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An international initiative of predicting the SARS-CoV-2 pandemic using ensemble data assimilation
1. | NORCE and NERSC, Bergen, Norway |
2. | Dept. of Meteorology University of Reading and NCEO, UK |
3. | CEREA, joint laboratory École des Ponts ParisTech and EDF R & D Université Paris-Est, Champs-sur-Marne, France |
4. | Mathematical Institute University of Utrecht, Netherlands |
5. | Environment and Climate Change Canada Dorval, Québec, Canada |
6. | Renaissance Computing Institute University of North Carolina, Chapel Hill, USA |
7. | Department of Geoscience and Engineering Delft University of Technology, Delft, Netherlands |
8. | FaCENA, UNNE and IMIT, CONICET Corrientes, Argentina |
This work demonstrates the efficiency of using iterative ensemble smoothers to estimate the parameters of an SEIR model. We have extended a standard SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition the model on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters yielding model predictions in close agreement with the observations. The updated ensemble of model simulations has predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development assuming knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries, i.e., the four European countries Norway, England, The Netherlands, and France; the province of Quebec in Canada; the South American countries Argentina and Brazil; and the four US states Alabama, North Carolina, California, and New York. These countries and states all have vastly different developments of the epidemic, and we could accurately model the SARS-CoV-2 outbreak in all of them. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.
References:
[1] |
S. I. Aanonsen, G. Nævdal, D. S. Oliver, A. C. Reynolds and B. Vallès,
Ensemble Kalman filter in reservoir engineering – A review, SPE Journal, 14 (2009), 393-412.
doi: 10.2118/117274-PA. |
[2] |
S. Abrams, The analysis of multivariate serological data, in |
[3] |
J. L. Anderson and S. L. Anderson,
A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127 (1999), 2741-2758.
doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2. |
[4] |
E. Armstrong, M. Runge and J. Gerardin, Identifying the measurements required to estimate rates of COVID-19 transmission, infection, and detection, using variational data assimilation, Infectious Disease Modelling, to appear.
doi: 10.1101/2020.05.27.20112987. |
[5] |
M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974546.pt1. |
[6] |
L. M. A. Bettencourt, R. M. Ribeiro, G. Chowell, T. Lant and C. Castillo-Chavez, Towards real time epidemiology: Data assimilation, modeling and anomaly detection of health surveillance data streams, in Intelligence and Security Informatics: Biosurveillance, Lecture Notes in Computer Science, 4506, Springer, 2007, 79–90.
doi: 10.1007/978-3-540-72608-1_8. |
[7] |
J. C. Blackwood and L. M. Childs,
An introduction to compartmental modeling for the budding infectious disease modeler, Lett. Biomath., 5 (2018), 195-221.
doi: 10.30707/LiB5.1Blackwood. |
[8] |
M. Bocquet and P. Sakov, An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140 (2014), 1521-1535. Google Scholar |
[9] |
M. Bocquet and P. Sakov,
Joint state and parameter estimation with an iterative ensemble Kalman smoother, Nonlin. Processes Geophys., 20 (2013), 803-818.
doi: 10.5194/npg-20-803-2013. |
[10] |
C. {B}rasil, Estimativa de Casos de COVID-19, 2020. Available from: https://ciis.fmrp.usp.br/covid19-subnotificacao/. Google Scholar |
[11] |
R. Buizza, M. Milleer and T. N. Palmer,
Stochastic representation of model uncertainties in the ECMWF ensemble prediction system, Q. J. R. Meteorol. Soc., 125 (1999), 2887-2908.
doi: 10.1002/qj.49712556006. |
[12] |
G. Burgers, P. J. van Leeuwen and G. Evensen,
Analysis scheme in the ensemble Kalman filter, Mon. Weather Rev., 126 (1998), 1719-1724.
doi: 10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2. |
[13] |
H. Cao and Y. Zhou,
The discrete age-structured SEIT model with application to tuberculosis transmission in China, Math. Comput. Modelling, 55 (2012), 385-395.
doi: 10.1016/j.mcm.2011.08.017. |
[14] |
A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the Geosciences: An overview on methods, issues and perspectives, WIREs Climate Change, 9 (2018), 50pp.
doi: 10.1002/wcc.535. |
[15] |
CBS, Bevolkingspyramide, Statistics Netherlands (CBS), 2020. Available from: https://www.cbs.nl/nl-nl/visualisaties/bevolkingspiramide. Google Scholar |
[16] |
CBS, Nearly 9 Thousand More Deaths in First 9 Weeks of COVID-19, Statistics Netherlands (CBS), 2020. Available from: https://www.cbs.nl/en-gb/news/2020/20/nearly-9-thousand-more-deaths-in-first-9-weeks-of-covid-19. Google Scholar |
[17] |
N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble Kalman inversion, Inverse Problems, 34 (2018), 31pp.
doi: 10.1088/1361-6420/aab6d9. |
[18] |
Y. Chen and D. S. Oliver,
Ensemble randomized maximum likelihood method as an iterative ensemble smoother, Math. Geosci., 44 (2012), 1-26.
doi: 10.1007/s11004-011-9376-z. |
[19] |
Y. Chen and D. S. Oliver,
Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification, Comput. Geosci., 17 (2013), 689-703.
doi: 10.1007/s10596-013-9351-5. |
[20] |
COVID-19 in Brazil: "So what?", The Lancet, 395 (2020).
doi: 10.1016/S0140-6736(20)31095-3. |
[21] |
A. A. Emerick and A. C. Reynolds,
Ensemble smoother with multiple data assimilation, Comput. Geosci., 55 (2013), 3-15.
doi: 10.1016/j.cageo.2012.03.011. |
[22] |
R. Engbert, M. M. Rabe, R. Kliegl and S. Reich, Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics, Bull. Math. Biol., 83 (2021).
doi: 10.1007/s11538-020-00834-8. |
[23] |
G. Evensen,
Accounting for model errors in iterative ensemble smoothers, Comput. Geosci., 23 (2019), 761-775.
doi: 10.1007/s10596-019-9819-z. |
[24] |
G. Evensen,
Analysis of iterative ensemble smoothers for solving inverse problems, Comput. Geosci., 22 (2018), 885-908.
doi: 10.1007/s10596-018-9731-y. |
[25] |
G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[26] |
G. Evensen,
The ensemble Kalman filter for combined state and parameter estimation: Monte Carlo techniques for data assimilation in large systems, IEEE Control Syst. Mag., 29 (2009), 83-104.
doi: 10.1109/MCS.2009.932223. |
[27] |
G. Evensen, Formulating the history matching problem with consistent error statistics, Comput. Geosci., to appear. Google Scholar |
[28] |
G. Evensen,
Sampling strategies and square root analysis schemes for the EnKF, Ocean Dynamics, 54 (2004), 539-560.
doi: 10.1007/s10236-004-0099-2. |
[29] |
G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99 (1994).
doi: 10.1029/94JC00572. |
[30] |
G. Evensen, P. N. Raanes, A. S. Stordal and J. Hove, Efficient implementation of an iterative ensemble smoother for data assimilation and reservoir history matching, Front. Appl. Math. Stat., 5 (2019), 47pp.
doi: 10.3389/fams.2019.00047. |
[31] |
S. Flaxman, S. Mishra, A. Gandy, H. Unwin and H. Coupland, et al., Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries, 2020. Available from: https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-13-europe-npi-impact/. Google Scholar |
[32] |
Gouvernement de la République Française, COVID-19: Carte et Données, 2020. Available from: https://www.gouvernement.fr/info-coronavirus/carte-et-donnees. Google Scholar |
[33] |
H. Gupta, K. K. Verma and P. Sharma, Using data assimilation technique and epidemic model to predict TB epidemic, Internat. J. Comput. Appl., 128 (2015), 5pp.
doi: 10.5120/ijca2015906625. |
[34] |
P. L. Houtekamer and H. L. Mitchell,
Data assimilation using an ensemble Kalman filter technique, Mon. Weather Rev., 126 (1998), 796-811.
doi: 10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2. |
[35] |
P. L. Houtekamer and F. Zhang,
Review of the ensemble Kalman filter for atmospheric data assimilation, Mon. Weather Rev., 144 (2016), 4489-4532.
doi: 10.1175/MWR-D-15-0440.1. |
[36] |
M. A. Iglesias, K. J. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 20pp.
doi: 10.1088/0266-5611/29/4/045001. |
[37] |
Imperial College COVID-19 Response Team, Short-term forecasts of COVID-19 deaths in multiple countries, 2020. Available from: https://mrc-ide.github.io/covid19-short-term-forecasts/index.html. Google Scholar |
[38] |
A. J. Ing, C. Cocks and J. P. Green,
COVID-19: In the footsteps of Ernest Shackleton, Thorax, 75 (2020), 613-613.
doi: 10.1136/thoraxjnl-2020-215091. |
[39] |
Institut de la Statistique Québec, 2020. Available from: https://www.stat.gouv.qc.ca/statistiques/population-demographie/deces-mortalite/nombre-hebdomadaire-deces.html., Google Scholar |
[40] |
Institut de la Statistique Québec: Population Data, 2019. Available from: https://www.stat.gouv.qc.ca/statistiques/population-demographie/structure/population-quebec-age-sexe.html#tri_pop=20., Google Scholar |
[41] |
Institut National de Santé Publique Québec, 2020. Available from: https://www.inspq.qc.ca/covid-19/donnees., Google Scholar |
[42] |
C. Jarvis, K. Van Zandvoort and A. Gimma, et al., Quantifying the impact of physical distance measures on the transmission of COVID-19 in the UK, BMC Med, 18 (2020), 1416-1430.
doi: 10.1186/s12916-020-01597-8. |
[43] |
M. A. Jorden, S. L. Rudman, E. Villarino, S. Hoferka and M. T. Patel, et al., Evidence for limited early spread of COVID-19 within the United States, January-February 2020, Morbid. Mortal. Weekly Rep. (MMWR), 69 (2020), 680-684,
doi: 10.15585/mmwr.mm6922e1. |
[44] |
A. A. King, E. L. Ionides, M. Pascual and M. J. Bouma,
Inapparent infections and cholera dynamics, Nature, 454 (2008), 877-880.
doi: 10.1038/nature07084. |
[45] |
R. Li, S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang and J. Shaman,
Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2), Science, 368 (2020), 489-493.
doi: 10.1126/science.abb3221. |
[46] |
T. A. Mellan, H. H. Hoeltgebaum, S. Mishra, C. Whittaker and R. Schnekenberg, et al., Report 21: Estimating COVID-19 cases and reproduction number in Brazil, (2020).
doi: 10.25561/78872. |
[47] |
J. Mossong, N. Hens, M. Jit, P. Beutels and K. Auranen, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med, 5.
doi: 10.1371/journal.pmed.0050074. |
[48] |
C. J. L. Murray, Forecasting the impact of the first wave of the COVID-19 pandemic on hospital demand and deaths for the USA and European economic area countries, preprint.
doi: 10.1101/2020.04.21.20074732. |
[49] |
National Health Service, Covid-19 Daily Deaths, 2020. Available from: https://www.england.nhs.uk/statistics/statistical-work-areas/covid-19-daily-deaths/. Google Scholar |
[50] |
R. M. Neal,
Sampling from multimodal distributions using tempered transitions, Statist. Comput., 6 (1996), 353-366.
doi: 10.1007/BF00143556. |
[51] |
NICE, COVID-19 Infecties op de IC's, Nationale Intensive Care Evaluatie, 2020. Accessed from: https://www.stichting-nice.nl/. Google Scholar |
[52] |
NICE, COVID-19 Infecties op de Verpleegadeling, Nationale Intensive Care Evaluatie, 2020. Available from: https://www.stichting-nice.nl/covid-19-op-de-zkh.jsp/ Google Scholar |
[53] |
D. Pasetto, F. Finger, A. Rinaldo and E. Bertuzzo,
Real-time projections of cholera outbreaks through data assimilation and rainfall forecasting, Adv. Water Res., 108 (2017), 345-356.
doi: 10.1016/j.advwatres.2016.10.004. |
[54] |
Public Health, England, The health protection (coronavirus, business closure) (England) regulations 2020, 2020. Available from: https://web.archive.org/web/20200323004800/http://www.legislation.gov.uk/uksi/2020/327/pdfs/uksi_20200327_en.pdf. Google Scholar |
[55] |
P. N. Raanes, A. S. Stordal and G. Evensen,
Revising the stochastic iterative ensemble smoother, Nonlin. Processes Geophys, 26 (2019), 325-338.
doi: 10.5194/npg-26-325-2019. |
[56] |
Registro Civil, Portal da Transparencia - Especial COVID-19, 2020. Available from: https://transparencia.registrocivil.org.br/especial-covid. Google Scholar |
[57] |
C. J. Rhodes and T. D. Hollingsworth,
Variational data assimilation with epidemic models, J. Theoret. Biol., 258 (2009), 591-602.
doi: 10.1016/j.jtbi.2009.02.017. |
[58] |
RIVM, Briefing Update Coronavirus Tweede Kamer 20 Mei 2020, National Institute for Public Health and the Environment, 2020. Available from: https://www.tweedekamer.nl/sites/default/files/atoms/files/presentatie_jaap_van_dissel_-_technische_briefing_20_mei_2020.pdf. Google Scholar |
[59] |
RIVM, Excess Mortality Caused by the Novel Coronavirus (COVID-19), National Institute for Public Health and the Environment, 2020. Available from: https://www.rivm.nl/node/155011. Google Scholar |
[60] |
RIVM, Ontwikkeling COVID-19 in Grafieken, National Institute for Public Health and the Environment, 2020. Available from: https://www.rivm.nl/coronavirus-covid-19/grafieken. Google Scholar |
[61] |
H. Salje, C. Tran Kiem, N. Lefrancq, N. Courtejoie and P. Bosetti, et al., Estimating the burden of SARS-CoV-2 in France, Science, 369 (2020), 208-211.
doi: 10.1126/science.abc3517. |
[62] |
J. L. Sesterhenn, Adjoint-based data assimilation of an epidemiology model for the COVID-19 pandemic in 2020, preprint, arXiv: 2003.13071. Google Scholar |
[63] |
J. Shaman, A. Karspeck, W. Yang, J. Tamerius and M. Lipsitch,
Real-time influenza forecasts during the 2012–2013 season, Nature Commu., 4 (2013), 1-10.
doi: 10.1038/ncomms3837. |
[64] |
A. S. Stordal and A. H. Elsheikh,
Iterative ensemble smoothers in the annealed importance sampling framework, Adv. Water Res., 86 (2015), 231-239.
doi: 10.1016/j.advwatres.2015.09.030. |
[65] |
UK Government, Coronavirus (COVID-19) in the UK, 2020. Available from: https://coronavirus.data.gov.uk. Google Scholar |
[66] |
UK Government, National COVID-19 Surveillance Reports, 2020. Available from: https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/. Google Scholar |
[67] |
UK Government, Slides, Datasets and Transcripts to Accompany Coronavirus Press Conferences, 2020. Available from: https://www.gov.uk/government/collections/slides-and-datasets-to-accompany-coronavirus-press-conferences/. Google Scholar |
[68] |
UK Office for National Statistics, Dataset: Deaths Registered Weekly in England and Wales, Provisional, 2020., Available from: https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/datasets/weeklyprovisionalfiguresondeathsregisteredinenglandandwales. Google Scholar |
[69] |
J. van Wees, S. Osinga, M. van der Kuip, M. Tanck and M. Hanegraaf, et al., Forecasting hospitalization and ICU rates of the COVID-19 outbreak: An efficient SEIR model, Bull. World Health Org., (2020).
doi: 10.2471/BLT.20.256743. |
[70] |
J. S. Whitaker and T. M. Hamill,
Evaluating methods to account for system errors in ensemble data assimilation, Mon. Weather. Rev., 140 (2012), 3078-3089.
doi: 10.1175/MWR-D-11-00276.1. |
[71] |
WHO, Coronavirus Disease (COVID-19): Similarities and Differences with Influenza, 2020. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/question-and-answers-hub/q-a-detail/q-a-similarities-and-differences-covid-19-and-influenza. Google Scholar |
[72] |
W. Yang, M. Lipsitch and J. Shaman,
Inference of seasonal and pandemic influenza transmission dynamics, PNAS, 112 (2015), 2723-2728.
doi: 10.1073/pnas.1415012112. |
[73] |
W. Yang, W. Zhang, D. Kargbo, R. Yang and Y. Chen, et al., Transmission network of the 2014–2015 Ebola epidemic in Sierra Leone, J. Roy. Soc. Interface, 12 (2015).
doi: 10.1098/rsif.2015.0536. |
show all references
References:
[1] |
S. I. Aanonsen, G. Nævdal, D. S. Oliver, A. C. Reynolds and B. Vallès,
Ensemble Kalman filter in reservoir engineering – A review, SPE Journal, 14 (2009), 393-412.
doi: 10.2118/117274-PA. |
[2] |
S. Abrams, The analysis of multivariate serological data, in |
[3] |
J. L. Anderson and S. L. Anderson,
A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127 (1999), 2741-2758.
doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2. |
[4] |
E. Armstrong, M. Runge and J. Gerardin, Identifying the measurements required to estimate rates of COVID-19 transmission, infection, and detection, using variational data assimilation, Infectious Disease Modelling, to appear.
doi: 10.1101/2020.05.27.20112987. |
[5] |
M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974546.pt1. |
[6] |
L. M. A. Bettencourt, R. M. Ribeiro, G. Chowell, T. Lant and C. Castillo-Chavez, Towards real time epidemiology: Data assimilation, modeling and anomaly detection of health surveillance data streams, in Intelligence and Security Informatics: Biosurveillance, Lecture Notes in Computer Science, 4506, Springer, 2007, 79–90.
doi: 10.1007/978-3-540-72608-1_8. |
[7] |
J. C. Blackwood and L. M. Childs,
An introduction to compartmental modeling for the budding infectious disease modeler, Lett. Biomath., 5 (2018), 195-221.
doi: 10.30707/LiB5.1Blackwood. |
[8] |
M. Bocquet and P. Sakov, An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140 (2014), 1521-1535. Google Scholar |
[9] |
M. Bocquet and P. Sakov,
Joint state and parameter estimation with an iterative ensemble Kalman smoother, Nonlin. Processes Geophys., 20 (2013), 803-818.
doi: 10.5194/npg-20-803-2013. |
[10] |
C. {B}rasil, Estimativa de Casos de COVID-19, 2020. Available from: https://ciis.fmrp.usp.br/covid19-subnotificacao/. Google Scholar |
[11] |
R. Buizza, M. Milleer and T. N. Palmer,
Stochastic representation of model uncertainties in the ECMWF ensemble prediction system, Q. J. R. Meteorol. Soc., 125 (1999), 2887-2908.
doi: 10.1002/qj.49712556006. |
[12] |
G. Burgers, P. J. van Leeuwen and G. Evensen,
Analysis scheme in the ensemble Kalman filter, Mon. Weather Rev., 126 (1998), 1719-1724.
doi: 10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2. |
[13] |
H. Cao and Y. Zhou,
The discrete age-structured SEIT model with application to tuberculosis transmission in China, Math. Comput. Modelling, 55 (2012), 385-395.
doi: 10.1016/j.mcm.2011.08.017. |
[14] |
A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the Geosciences: An overview on methods, issues and perspectives, WIREs Climate Change, 9 (2018), 50pp.
doi: 10.1002/wcc.535. |
[15] |
CBS, Bevolkingspyramide, Statistics Netherlands (CBS), 2020. Available from: https://www.cbs.nl/nl-nl/visualisaties/bevolkingspiramide. Google Scholar |
[16] |
CBS, Nearly 9 Thousand More Deaths in First 9 Weeks of COVID-19, Statistics Netherlands (CBS), 2020. Available from: https://www.cbs.nl/en-gb/news/2020/20/nearly-9-thousand-more-deaths-in-first-9-weeks-of-covid-19. Google Scholar |
[17] |
N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble Kalman inversion, Inverse Problems, 34 (2018), 31pp.
doi: 10.1088/1361-6420/aab6d9. |
[18] |
Y. Chen and D. S. Oliver,
Ensemble randomized maximum likelihood method as an iterative ensemble smoother, Math. Geosci., 44 (2012), 1-26.
doi: 10.1007/s11004-011-9376-z. |
[19] |
Y. Chen and D. S. Oliver,
Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification, Comput. Geosci., 17 (2013), 689-703.
doi: 10.1007/s10596-013-9351-5. |
[20] |
COVID-19 in Brazil: "So what?", The Lancet, 395 (2020).
doi: 10.1016/S0140-6736(20)31095-3. |
[21] |
A. A. Emerick and A. C. Reynolds,
Ensemble smoother with multiple data assimilation, Comput. Geosci., 55 (2013), 3-15.
doi: 10.1016/j.cageo.2012.03.011. |
[22] |
R. Engbert, M. M. Rabe, R. Kliegl and S. Reich, Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics, Bull. Math. Biol., 83 (2021).
doi: 10.1007/s11538-020-00834-8. |
[23] |
G. Evensen,
Accounting for model errors in iterative ensemble smoothers, Comput. Geosci., 23 (2019), 761-775.
doi: 10.1007/s10596-019-9819-z. |
[24] |
G. Evensen,
Analysis of iterative ensemble smoothers for solving inverse problems, Comput. Geosci., 22 (2018), 885-908.
doi: 10.1007/s10596-018-9731-y. |
[25] |
G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[26] |
G. Evensen,
The ensemble Kalman filter for combined state and parameter estimation: Monte Carlo techniques for data assimilation in large systems, IEEE Control Syst. Mag., 29 (2009), 83-104.
doi: 10.1109/MCS.2009.932223. |
[27] |
G. Evensen, Formulating the history matching problem with consistent error statistics, Comput. Geosci., to appear. Google Scholar |
[28] |
G. Evensen,
Sampling strategies and square root analysis schemes for the EnKF, Ocean Dynamics, 54 (2004), 539-560.
doi: 10.1007/s10236-004-0099-2. |
[29] |
G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99 (1994).
doi: 10.1029/94JC00572. |
[30] |
G. Evensen, P. N. Raanes, A. S. Stordal and J. Hove, Efficient implementation of an iterative ensemble smoother for data assimilation and reservoir history matching, Front. Appl. Math. Stat., 5 (2019), 47pp.
doi: 10.3389/fams.2019.00047. |
[31] |
S. Flaxman, S. Mishra, A. Gandy, H. Unwin and H. Coupland, et al., Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries, 2020. Available from: https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-13-europe-npi-impact/. Google Scholar |
[32] |
Gouvernement de la République Française, COVID-19: Carte et Données, 2020. Available from: https://www.gouvernement.fr/info-coronavirus/carte-et-donnees. Google Scholar |
[33] |
H. Gupta, K. K. Verma and P. Sharma, Using data assimilation technique and epidemic model to predict TB epidemic, Internat. J. Comput. Appl., 128 (2015), 5pp.
doi: 10.5120/ijca2015906625. |
[34] |
P. L. Houtekamer and H. L. Mitchell,
Data assimilation using an ensemble Kalman filter technique, Mon. Weather Rev., 126 (1998), 796-811.
doi: 10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2. |
[35] |
P. L. Houtekamer and F. Zhang,
Review of the ensemble Kalman filter for atmospheric data assimilation, Mon. Weather Rev., 144 (2016), 4489-4532.
doi: 10.1175/MWR-D-15-0440.1. |
[36] |
M. A. Iglesias, K. J. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 20pp.
doi: 10.1088/0266-5611/29/4/045001. |
[37] |
Imperial College COVID-19 Response Team, Short-term forecasts of COVID-19 deaths in multiple countries, 2020. Available from: https://mrc-ide.github.io/covid19-short-term-forecasts/index.html. Google Scholar |
[38] |
A. J. Ing, C. Cocks and J. P. Green,
COVID-19: In the footsteps of Ernest Shackleton, Thorax, 75 (2020), 613-613.
doi: 10.1136/thoraxjnl-2020-215091. |
[39] |
Institut de la Statistique Québec, 2020. Available from: https://www.stat.gouv.qc.ca/statistiques/population-demographie/deces-mortalite/nombre-hebdomadaire-deces.html., Google Scholar |
[40] |
Institut de la Statistique Québec: Population Data, 2019. Available from: https://www.stat.gouv.qc.ca/statistiques/population-demographie/structure/population-quebec-age-sexe.html#tri_pop=20., Google Scholar |
[41] |
Institut National de Santé Publique Québec, 2020. Available from: https://www.inspq.qc.ca/covid-19/donnees., Google Scholar |
[42] |
C. Jarvis, K. Van Zandvoort and A. Gimma, et al., Quantifying the impact of physical distance measures on the transmission of COVID-19 in the UK, BMC Med, 18 (2020), 1416-1430.
doi: 10.1186/s12916-020-01597-8. |
[43] |
M. A. Jorden, S. L. Rudman, E. Villarino, S. Hoferka and M. T. Patel, et al., Evidence for limited early spread of COVID-19 within the United States, January-February 2020, Morbid. Mortal. Weekly Rep. (MMWR), 69 (2020), 680-684,
doi: 10.15585/mmwr.mm6922e1. |
[44] |
A. A. King, E. L. Ionides, M. Pascual and M. J. Bouma,
Inapparent infections and cholera dynamics, Nature, 454 (2008), 877-880.
doi: 10.1038/nature07084. |
[45] |
R. Li, S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang and J. Shaman,
Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2), Science, 368 (2020), 489-493.
doi: 10.1126/science.abb3221. |
[46] |
T. A. Mellan, H. H. Hoeltgebaum, S. Mishra, C. Whittaker and R. Schnekenberg, et al., Report 21: Estimating COVID-19 cases and reproduction number in Brazil, (2020).
doi: 10.25561/78872. |
[47] |
J. Mossong, N. Hens, M. Jit, P. Beutels and K. Auranen, et al., Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Med, 5.
doi: 10.1371/journal.pmed.0050074. |
[48] |
C. J. L. Murray, Forecasting the impact of the first wave of the COVID-19 pandemic on hospital demand and deaths for the USA and European economic area countries, preprint.
doi: 10.1101/2020.04.21.20074732. |
[49] |
National Health Service, Covid-19 Daily Deaths, 2020. Available from: https://www.england.nhs.uk/statistics/statistical-work-areas/covid-19-daily-deaths/. Google Scholar |
[50] |
R. M. Neal,
Sampling from multimodal distributions using tempered transitions, Statist. Comput., 6 (1996), 353-366.
doi: 10.1007/BF00143556. |
[51] |
NICE, COVID-19 Infecties op de IC's, Nationale Intensive Care Evaluatie, 2020. Accessed from: https://www.stichting-nice.nl/. Google Scholar |
[52] |
NICE, COVID-19 Infecties op de Verpleegadeling, Nationale Intensive Care Evaluatie, 2020. Available from: https://www.stichting-nice.nl/covid-19-op-de-zkh.jsp/ Google Scholar |
[53] |
D. Pasetto, F. Finger, A. Rinaldo and E. Bertuzzo,
Real-time projections of cholera outbreaks through data assimilation and rainfall forecasting, Adv. Water Res., 108 (2017), 345-356.
doi: 10.1016/j.advwatres.2016.10.004. |
[54] |
Public Health, England, The health protection (coronavirus, business closure) (England) regulations 2020, 2020. Available from: https://web.archive.org/web/20200323004800/http://www.legislation.gov.uk/uksi/2020/327/pdfs/uksi_20200327_en.pdf. Google Scholar |
[55] |
P. N. Raanes, A. S. Stordal and G. Evensen,
Revising the stochastic iterative ensemble smoother, Nonlin. Processes Geophys, 26 (2019), 325-338.
doi: 10.5194/npg-26-325-2019. |
[56] |
Registro Civil, Portal da Transparencia - Especial COVID-19, 2020. Available from: https://transparencia.registrocivil.org.br/especial-covid. Google Scholar |
[57] |
C. J. Rhodes and T. D. Hollingsworth,
Variational data assimilation with epidemic models, J. Theoret. Biol., 258 (2009), 591-602.
doi: 10.1016/j.jtbi.2009.02.017. |
[58] |
RIVM, Briefing Update Coronavirus Tweede Kamer 20 Mei 2020, National Institute for Public Health and the Environment, 2020. Available from: https://www.tweedekamer.nl/sites/default/files/atoms/files/presentatie_jaap_van_dissel_-_technische_briefing_20_mei_2020.pdf. Google Scholar |
[59] |
RIVM, Excess Mortality Caused by the Novel Coronavirus (COVID-19), National Institute for Public Health and the Environment, 2020. Available from: https://www.rivm.nl/node/155011. Google Scholar |
[60] |
RIVM, Ontwikkeling COVID-19 in Grafieken, National Institute for Public Health and the Environment, 2020. Available from: https://www.rivm.nl/coronavirus-covid-19/grafieken. Google Scholar |
[61] |
H. Salje, C. Tran Kiem, N. Lefrancq, N. Courtejoie and P. Bosetti, et al., Estimating the burden of SARS-CoV-2 in France, Science, 369 (2020), 208-211.
doi: 10.1126/science.abc3517. |
[62] |
J. L. Sesterhenn, Adjoint-based data assimilation of an epidemiology model for the COVID-19 pandemic in 2020, preprint, arXiv: 2003.13071. Google Scholar |
[63] |
J. Shaman, A. Karspeck, W. Yang, J. Tamerius and M. Lipsitch,
Real-time influenza forecasts during the 2012–2013 season, Nature Commu., 4 (2013), 1-10.
doi: 10.1038/ncomms3837. |
[64] |
A. S. Stordal and A. H. Elsheikh,
Iterative ensemble smoothers in the annealed importance sampling framework, Adv. Water Res., 86 (2015), 231-239.
doi: 10.1016/j.advwatres.2015.09.030. |
[65] |
UK Government, Coronavirus (COVID-19) in the UK, 2020. Available from: https://coronavirus.data.gov.uk. Google Scholar |
[66] |
UK Government, National COVID-19 Surveillance Reports, 2020. Available from: https://www.gov.uk/government/publications/national-covid-19-surveillance-reports/. Google Scholar |
[67] |
UK Government, Slides, Datasets and Transcripts to Accompany Coronavirus Press Conferences, 2020. Available from: https://www.gov.uk/government/collections/slides-and-datasets-to-accompany-coronavirus-press-conferences/. Google Scholar |
[68] |
UK Office for National Statistics, Dataset: Deaths Registered Weekly in England and Wales, Provisional, 2020., Available from: https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/datasets/weeklyprovisionalfiguresondeathsregisteredinenglandandwales. Google Scholar |
[69] |
J. van Wees, S. Osinga, M. van der Kuip, M. Tanck and M. Hanegraaf, et al., Forecasting hospitalization and ICU rates of the COVID-19 outbreak: An efficient SEIR model, Bull. World Health Org., (2020).
doi: 10.2471/BLT.20.256743. |
[70] |
J. S. Whitaker and T. M. Hamill,
Evaluating methods to account for system errors in ensemble data assimilation, Mon. Weather. Rev., 140 (2012), 3078-3089.
doi: 10.1175/MWR-D-11-00276.1. |
[71] |
WHO, Coronavirus Disease (COVID-19): Similarities and Differences with Influenza, 2020. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/question-and-answers-hub/q-a-detail/q-a-similarities-and-differences-covid-19-and-influenza. Google Scholar |
[72] |
W. Yang, M. Lipsitch and J. Shaman,
Inference of seasonal and pandemic influenza transmission dynamics, PNAS, 112 (2015), 2723-2728.
doi: 10.1073/pnas.1415012112. |
[73] |
W. Yang, W. Zhang, D. Kargbo, R. Yang and Y. Chen, et al., Transmission network of the 2014–2015 Ebola epidemic in Sierra Leone, J. Roy. Soc. Interface, 12 (2015).
doi: 10.1098/rsif.2015.0536. |






















Parameter | First guess | Description |
5.5 | Incubation period | |
3.8 | Infection time | |
14.0 | Recovery time mild cases | |
5.0 | Recovery time severe cases | |
6.0 | Time until hospitalization | |
16.0 | Time until death | |
0.009 | Case fatality rate | |
0.039 | Hospitalization rate (severe cases) | |
0.4 | Fraction of fatally ill going to hospital |
Parameter | First guess | Description |
5.5 | Incubation period | |
3.8 | Infection time | |
14.0 | Recovery time mild cases | |
5.0 | Recovery time severe cases | |
6.0 | Time until hospitalization | |
16.0 | Time until death | |
0.009 | Case fatality rate | |
0.039 | Hospitalization rate (severe cases) | |
0.4 | Fraction of fatally ill going to hospital |
Age group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
Age range | 0–5 | 6–12 | 13–19 | 20–29 | 30–39 | 40–49 | 50–59 | 60–69 | 70–79 | 80–89 | 90–105 |
Population | 351159 | 451246 | 446344 | 711752 | 730547 | 723663 | 703830 | 582495 | 435834 | 185480 | 45230 |
p–mild | 1.0000 | 1.0000 | 0.9998 | 0.9913 | 0.9759 | 0.9686 | 0.9369 | 0.9008 | 0.8465 | 0.8183 | 0.8183 |
p–severe | 0.0000 | 0.0000 | 0.0002 | 0.0078 | 0.0232 | 0.0295 | 0.0570 | 0.0823 | 0.1160 | 0.1160 | 0.1160 |
p–fatal | 0.0000 | 0.0000 | 0.0000 | 0.0009 | 0.0009 | 0.0019 | 0.0061 | 0.0169 | 0.0375 | 0.0656 | 0.0656 |
Age group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
Age range | 0–5 | 6–12 | 13–19 | 20–29 | 30–39 | 40–49 | 50–59 | 60–69 | 70–79 | 80–89 | 90–105 |
Population | 351159 | 451246 | 446344 | 711752 | 730547 | 723663 | 703830 | 582495 | 435834 | 185480 | 45230 |
p–mild | 1.0000 | 1.0000 | 0.9998 | 0.9913 | 0.9759 | 0.9686 | 0.9369 | 0.9008 | 0.8465 | 0.8183 | 0.8183 |
p–severe | 0.0000 | 0.0000 | 0.0002 | 0.0078 | 0.0232 | 0.0295 | 0.0570 | 0.0823 | 0.1160 | 0.1160 | 0.1160 |
p–fatal | 0.0000 | 0.0000 | 0.0000 | 0.0009 | 0.0009 | 0.0019 | 0.0061 | 0.0169 | 0.0375 | 0.0656 | 0.0656 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 1.8 | 1.8 | 1.3 | 1.3 | 1.0 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
2 | 1.8 | 1.8 | 1.3 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
3 | 1.8 | 1.8 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
4 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
5 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
6 | 1.0 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
7 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
8 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
10 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
11 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 1.8 | 1.8 | 1.3 | 1.3 | 1.0 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
2 | 1.8 | 1.8 | 1.3 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
3 | 1.8 | 1.8 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
4 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
5 | 1.3 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
6 | 1.0 | 1.3 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
7 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
8 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
10 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |
11 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 1.5 | 1.5 | 1.0 | 1.5 | 0.5 | 0.5 | 0.5 | 0.4 | 0.4 | 0.4 | |
2 | 0.5 | 6.0 | 2.0 | 2.5 | 2.5 | 1.5 | 1.4 | 0.9 | 0.9 | 0.9 | |
3 | 0.5 | 6.0 | 2.0 | 2.5 | 2.5 | 1.5 | 1.4 | 0.9 | 0.9 | 0.9 | |
4 | 0.5 | 2.5 | 2.5 | 2.0 | 2.0 | 1.9 | 1.5 | 0.9 | 0.9 | 0.9 | |
5 | 1.2 | 2.5 | 2.5 | 2.0 | 2.0 | 1.9 | 1.8 | 0.5 | 0.5 | 0.5 | |
6 | 0.5 | 2.3 | 2.3 | 2.0 | 2.0 | 1.9 | 1.5 | 1.4 | 1.4 | 1.4 | |
7 | 0.5 | 2.0 | 2.0 | 1.5 | 1.5 | 1.5 | 1.5 | 0.9 | 0.9 | 0.9 | |
8 | 0.5 | 1.9 | 1.9 | 1.0 | 1.2 | 1.2 | 1.9 | 0.9 | 0.9 | 0.9 | |
9 | 0.5 | 1.5 | 1.5 | 0.9 | 0.9 | 1.2 | 1.0 | 1.5 | 1.5 | 1.5 | |
10 | 0.4 | 1.0 | 1.0 | 0.9 | 0.7 | 1.2 | 1.0 | 1.0 | 1.5 | 1.5 | |
11 | 0.4 | 0.9 | 0.9 | 0.9 | 0.7 | 1.2 | 1.0 | 1.0 | 1.5 | 1.5 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 1.5 | 1.5 | 1.0 | 1.5 | 0.5 | 0.5 | 0.5 | 0.4 | 0.4 | 0.4 | |
2 | 0.5 | 6.0 | 2.0 | 2.5 | 2.5 | 1.5 | 1.4 | 0.9 | 0.9 | 0.9 | |
3 | 0.5 | 6.0 | 2.0 | 2.5 | 2.5 | 1.5 | 1.4 | 0.9 | 0.9 | 0.9 | |
4 | 0.5 | 2.5 | 2.5 | 2.0 | 2.0 | 1.9 | 1.5 | 0.9 | 0.9 | 0.9 | |
5 | 1.2 | 2.5 | 2.5 | 2.0 | 2.0 | 1.9 | 1.8 | 0.5 | 0.5 | 0.5 | |
6 | 0.5 | 2.3 | 2.3 | 2.0 | 2.0 | 1.9 | 1.5 | 1.4 | 1.4 | 1.4 | |
7 | 0.5 | 2.0 | 2.0 | 1.5 | 1.5 | 1.5 | 1.5 | 0.9 | 0.9 | 0.9 | |
8 | 0.5 | 1.9 | 1.9 | 1.0 | 1.2 | 1.2 | 1.9 | 0.9 | 0.9 | 0.9 | |
9 | 0.5 | 1.5 | 1.5 | 0.9 | 0.9 | 1.2 | 1.0 | 1.5 | 1.5 | 1.5 | |
10 | 0.4 | 1.0 | 1.0 | 0.9 | 0.7 | 1.2 | 1.0 | 1.0 | 1.5 | 1.5 | |
11 | 0.4 | 0.9 | 0.9 | 0.9 | 0.7 | 1.2 | 1.0 | 1.0 | 1.5 | 1.5 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 0.9 | 0.9 | 0.8 | 1.0 | 0.5 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
2 | 0.5 | 1.5 | 0.9 | 1.0 | 1.0 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
3 | 0.5 | 1.5 | 0.9 | 1.0 | 1.0 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
4 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
5 | 0.8 | 1.0 | 1.0 | 0.9 | 0.9 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
6 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
7 | 0.5 | 0.6 | 0.6 | 0.9 | 0.9 | 0.9 | 0.7 | 0.5 | 0.5 | 0.5 | |
8 | 0.5 | 0.6 | 0.6 | 0.8 | 0.9 | 1.0 | 1.0 | 0.5 | 0.5 | 0.5 | |
9 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 | |
10 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 | |
11 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 |
Age | |||||||||||
groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
1 | 0.9 | 0.9 | 0.8 | 1.0 | 0.5 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
2 | 0.5 | 1.5 | 0.9 | 1.0 | 1.0 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
3 | 0.5 | 1.5 | 0.9 | 1.0 | 1.0 | 0.5 | 0.4 | 0.3 | 0.3 | 0.3 | |
4 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
5 | 0.8 | 1.0 | 1.0 | 0.9 | 0.9 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
6 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 | 0.5 | 0.4 | 0.3 | 0.3 | |
7 | 0.5 | 0.6 | 0.6 | 0.9 | 0.9 | 0.9 | 0.7 | 0.5 | 0.5 | 0.5 | |
8 | 0.5 | 0.6 | 0.6 | 0.8 | 0.9 | 1.0 | 1.0 | 0.5 | 0.5 | 0.5 | |
9 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 | |
10 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 | |
11 | 0.5 | 0.6 | 0.6 | 0.6 | 0.5 | 1.0 | 0.9 | 0.9 | 1.1 | 1.1 |
Parameters | Prior | Posterior D | Posterior DH | Posterior DHC |
59.97 (6.06) | 61.32 (6.01) | 61.95 (6.01) | 60.08 (5.98) | |
240.64 (24.17) | 246.68 (23.85) | 251.56 (23.62) | 239.43 (23.46) | |
3.80 (0.50) | 2.73 (0.33) | 3.13 (0.33) | 2.83 (0.28) | |
5.50 (0.50) | 4.62 (0.40) | 4.79 (0.40) | 5.14 (0.33) | |
13.98 (0.49) | 13.99 (0.49) | 13.94 (0.49) | 13.94 (0.49) | |
4.99 (0.41) | 4.98 (0.40) | 4.13 (0.31) | 3.57 (0.31) | |
5.99 (0.51) | 5.54 (0.49) | 5.35 (0.48) | 4.79 (0.39) | |
15.99 (0.50) | 15.64 (0.50) | 15.13 (0.47) | 14.43 (0.44) | |
0.009 (0.001) | 0.009 (0.001) | 0.014 (0.0009) | 0.014 (0.0002) | |
0.039 (0.004) | 0.039 (0.003) | 0.011 (0.002) | 0.015 (0.002) |
Parameters | Prior | Posterior D | Posterior DH | Posterior DHC |
59.97 (6.06) | 61.32 (6.01) | 61.95 (6.01) | 60.08 (5.98) | |
240.64 (24.17) | 246.68 (23.85) | 251.56 (23.62) | 239.43 (23.46) | |
3.80 (0.50) | 2.73 (0.33) | 3.13 (0.33) | 2.83 (0.28) | |
5.50 (0.50) | 4.62 (0.40) | 4.79 (0.40) | 5.14 (0.33) | |
13.98 (0.49) | 13.99 (0.49) | 13.94 (0.49) | 13.94 (0.49) | |
4.99 (0.41) | 4.98 (0.40) | 4.13 (0.31) | 3.57 (0.31) | |
5.99 (0.51) | 5.54 (0.49) | 5.35 (0.48) | 4.79 (0.39) | |
15.99 (0.50) | 15.64 (0.50) | 15.13 (0.47) | 14.43 (0.44) | |
0.009 (0.001) | 0.009 (0.001) | 0.014 (0.0009) | 0.014 (0.0002) | |
0.039 (0.004) | 0.039 (0.003) | 0.011 (0.002) | 0.015 (0.002) |
Parameters | Prior(Std Dev) | DHC | DH | D |
3.0(0.6) | - | - | - | |
1.0(0.5) | - | - | - | |
100.0(20.0) | 67 | 98 | 96 | |
240.0(48.0) | 167 | 204 | 235 | |
5.5(1.0) | 5.2 | 3.4 | 3.8 | |
3.8(0.6) | 1.8 | 1.9 | 2.7 | |
14.0(2.0) | 14.1 | 12.8 | 14.8 | |
5.0(1.0) | 6.9 | 6.8 | 5.5 | |
6.0(1.2) | 5.9 | 5.8 | 6.7 | |
10.0(2.0) | 5.8 | 3.4 | 10.4 | |
0.020(0.004) | 0.020 | 0.021 | 0.023 | |
0.039(0.006) | 0.040 | 0.047 | 0.038 | |
0.5(0) | - | - | - |
Parameters | Prior(Std Dev) | DHC | DH | D |
3.0(0.6) | - | - | - | |
1.0(0.5) | - | - | - | |
100.0(20.0) | 67 | 98 | 96 | |
240.0(48.0) | 167 | 204 | 235 | |
5.5(1.0) | 5.2 | 3.4 | 3.8 | |
3.8(0.6) | 1.8 | 1.9 | 2.7 | |
14.0(2.0) | 14.1 | 12.8 | 14.8 | |
5.0(1.0) | 6.9 | 6.8 | 5.5 | |
6.0(1.2) | 5.9 | 5.8 | 6.7 | |
10.0(2.0) | 5.8 | 3.4 | 10.4 | |
0.020(0.004) | 0.020 | 0.021 | 0.023 | |
0.039(0.006) | 0.040 | 0.047 | 0.038 | |
0.5(0) | - | - | - |
Parameter | First guess | Std. Dev. | Description |
500.0 | 50.0 | Initially exposed | |
400.0 | 40.0 | Initially infectious | |
3.8 | 0.05 | Reproduction number before interventions | |
(Case 1DH and Case 1DI) | |||
0.8 | 0.01 | Reproduction number after first nation-wide | |
intervention (Case 1DH and Case 1DI) | |||
1.0 | 0.75 | Reproduction number (Case 2DH) | |
0.010 | 0.0001 | Hospitalization rate | |
(Case 1DI) | |||
0.039 | 0.0039 | Hospitalization rate | |
(Case 1DH, Case 2DH and Case 3DH) | |||
0.5 | Fraction of fatally ill going to hospital | ||
(Case 1DI) | |||
0.6 | Fraction of fatally ill going to hospital | ||
(Case 1DH, Case 2DH and Case 3DH) |
Parameter | First guess | Std. Dev. | Description |
500.0 | 50.0 | Initially exposed | |
400.0 | 40.0 | Initially infectious | |
3.8 | 0.05 | Reproduction number before interventions | |
(Case 1DH and Case 1DI) | |||
0.8 | 0.01 | Reproduction number after first nation-wide | |
intervention (Case 1DH and Case 1DI) | |||
1.0 | 0.75 | Reproduction number (Case 2DH) | |
0.010 | 0.0001 | Hospitalization rate | |
(Case 1DI) | |||
0.039 | 0.0039 | Hospitalization rate | |
(Case 1DH, Case 2DH and Case 3DH) | |||
0.5 | Fraction of fatally ill going to hospital | ||
(Case 1DI) | |||
0.6 | Fraction of fatally ill going to hospital | ||
(Case 1DH, Case 2DH and Case 3DH) |
Case | Assimilated data | Description |
1DI | deaths, ICU patients | prior |
1DH | deaths, hospitalized | prior |
2DH | deaths, hospitalized | prior |
3DH | deaths, hospitalized | prior |
and gradually ramps down to 0.8 |
Case | Assimilated data | Description |
1DI | deaths, ICU patients | prior |
1DH | deaths, hospitalized | prior |
2DH | deaths, hospitalized | prior |
3DH | deaths, hospitalized | prior |
and gradually ramps down to 0.8 |
Parameter | First guess | Std. Dev. | Description |
February 16th | - | Start date of simulation | |
March 17th | - | Start date of intervention | |
May 11th | - | End of lockdown | |
3.5 | 0.20 | ||
0.65 | 0.20 | ||
0.85 | 0.20 | ||
500 | 500 | Initial Exposed | |
200 | 200 | Initial Infectious | |
20 | 2 | Recovery time severe cases | |
6 | 0.5 | Time until hospitalization | |
7 | 1 | Time until death | |
0.02 | 0.02 | Case fatality rate | |
0.039 | 0.03 | Hospitalization rate for severe cases | |
- | Fraction of |
Parameter | First guess | Std. Dev. | Description |
February 16th | - | Start date of simulation | |
March 17th | - | Start date of intervention | |
May 11th | - | End of lockdown | |
3.5 | 0.20 | ||
0.65 | 0.20 | ||
0.85 | 0.20 | ||
500 | 500 | Initial Exposed | |
200 | 200 | Initial Infectious | |
20 | 2 | Recovery time severe cases | |
6 | 0.5 | Time until hospitalization | |
7 | 1 | Time until death | |
0.02 | 0.02 | Case fatality rate | |
0.039 | 0.03 | Hospitalization rate for severe cases | |
- | Fraction of |
Parameter | Initial value | Std dev | |
March 10th | - | Start date of simulation | |
March 23rd | - | Start date of interventions | |
June 1st | - | Start date of prediction | |
656.0 | 65.6 | Initial Exposed | |
164.0 | 16.4 | Initial Infectious | |
4.0 | 0.4 | Prior |
|
1.0 | 0.1 | Prior |
|
1.0 | 0.1 | Prior |
|
0.2 | - | Ratio of fatally sick hospitalised | |
0.065 | 0.0065 | Case fatality rate (CFR) |
Parameter | Initial value | Std dev | |
March 10th | - | Start date of simulation | |
March 23rd | - | Start date of interventions | |
June 1st | - | Start date of prediction | |
656.0 | 65.6 | Initial Exposed | |
164.0 | 16.4 | Initial Infectious | |
4.0 | 0.4 | Prior |
|
1.0 | 0.1 | Prior |
|
1.0 | 0.1 | Prior |
|
0.2 | - | Ratio of fatally sick hospitalised | |
0.065 | 0.0065 | Case fatality rate (CFR) |
Param | Prior | Std. Dev. | Description |
20/2-2020 | - | Start Date | |
50(NY, CA), 10(AL), 20(NC) | 10(NY, CA, NC), 2.0(AL) | Initial Exposed | |
10(NY, CA), 1(AL), 2(NC) | 5(NY, CA, NC), 1.0(AL) | Initial infectious | |
0.18(NY), 0.009(CA, NC, AL) | 0.001 | CFR |
Param | Prior | Std. Dev. | Description |
20/2-2020 | - | Start Date | |
50(NY, CA), 10(AL), 20(NC) | 10(NY, CA, NC), 2.0(AL) | Initial Exposed | |
10(NY, CA), 1(AL), 2(NC) | 5(NY, CA, NC), 1.0(AL) | Initial infectious | |
0.18(NY), 0.009(CA, NC, AL) | 0.001 | CFR |
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