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

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

doi:10.3934/fods.2021001

Article Contents

2021, Volume 3, Issue 3: 413-477. Doi: 10.3934/fods.2021001 open access

This issue Previous Article Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds Next Article A study of disproportionately affected populations by race/ethnicity during the SARS-CoV-2 pandemic using multi-population SEIR modeling and ensemble data assimilation

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

^*Corresponding author: Geir Evensen
^*Corresponding author: Geir Evensen

Received: May 31, 2020

Revised: October 31, 2020

Early access: December 2020

Published: September 2021

The first author is supported by NORCE

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Abstract

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:

Mathematics Subject Classification: Primary:65K10, 65K99.

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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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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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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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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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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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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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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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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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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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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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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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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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