American Institute of Mathematical Sciences

Advanced
doi: 10.3934/fods.2021001

An international initiative of predicting the SARS-CoV-2 pandemic using ensemble data assimilation

Geir Evensen 1,, , Javier Amezcua 2, , Marc Bocquet 3, , Alberto Carrassi 2,4, , Alban Farchi 3, , Alison Fowler 2, , Pieter L. Houtekamer 5, , Christopher K. Jones 6, , Rafael J. de Moraes 7, , Manuel Pulido 8, , Christian Sampson 6, and  Femke C. Vossepoel 7,

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

*Corresponding author: Geir Evensen

Received  June 2020 Revised  November 2020 Published  December 2020

Fund Project: The first author is supported by NORCE

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.

Keywords: SARS-CoV-2, ensemble data assimilation, ESMDA, parameter estimation, model calibration.
Mathematics Subject Classification: Primary:65K10, 65K99.
Citation: Geir Evensen, Javier Amezcua, Marc Bocquet, Alberto Carrassi, Alban Farchi, Alison Fowler, Pieter L. Houtekamer, Christopher K. Jones, Rafael J. de Moraes, Manuel Pulido, Christian Sampson, Femke C. Vossepoel. An international initiative of predicting the SARS-CoV-2 pandemic using ensemble data assimilation. Foundations of Data Science, doi: 10.3934/fods.2021001
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.  Google Scholar

[2] S. Abrams, The analysis of multivariate serological data, in Handbook of Infectious Disease Data Analysis, CRC Press, 2019.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

COVID-19 in Brazil: "So what?", The Lancet, 395 (2020). doi: 10.1016/S0140-6736(20)31095-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Evensen, Accounting for model errors in iterative ensemble smoothers, Comput. Geosci., 23 (2019), 761-775.  doi: 10.1007/s10596-019-9819-z.  Google Scholar

[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.  Google Scholar

[25]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2] S. Abrams, The analysis of multivariate serological data, in Handbook of Infectious Disease Data Analysis, CRC Press, 2019.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

COVID-19 in Brazil: "So what?", The Lancet, 395 (2020). doi: 10.1016/S0140-6736(20)31095-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Evensen, Accounting for model errors in iterative ensemble smoothers, Comput. Geosci., 23 (2019), 761-775.  doi: 10.1007/s10596-019-9819-z.  Google Scholar

[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.  Google Scholar

[25]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Flow diagram of the SEIR model
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Figure 2.  Norway: This figure summarizes scenarios related to opening up kindergartens and schools on the 20th of April. The left plots show the ensemble means and the 100 first ensemble realizations, for the number of hospitalized and the accumulated amount of deaths for different scenarios of future $ R(t) = 0.8,\, 1.0 $, and $ 1.2 $. The right plots show the prior and posterior ensembles of $ R(t) $. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 3.  Norway (base case): For the two cases (left and right plots), the only difference is the prior-guess for $ R(t) $ after starting the interventions. For the first two rows of plots, we show the posterior ensemble means and the 100 first realizations of the posterior solution from ESMDA. The blue lines are the total number of cases, while the gray lines give the number of active cases. The red curves denote the number of hospitalized, and green lines show the total number of deaths. The upper plot uses a log $ y $-axis. The second row is a zoom of the upper plot using a linear $ y $-axis. The lower plots show the corresponding prior and posterior estimates of $ R(t) $ for the two cases. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 4.  England: The plots show the data available for assimilation. Three different agencies report the number of deaths, the UK government press conference publishes the number of people in hospitals, CHESS reports the daily hospital admissions, and PHE presents the number of new cases
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Figure 5.  England. Left panels: ESMDA posterior estimates of the accumulated number of deaths (green), the daily number of hospitalizations (red), active (gray), and total cases (blue). Observations are displayed in black. Right panels: Prior and ESMDA posterior estimates of the effective reproduction number $ R(t) $. Ensemble members (thin lines) and ensemble means (thick lines). Assimilation experiment D (top panels), DH (mid panels) and DHC (bottom panels). The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 6.  Québec: From top to bottom, the plots show the results from the three assimilation experiments DHC, DH, and D. The left column presents the accumulated number of deaths and the number of hospitalizations, and the right column shows corresponding the reproductive number $ R(t) $ for the experiments. Time is since the start of the epidemic on March 8th. Observations are indicated with points when used to obtain the model fit. The solid lines are for the ensemble mean posterior estimates. After May 28th, we kept the realizations of $ R(t) $ constant and equal to the latest values for the remainder of the simulation. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 7.  Québec: The plots show verification of retroactive week-two forecasts for experiments DHC, DH, and D. For the predictions, issued one week apart, we show the mean estimate with the full line, and the dashed lines give the mean value plus or minus one standard deviation. We indicate the reported values with crosses
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Figure 8.  Québec: The plots present verification of the week-two forecasts (retroactively) issued on April 1st for experiment D (left) and the experiment DHC (right). The solid line denotes the mean prediction. The reported values used to fit the model parameters are indicated with circles, while the triangles are the values used for verification
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Figure 9.  The Netherlands: The plots show the results from the Cases 1I, 1H, 2H, and 3H, from top to bottom. The left plots include model estimates of the number of hospitalized patients and dead, in addition to the total number of cases as well as the number of active cases. The right plots show the corresponding estimates of $ R(t) $. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 10.  France: The plot shows official data curves for France from Santé Publique France up to May, 31
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Figure 11.  France: The figure shows the reference case (left) and case with an unknown intervention (right). The upper plots show the number of deaths at hospital, hospitalized patients, the total number of cases, and the currently active cases. The lower plots show the ensemble of effective reproduction number $ R(t) $. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 12.  France: The plot presents forecasts of the reference case with three distinct scenarios after intervention setting the reproduction number to $ R_3 = 0.75,\, 0.85,\, 1.00 $. We have plotted the posterior values for the number of deaths at hospitals (green lines), and the hospitalizations (red lines), together with the first hundred realizations for each case
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Figure 13.  Brazil: The plots illustrate the evolution of SARSCoV-2 in terms of the number of confirmed cases, deaths, and mortality rate (CFR), starting from the day of the first reported death due to COVID-19. The reported states are São Paulo (SP), from the Southeast geopolitical region; Bahia (BA), from the Northeast; Para (PA), North; Rio Grande do Sul (RS), South; and Goiás (GO), Mid West. Each state represents a different evolution of the disease. We have shifted the curves in time to correspond to the first confirmed case.
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Figure 14.  Brazil: The figure shows the simulation of the SARS-CoV-2 evolution in the {São} Paulo Brazilian State. In the left plots, we show a neutral case where the reproductive number $ R(t)\sim 1.0 $ with a standard deviation of $ 0.1 $ for the prediction. The right plots present a stable situation where $ R(t)\sim 0.6 $ with a standard deviation of 0.06, after a second intervention. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 15.  Brazil: The plots compare simulations considering relative observation errors of 5% and 20%. The upper and lower rows represent a neutral scenario with $ R_3 = 1.0 $. They differ in the standard deviations or $ R(t) $ (0.1 and 0.2). The middle plots show a stable situation with $ R_3 = 0.6 $ and a standard deviation of 0.06, following a second assumed intervention. The left plots show the deaths, and to the right, we present $ R(t) $ using different measurement errors
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Figure 16.  Argentina: The dots are the observations. In Case DC (left plots) both the accumulated deaths and the estimated number of cases were conditioned on, while in Case D (right plots) we only conditioned on the total number of deaths. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 17.  Argentina: Same as Figure 16 but for an experiment (Exp. 2) focused on a probabilistic prediction in which three scenarios, with different effective reproductive numbers imposed from June 1st, $ R(t) = 1.7 $, 1.3, and 0.9. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 18.  US: Mobilites for each of the states considered
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Figure 19.  US Case 1: Large uncertainty in $ R(t) $ The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 20.  US Case 2: Assuming voluntary federal guidance was immediately observed by the citizens on March 16th. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 21.  US Case 3: A gradual step down in $ R(t) $ with the first and intermediate values chosen so that the prior mean closely follows the data until the time for which $ R(t) $ is guessed to be one. The red thin line in the plots for $ R(t) $ is an indication of the value $ R(t) = 1 $ for easier identification
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Figure 22.  US: Forecasts for Case 3: Here we also show the prior forecast mean to highlight the difference between the idealized scenario and after analysis. In this case, only New York was able to achieve a trajectory that predicts fewer deaths than the idealized scenario. Furthermore, New York also is the only state in this study to make an end of the outbreak by late August
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Table 1.  The table gives a set of first-guess model parameters. As we could not find scientific estimates of these parameters, we set their values based on available information from the internet and initial model-tuning experiments. We leave it to the data assimilation system to fine-tune the parameter values
Parameter First guess Description
$ \tau_ \rm{inc} $ 5.5 Incubation period
$ \tau_ \rm{inf} $ 3.8 Infection time
$ \tau_ \rm{recm} $ 14.0 Recovery time mild cases
$ \tau_ \rm{recs} $ 5.0 Recovery time severe cases
$ \tau_ \rm{hosp} $ 6.0 Time until hospitalization
$ \tau_ \rm{death} $ 16.0 Time until death
$ p_ \mathrm{f} $ 0.009 Case fatality rate
$ p_ \mathrm{s} $ 0.039 Hospitalization rate (severe cases)
$ p_ \mathrm{h} $ 0.4 Fraction of fatally ill going to hospital
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Parameter First guess Description
$ \tau_ \rm{inc} $ 5.5 Incubation period
$ \tau_ \rm{inf} $ 3.8 Infection time
$ \tau_ \rm{recm} $ 14.0 Recovery time mild cases
$ \tau_ \rm{recs} $ 5.0 Recovery time severe cases
$ \tau_ \rm{hosp} $ 6.0 Time until hospitalization
$ \tau_ \rm{death} $ 16.0 Time until death
$ p_ \mathrm{f} $ 0.009 Case fatality rate
$ p_ \mathrm{s} $ 0.039 Hospitalization rate (severe cases)
$ p_ \mathrm{h} $ 0.4 Fraction of fatally ill going to hospital
Table 2.  The $ p $ numbers indicate the fraction of sick people in an age group ending up with mild symptoms, severe symptoms (hospitalized), and fatal infection. The population-weighted averages (for the Norwegian population) of the case-fatality rate is 0.0090, and the rate of severe (hospitalized) cases is 0.039
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
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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
Table 3.  Norway: This $ \hat{ {\bf{R}}} $-matrix increases transmissions among children after opening kindergartens and schools on April 20th. We chose the numbers ad-hoc to give a qualitative impact of opening kindergartens and schools. To estimate these transmissions' correct values, we will need access to additional data that are not yet available
Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 3.3$ 1.8 1.8 1.3 1.3 1.0 0.9 0.9 0.9 0.9 0.9
2 1.8 $\bf 3.3$ 1.8 1.3 1.3 1.3 0.9 0.9 0.9 0.9 0.9
3 1.8 1.8 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
4 1.3 1.3 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9 0.9
5 1.3 1.3 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9
6 1.0 1.3 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9
7 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9
8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9
9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9
10 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9
11 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$
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Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 3.3$ 1.8 1.8 1.3 1.3 1.0 0.9 0.9 0.9 0.9 0.9
2 1.8 $\bf 3.3$ 1.8 1.3 1.3 1.3 0.9 0.9 0.9 0.9 0.9
3 1.8 1.8 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
4 1.3 1.3 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9 0.9
5 1.3 1.3 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9 0.9
6 1.0 1.3 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9 0.9
7 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9 0.9
8 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9 0.9
9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9 0.9
10 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$ 0.9
11 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 $\bf 0.9$
Table 4.  England: The contact matrix $ \hat{ {\bf{R}}}_1 $ used to describe the transmission between different age groups in England before the enforced lockdown on March 23rd. The same contact matrix is used for the prediction from June 1st. See the right panel of Figure 2A in [42] for a heat-map representation of the original matrix
Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 2.0$ 1.5 1.5 1.0 1.5 0.5 0.5 0.5 0.4 0.4 0.4
2 0.5 $\bf 8.0$ 6.0 2.0 2.5 2.5 1.5 1.4 0.9 0.9 0.9
3 0.5 6.0 $\bf 8.0$ 2.0 2.5 2.5 1.5 1.4 0.9 0.9 0.9
4 0.5 2.5 2.5 $\bf 6.0$ 2.0 2.0 1.9 1.5 0.9 0.9 0.9
5 1.2 2.5 2.5 2.0 $\bf 3.0$ 2.0 1.9 1.8 0.5 0.5 0.5
6 0.5 2.3 2.3 2.0 2.0 $\bf 3.0$ 1.9 1.5 1.4 1.4 1.4
7 0.5 2.0 2.0 1.5 1.5 1.5 $\bf 2.0$ 1.5 0.9 0.9 0.9
8 0.5 1.9 1.9 1.0 1.2 1.2 1.9 $\bf 1.5$ 0.9 0.9 0.9
9 0.5 1.5 1.5 0.9 0.9 1.2 1.0 1.5 $\bf 1.5$ 1.5 1.5
10 0.4 1.0 1.0 0.9 0.7 1.2 1.0 1.0 1.5 $\bf 1.5$ 1.5
11 0.4 0.9 0.9 0.9 0.7 1.2 1.0 1.0 1.5 1.5 $\bf 1.5$
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Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 2.0$ 1.5 1.5 1.0 1.5 0.5 0.5 0.5 0.4 0.4 0.4
2 0.5 $\bf 8.0$ 6.0 2.0 2.5 2.5 1.5 1.4 0.9 0.9 0.9
3 0.5 6.0 $\bf 8.0$ 2.0 2.5 2.5 1.5 1.4 0.9 0.9 0.9
4 0.5 2.5 2.5 $\bf 6.0$ 2.0 2.0 1.9 1.5 0.9 0.9 0.9
5 1.2 2.5 2.5 2.0 $\bf 3.0$ 2.0 1.9 1.8 0.5 0.5 0.5
6 0.5 2.3 2.3 2.0 2.0 $\bf 3.0$ 1.9 1.5 1.4 1.4 1.4
7 0.5 2.0 2.0 1.5 1.5 1.5 $\bf 2.0$ 1.5 0.9 0.9 0.9
8 0.5 1.9 1.9 1.0 1.2 1.2 1.9 $\bf 1.5$ 0.9 0.9 0.9
9 0.5 1.5 1.5 0.9 0.9 1.2 1.0 1.5 $\bf 1.5$ 1.5 1.5
10 0.4 1.0 1.0 0.9 0.7 1.2 1.0 1.0 1.5 $\bf 1.5$ 1.5
11 0.4 0.9 0.9 0.9 0.7 1.2 1.0 1.0 1.5 1.5 $\bf 1.5$
Table 5.  England: The contact matrix $ \hat{ {\bf{R}}}_2 $ used to describe the transmission between different age groups in England during the lockdown from March 23rd to May 31st. See left hand panel of Figurek__ge 2A in [42] for a heat-map representation
Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 1.0$ 0.9 0.9 0.8 1.0 0.5 0.5 0.4 0.3 0.3 0.3
2 0.5 $\bf 2.0$ 1.5 0.9 1.0 1.0 0.5 0.4 0.3 0.3 0.3
3 0.5 1.5 $\bf 2.0$ 0.9 1.0 1.0 0.5 0.4 0.3 0.3 0.3
4 0.5 1.0 1.0 $\bf 1.2$ 1.0 1.0 0.9 0.5 0.4 0.3 0.3
5 0.8 1.0 1.0 0.9 $\bf 1.1$ 0.9 0.9 0.5 0.4 0.3 0.3
6 0.5 1.0 1.0 1.0 1.0 $\bf 1.1$ 0.9 0.5 0.4 0.3 0.3
7 0.5 0.6 0.6 0.9 0.9 0.9 $\bf 1.0$ 0.7 0.5 0.5 0.5
8 0.5 0.6 0.6 0.8 0.9 1.0 1.0 $\bf 1.0$ 0.5 0.5 0.5
9 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 $\bf 1.1$ 1.1 1.1
10 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 1.1 $\bf 1.1$ 1.1
11 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 1.1 1.1 $\bf 1.1$
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Age
groups 1 2 3 4 5 6 7 8 9 10 11
1 $\bf 1.0$ 0.9 0.9 0.8 1.0 0.5 0.5 0.4 0.3 0.3 0.3
2 0.5 $\bf 2.0$ 1.5 0.9 1.0 1.0 0.5 0.4 0.3 0.3 0.3
3 0.5 1.5 $\bf 2.0$ 0.9 1.0 1.0 0.5 0.4 0.3 0.3 0.3
4 0.5 1.0 1.0 $\bf 1.2$ 1.0 1.0 0.9 0.5 0.4 0.3 0.3
5 0.8 1.0 1.0 0.9 $\bf 1.1$ 0.9 0.9 0.5 0.4 0.3 0.3
6 0.5 1.0 1.0 1.0 1.0 $\bf 1.1$ 0.9 0.5 0.4 0.3 0.3
7 0.5 0.6 0.6 0.9 0.9 0.9 $\bf 1.0$ 0.7 0.5 0.5 0.5
8 0.5 0.6 0.6 0.8 0.9 1.0 1.0 $\bf 1.0$ 0.5 0.5 0.5
9 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 $\bf 1.1$ 1.1 1.1
10 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 1.1 $\bf 1.1$ 1.1
11 0.5 0.6 0.6 0.6 0.5 1.0 0.9 0.9 1.1 1.1 $\bf 1.1$
Table 6.  England: Prior and posterior mean and standard deviation for the time independent parameters estimated with ESDMA in the experiments D, DH and DHC
Parameters Prior Posterior D Posterior DH Posterior DHC
$ I_0 $ 59.97 (6.06) 61.32 (6.01) 61.95 (6.01) 60.08 (5.98)
$ E_0 $ 240.64 (24.17) 246.68 (23.85) 251.56 (23.62) 239.43 (23.46)
$ \tau_ \rm{inf} $ 3.80 (0.50) 2.73 (0.33) 3.13 (0.33) 2.83 (0.28)
$ \tau_ \rm{inc} $ 5.50 (0.50) 4.62 (0.40) 4.79 (0.40) 5.14 (0.33)
$ \tau_ \rm{recm} $ 13.98 (0.49) 13.99 (0.49) 13.94 (0.49) 13.94 (0.49)
$ \tau_ \rm{recs} $ 4.99 (0.41) 4.98 (0.40) 4.13 (0.31) 3.57 (0.31)
$ \tau_ \rm{hosp} $ 5.99 (0.51) 5.54 (0.49) 5.35 (0.48) 4.79 (0.39)
$ \tau_ \rm{death} $ 15.99 (0.50) 15.64 (0.50) 15.13 (0.47) 14.43 (0.44)
$ p_ \mathrm{f} $ 0.009 (0.001) 0.009 (0.001) 0.014 (0.0009) 0.014 (0.0002)
$ p_ \mathrm{s} $ 0.039 (0.004) 0.039 (0.003) 0.011 (0.002) 0.015 (0.002)
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Parameters Prior Posterior D Posterior DH Posterior DHC
$ I_0 $ 59.97 (6.06) 61.32 (6.01) 61.95 (6.01) 60.08 (5.98)
$ E_0 $ 240.64 (24.17) 246.68 (23.85) 251.56 (23.62) 239.43 (23.46)
$ \tau_ \rm{inf} $ 3.80 (0.50) 2.73 (0.33) 3.13 (0.33) 2.83 (0.28)
$ \tau_ \rm{inc} $ 5.50 (0.50) 4.62 (0.40) 4.79 (0.40) 5.14 (0.33)
$ \tau_ \rm{recm} $ 13.98 (0.49) 13.99 (0.49) 13.94 (0.49) 13.94 (0.49)
$ \tau_ \rm{recs} $ 4.99 (0.41) 4.98 (0.40) 4.13 (0.31) 3.57 (0.31)
$ \tau_ \rm{hosp} $ 5.99 (0.51) 5.54 (0.49) 5.35 (0.48) 4.79 (0.39)
$ \tau_ \rm{death} $ 15.99 (0.50) 15.64 (0.50) 15.13 (0.47) 14.43 (0.44)
$ p_ \mathrm{f} $ 0.009 (0.001) 0.009 (0.001) 0.014 (0.0009) 0.014 (0.0002)
$ p_ \mathrm{s} $ 0.039 (0.004) 0.039 (0.003) 0.011 (0.002) 0.015 (0.002)
Table 7.  Québec: The set of prior model parameters and their standard deviations (a zero std dev denotes that the parameter is kept fixed). Columns DHC, DH and D show posterior values for, respectively, experiments DHC, DH and D. Note that $ p_{h} $ was supplied externally. The curves for $ R_1 $ and $ R_2 $ are shown in Figure 6
Parameters Prior(Std Dev) DHC DH D
$ R_1 $ 3.0(0.6) - - -
$ R_2 $ 1.0(0.5) - - -
$ I_0 $ 100.0(20.0) 67 98 96
$ E_0 $ 240.0(48.0) 167 204 235
$ \tau_ \rm{inc} $ 5.5(1.0) 5.2 3.4 3.8
$ \tau_ \rm{inf} $ 3.8(0.6) 1.8 1.9 2.7
$ \tau_ \rm{recm} $ 14.0(2.0) 14.1 12.8 14.8
$ \tau_ \rm{recs} $ 5.0(1.0) 6.9 6.8 5.5
$ \tau_ \rm{hosp} $ 6.0(1.2) 5.9 5.8 6.7
$ \tau_ \rm{death} $ 10.0(2.0) 5.8 3.4 10.4
$ p_ \mathrm{f} $ 0.020(0.004) 0.020 0.021 0.023
$ p_ \mathrm{s} $ 0.039(0.006) 0.040 0.047 0.038
$ p_{h} $ 0.5(0) - - -
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Parameters Prior(Std Dev) DHC DH D
$ R_1 $ 3.0(0.6) - - -
$ R_2 $ 1.0(0.5) - - -
$ I_0 $ 100.0(20.0) 67 98 96
$ E_0 $ 240.0(48.0) 167 204 235
$ \tau_ \rm{inc} $ 5.5(1.0) 5.2 3.4 3.8
$ \tau_ \rm{inf} $ 3.8(0.6) 1.8 1.9 2.7
$ \tau_ \rm{recm} $ 14.0(2.0) 14.1 12.8 14.8
$ \tau_ \rm{recs} $ 5.0(1.0) 6.9 6.8 5.5
$ \tau_ \rm{hosp} $ 6.0(1.2) 5.9 5.8 6.7
$ \tau_ \rm{death} $ 10.0(2.0) 5.8 3.4 10.4
$ p_ \mathrm{f} $ 0.020(0.004) 0.020 0.021 0.023
$ p_ \mathrm{s} $ 0.039(0.006) 0.040 0.047 0.038
$ p_{h} $ 0.5(0) - - -
Table 8.  The Netherlands: The table gives values of the parameters used in Case 1, for the parameters that are different from those indicated in Table 1. The starting date of the simulations is February 20th, 2020
Parameter First guess Std. Dev. Description
$ E_0 $ 500.0 50.0 Initially exposed
$ I_0 $ 400.0 40.0 Initially infectious
$ R_1 $ 3.8 0.05 Reproduction number before interventions
(Case 1DH and Case 1DI)
$ R_1 $ 0.8 0.01 Reproduction number after first nation-wide
intervention (Case 1DH and Case 1DI)
$ R_1 $ 1.0 0.75 Reproduction number (Case 2DH)
$ p_ \mathrm{s} $ 0.010 0.0001 Hospitalization rate
(Case 1DI)
$ p_ \mathrm{s} $ 0.039 0.0039 Hospitalization rate
(Case 1DH, Case 2DH and Case 3DH)
$ p_ \mathrm{h} $ 0.5 Fraction of fatally ill going to hospital
(Case 1DI)
$ p_ \mathrm{h} $ 0.6 Fraction of fatally ill going to hospital
(Case 1DH, Case 2DH and Case 3DH)
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Parameter First guess Std. Dev. Description
$ E_0 $ 500.0 50.0 Initially exposed
$ I_0 $ 400.0 40.0 Initially infectious
$ R_1 $ 3.8 0.05 Reproduction number before interventions
(Case 1DH and Case 1DI)
$ R_1 $ 0.8 0.01 Reproduction number after first nation-wide
intervention (Case 1DH and Case 1DI)
$ R_1 $ 1.0 0.75 Reproduction number (Case 2DH)
$ p_ \mathrm{s} $ 0.010 0.0001 Hospitalization rate
(Case 1DI)
$ p_ \mathrm{s} $ 0.039 0.0039 Hospitalization rate
(Case 1DH, Case 2DH and Case 3DH)
$ p_ \mathrm{h} $ 0.5 Fraction of fatally ill going to hospital
(Case 1DI)
$ p_ \mathrm{h} $ 0.6 Fraction of fatally ill going to hospital
(Case 1DH, Case 2DH and Case 3DH)
Table 9.  The Netherlands: overview of cases
Case Assimilated data Description
1DI deaths, ICU patients prior $ R(t) $ equals 3.8 before intervention, 0.8 after
1DH deaths, hospitalized prior $ R(t) $ equals 3.8 before intervention, 0.8 after
2DH deaths, hospitalized prior $ R(t) $ equals 1.0
3DH deaths, hospitalized prior $ R(t) $ equals 1.8 at start of simulation
and gradually ramps down to 0.8
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Case Assimilated data Description
1DI deaths, ICU patients prior $ R(t) $ equals 3.8 before intervention, 0.8 after
1DH deaths, hospitalized prior $ R(t) $ equals 3.8 before intervention, 0.8 after
2DH deaths, hospitalized prior $ R(t) $ equals 1.0
3DH deaths, hospitalized prior $ R(t) $ equals 1.8 at start of simulation
and gradually ramps down to 0.8
Table 10.  France: The table gives a set of "calibrated" first-guess (i.e., prior) model parameters and their standard deviations used for France. $ p_{\mathrm h} $ is set ot $ 1 $ to inform the model that care homes deaths are excluded from the death numbers. All other parameter settings are unchanged as compared to the ones given in Table 1
Parameter First guess Std. Dev. Description
$ t_0 $ February 16th - Start date of simulation
$ t_1 $ March 17th - Start date of intervention
$ t_2 $ May 11th - End of lockdown
$ R_1 $ 3.5 0.20 $ R(t) $ prior before intervention
$ R_2 $ 0.65 0.20 $ R(t) $ prior during lockdown
$ R_3 $ 0.85 0.20 $ R(t) $ prior after full lockdown
$ E_0 $ 500 500 Initial Exposed
$ I_0 $ 200 200 Initial Infectious
$ \tau_ \rm{recs} $ 20 2 Recovery time severe cases
$ \tau_ \rm{hosp} $ 6 0.5 Time until hospitalization
$ \tau_ \rm{death} $ 7 1 Time until death
$ p_ \mathrm{f} $ 0.02 0.02 Case fatality rate
$ p_ \mathrm{s} $ 0.039 0.03 Hospitalization rate for severe cases
$ p_{\mathrm h} $ $ 1 $ - Fraction of $ {\bf Q}_\mathrm{f} $ that go to hospital
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Parameter First guess Std. Dev. Description
$ t_0 $ February 16th - Start date of simulation
$ t_1 $ March 17th - Start date of intervention
$ t_2 $ May 11th - End of lockdown
$ R_1 $ 3.5 0.20 $ R(t) $ prior before intervention
$ R_2 $ 0.65 0.20 $ R(t) $ prior during lockdown
$ R_3 $ 0.85 0.20 $ R(t) $ prior after full lockdown
$ E_0 $ 500 500 Initial Exposed
$ I_0 $ 200 200 Initial Infectious
$ \tau_ \rm{recs} $ 20 2 Recovery time severe cases
$ \tau_ \rm{hosp} $ 6 0.5 Time until hospitalization
$ \tau_ \rm{death} $ 7 1 Time until death
$ p_ \mathrm{f} $ 0.02 0.02 Case fatality rate
$ p_ \mathrm{s} $ 0.039 0.03 Hospitalization rate for severe cases
$ p_{\mathrm h} $ $ 1 $ - Fraction of $ {\bf Q}_\mathrm{f} $ that go to hospital
Table 11.  Brazil: The table gives the parameter values used in the SARS-CoV-2 simulations for São Paulo, Brazil. All other parameters were kept unchanged as compared to the ones given in Table \protect1
Parameter Initial value Std dev
$ t_0 $ March 10th - Start date of simulation
$ t_1 $ March 23rd - Start date of interventions
$ t_2 $ June 1st - Start date of prediction
$ E_0 $ 656.0 65.6 Initial Exposed
$ I_0 $ 164.0 16.4 Initial Infectious
$ R_1 $ 4.0 0.4 Prior $ R(t) $ during spinup
$ R_2 $ 1.0 0.1 Prior $ R(t) $ during interventions
$ R_3 $ 1.0 0.1 Prior $ R(t) $ in prediction phase
$ p_ \mathrm{h} $ 0.2 - Ratio of fatally sick hospitalised
$ p_ \mathrm{f} $ 0.065 0.0065 Case fatality rate (CFR)
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Parameter Initial value Std dev
$ t_0 $ March 10th - Start date of simulation
$ t_1 $ March 23rd - Start date of interventions
$ t_2 $ June 1st - Start date of prediction
$ E_0 $ 656.0 65.6 Initial Exposed
$ I_0 $ 164.0 16.4 Initial Infectious
$ R_1 $ 4.0 0.4 Prior $ R(t) $ during spinup
$ R_2 $ 1.0 0.1 Prior $ R(t) $ during interventions
$ R_3 $ 1.0 0.1 Prior $ R(t) $ in prediction phase
$ p_ \mathrm{h} $ 0.2 - Ratio of fatally sick hospitalised
$ p_ \mathrm{f} $ 0.065 0.0065 Case fatality rate (CFR)
Table 12.  US: The parameters used in our experiments, the different values for the reproductive number at intervention steps are described in the sections for each case. Values without state indications are the same for all states. Any parameters not listed are the same as in Tab 1
Param Prior Std. Dev. Description
$ T_s $ 20/2-2020 - Start Date
$ E_0 $ 50(NY, CA), 10(AL), 20(NC) 10(NY, CA, NC), 2.0(AL) Initial Exposed
$ I_0 $ 10(NY, CA), 1(AL), 2(NC) 5(NY, CA, NC), 1.0(AL) Initial infectious
$ p_ \mathrm{f} $ 0.18(NY), 0.009(CA, NC, AL) 0.001 CFR
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Param Prior Std. Dev. Description
$ T_s $ 20/2-2020 - Start Date
$ E_0 $ 50(NY, CA), 10(AL), 20(NC) 10(NY, CA, NC), 2.0(AL) Initial Exposed
$ I_0 $ 10(NY, CA), 1(AL), 2(NC) 5(NY, CA, NC), 1.0(AL) Initial infectious
$ p_ \mathrm{f} $ 0.18(NY), 0.009(CA, NC, AL) 0.001 CFR
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