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Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds
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.
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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.
|
[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/. |
[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. |
[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. |
[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. |
[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/. |
[32] |
Gouvernement de la République Française, COVID-19: Carte et Données, 2020. Available from: https://www.gouvernement.fr/info-coronavirus/carte-et-donnees. |
[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] |
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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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