines, respectively). Also shown for clarity the hundredths calendar day ( d ) since the firstconfirmed case for each country. (*) For the case of India and Russia a 5% error was added to the death’s data to achieve asuccessful fitting. TABLE I. Best fit parameters of the Gompertz functions of the reconstructed infection curves and fitting results of the death’sdata obtained for all six countries. Also shown the 95% confidence interval (CI) for the total number of deaths for Aug 31,2020 and the estimated prevalences at the first day of each month.USA Brazil Mexico UK India a Russia a N (10 ) 11.6 ± ±
14 11.9 ± ± ± ± λ ( d − ) 0.109 ± ± ± ± ± ± t ( d ) 65.7 ± ±
25 103 ± ± ±
13 112.8 ± N (10 ) 12.8 ± ±
23 - 2.1 ± ± λ ( d − ) 0.026 ± ± ± ± t ( d ) 115 ±
11 135 ±
28 - 93 ±
24 180 ±
21 - N (10 ) 6 ± λ ( d − ) 0.07 ± t ( d ) 179 ±
18 - - - - - N d (95% CI) (10 ) b < < < < < < < < < n/n.d.f. c χ /p d Mar 07, 2020 Mar 21, 2020 Mar 28, 2020 Mar 14, 2020 Mar 28, 2020 Mar 28, 2020 a
5% error was added to the deaths data to achieve a successful fitting b Model estimates for Aug 31, 2020 c n.d.f. : number of degrees of freedom d These dates correspond to the first day of the first epidemiological week included in the fitting. P r o b a b i l i t y time (d)gamma(5.1,0.86)gamma(17.8,0.45)Prob(t) A cc u m u l a t e d d e a t h s Time since first confirmed case (d)
EUA Brazil UKMexicoindia Russia USA (model) Brazil (model) UK (model) Mexico (model) India (model) Russia (model) A cc u m u l t a e d i n f e c t i o n s FIG. 2. Upper panel: Model predictions for the accumulated number of infections for all six countries (solid lines). Lower Panel:Model predictions for the accumulated deaths (solid lines) versus the available data (data-points), which are presented within5 days’ time intervals for clarity. The insert of the lower panel shows the Monte Carlo generated distribution
Prob ( tt