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A study of disproportionately affected populations by race/ethnicity during the SARS-CoV-2 pandemic using multi-population SEIR modeling and ensemble data assimilation

The second author (CSS) is supported by the US Office of Naval Research under grant N00014-18-1-2204. The third author (DPM) is supported by the NSF GRFP DGE 1256260. The seventh author (GE) was supported by internal funding from NORCE
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  • The disparity in the impact of COVID-19 on minority populations in the United States has been well established in the available data on deaths, case counts, and adverse outcomes. However, critical metrics used by public health officials and epidemiologists, such as a time dependent viral reproductive number ($ R_t $), can be hard to calculate from this data especially for individual populations. Furthermore, disparities in the availability of testing, record keeping infrastructure, or government funding in disadvantaged populations can produce incomplete data sets. In this work, we apply ensemble data assimilation techniques which optimally combine model and data to produce a more complete data set providing better estimates of the critical metrics used by public health officials and epidemiologists. We employ a multi-population SEIR (Susceptible, Exposed, Infected and Recovered) model with a time dependent reproductive number and age stratified contact rate matrix for each population. We assimilate the daily death data for populations separated by ethnic/racial groupings using a technique called Ensemble Smoothing with Multiple Data Assimilation (ESMDA) to estimate model parameters and produce an $R_t(n)$ for the $n^{th}$ population. We do this with three distinct approaches, (1) using the same contact matrices and prior $R_t(n)$ for each population, (2) assigning contact matrices with increased contact rates for working age and older adults to populations experiencing disparity and (3) as in (2) but with a time-continuous update to $R_t(n)$. We make a study of 9 U.S. states and the District of Columbia providing a complete time series of the pandemic in each and, in some cases, identifying disparities not otherwise evident in the aggregate statistics.

    Mathematics Subject Classification: 91F99, 65K10, 65P99, 65K99.

    Citation:

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  • Figure 1.  The general form of the $ R_{ij}^A(n) $ contact matrix elements for contact rates between age groups $ i $ and $ j $ in a given sub-population. The contact matrix is then subdivided into different blocks where parameters $ \alpha, \beta, \eta, \gamma, \varepsilon, \zeta, \delta_1, \delta_2, \delta_3, \xi_1, \xi_2, \xi_3 $ and $ \xi_4 $ control the contact rates between different age groups which generate similar patterns for spreading the disease. In particular, we define $ \gamma, \delta_2, \xi_2 $ and $ \xi_3 $ to be the parameters for the contact rates of the working class age groups

    Figure 2.  $ \mathbf{R}^{\rm{A}} $ for the NDAP $ (i) $ and DAP $ (ii) $ in our runs without DA for each intervention period along with the $ \mathbf{R}^{\rm{C}} $ matrix $ (iii) $ used in these simulations. We also used the $ \mathbf{R}^{\rm{A}} $'s in $ (i) $ and $ (ii) $ for the DA runs

    Figure 3.  Results for non-DA runs with DAP and NDAP at about the same population

    Figure 4.  Results for non-DA runs with NDAP at $ 70\% $ of the total population

    Figure 5.  Examples of analysis runs for various values of $ R^C_{nm} $ from the District of Columbia (DC). Top: left $ R^C_{nm} = 0 $, right $ R^C_{nm} = 10^{-3} $. Bottom: left $ R^C_{nm} = 10^{-1} $, right $ R^C_{nm} = 6\times10^{-1} $

    Figure 6.  Analysis results for the District of Columbia (DC) with $ R^A_{ij}=1 $ and piece-wise updates to $ R_t(n) $

    Figure 7.  Analysis results for the District of Columbia (DC) with age stratified $ R^A $ and piece-wise updates to $ R_t(n) $

    Figure 8.  Analysis results for the continuous update case for the state of AK

    Figure 9.  Analysis results for the continuous update case for the state of CA

    Figure 10.  Analysis results for the continuous update case for the state of CT

    Figure 11.  Analysis results for the continuous update case for the state of DE

    Figure 12.  Analysis results for the continuous update case for the District of Columbia

    Figure 13.  Analysis results for the continuous update case for the state of HI

    Figure 14.  Analysis results for the continuous update case for the state of MD

    Figure 15.  Analysis results for the continuous update case for the state of MI

    Figure 16.  Analysis results for the continuous update case for the state of UT

    Figure 17.  Analysis results for the continuous update case for the state of WA

    Figure 18.  Analysis results when using piece-wise updates to $ R(t) $ for AK. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 19.  Analysis results when using piecewise updates to $ R(t) $ for CA. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 20.  Analysis results when using piecewise updates to $ R(t) $ for CT. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 21.  Analysis results when using piecewise updates to $ R(t) $ for DE. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 22.  Analysis results when using piecewise updates to $ R(t) $ for DC. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 23.  Analysis results when using piecewise updates to $ R(t) $ for HI. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 24.  Analysis results when using piecewise updates to $ R(t) $ for MD. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 25.  Analysis results when using piecewise updates to $ R(t) $ for MI. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 26.  Analysis results when using piecewise updates to $ R(t) $ for UT. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 27.  Analysis results when using piecewise updates to $ R(t) $ for WA. Top row: $ R^A $ with entries of all ones. Bottom Row: $ R^A $ for DAPs and NDAPs

    Figure 28.  Analysis results with a CFR prior of $ 0.009 $ ($ \sigma_{CFR} = 0.05 $) and the same $ R_t(n) $ ($ \sigma_{R(t)} = 1.5 $) prior for all populations. This figure corresponds to columns 2 and 3 in Table 5

    Figure 29.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.05 $) and the same $ R_t(n) $ ($ \sigma_{R(t)} = 1.5 $) prior for all populations. This figure corresponds to columns 4 and 5 in Table 5

    Figure 30.  Analysis results with a CFR prior of $ 0.001 $ ($ \sigma_{CFR} = 0.03 $) and the same $ R_t(n) $ ($ \sigma_{R(t)} = 1.5 $) prior for all populations. This figure corresponds to columns 6 and 7 in Table 5

    Figure 31.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.05 $) and prior $ R_t(n) $ ($ \sigma_{R(t)} = 0.5 $) curves coming from initial piece-wise assimilation also assuming a $ 0.020 $ CFR prior. This figure corresponds to columns 2 and 3 in Table 6

    Figure 32.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.002 $) and prior $ R_t(n) $ ($ \sigma_{R(t)} = 0.5 $) curves coming from initial piece-wise assimilation also assuming a $ 0.020 $ CFR prior. This figure corresponds to columns 4 and 5 in Table 6

    Figure 33.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.05 $) for the Black population only and a CFR Prior of $ 0.009 $ ($ \sigma_{CFR} = 0.05 $) for all other populations. The prior $ R_t(n) $ ($ \sigma_{R(t)} = 1.5 $) curves are the same for all populations. This figure corresponds to columns 2 and 3 in Table 7

    Figure 34.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.002 $) for the Black population only and a CFR Prior of $ 0.009 $ ($ \sigma_{CFR} = 0.0009 $) for all other populations. The prior $ R_t(n) $ ($ \sigma_{R(t)} = 0.5 $) curves coming from initial piece-wise assimilation also assuming a $ 0.020 $ CFR prior. This figure corresponds to columns 4 and 5 in Table 7 and is the only case where Black infections can be lower than that of the LatinX population and still have more deaths

    Figure 35.  Analysis results with a CFR prior of $ 0.020 $ ($ \sigma_{CFR} = 0.05 $) for the Black population only and a CFR Prior of $ 0.009 $ ($ \sigma_{CFR} = 0.05 $) for all other populations. The prior $ R_t(n) $ ($ \sigma_{R(t)} = 0.5 $) curves coming from initial piece-wise assimilation also assuming a $ 0.020 $ CFR prior. This figure corresponds to columns 6 and 7 in Table 7

    Figure 36.  Analysis results for the continuous update case for the state of AK

    Figure 37.  Analysis results for the continuous update case for the state of CA

    Figure 38.  Analysis results for the continuous update case for the state of CT

    Figure 39.  Analysis results for the continuous update case for the state of DC

    Figure 40.  Analysis results for the continuous update case for the state of DE

    Figure 41.  Analysis results for the continuous update case for the state of HI

    Figure 42.  Analysis results for the continuous update case for the state of MD

    Figure 43.  Analysis results for the continuous update case for the state of MI

    Figure 44.  Analysis results for the continuous update case for the state of UT

    Figure 45.  Analysis results for the continuous update case for the state of WA

    Table 1.  Matrix scaling parameters for DAP and NDAP workers in lockdown and post-lockdown time periods

    Scalings Matrix Parameters
    $\alpha$ $\beta$ $\gamma$ $\eta$ $\epsilon$ $\zeta$ $\delta_1$ $\delta_2$ $\delta_3$ $\xi_1$ $\xi_2$ $\xi_3$ $\xi_4$
    DAP lock 0.5 0.7 0.55 0.25 0.25 0.35 0.3 0.7 0.7 0.4 0.65 0.55 0.6
    DAP post-lock 0.7 0.8 0.7 0.3 0.3 0.4 0.6 0.85 0.7 0.85 0.7 0.65 0.65
    NDAP lock 0.5 0.7 0.5 0.2 0.2 0.3 0.3 0.6 0.7 0.4 0.6 0.5 0.6
    NDAP post-lock 0.7 0.8 0.7 0.25 0.25 0.35 0.6 0.7 0.75 0.7 0.7 0.65 0.65
     | Show Table
    DownLoad: CSV

    Table 2.  Date breakdown by intervention periods for all states and the District of Columbia

    Interventions Information on Intervention Periods by State and the District of Columbia
    AK CA CT DC DE HI MD MI UT WA
    Start date 3/8/20 2/25/20 2/27/20 2/27/20 3/1/20 2/28/20 2/25/20 2/21/20 2/29/20 1/9/20
    1st Phase 3/19/20 3/19/20 3/19/20 3/17/20 3/17/20 3/17/20 3/17/20 3/17/20 3/17/20 3/17/20
    2nd Phase 5/27/20 5/27/20 5/27/20 5/29/20 5/31/20 5/15/20 5/15/20 5/19/20 5/5/20 5/28/20
     | Show Table
    DownLoad: CSV

    Table 3.  Demographic breakdown by race/ethnicity (where data is available) for all states and the District of Columbia. Groups that meet the criteria to be a DAP are in bold. AIAN = American Indian and Alaska Native, NHPI = Native Hawaiian and Pacific Islander

    Race/Ethnicity Percentage of Population per State and the District of Columbia
    AK CA CT DC DE HI MD MI UT WA
    AIAN 0.15 0.076 X X X X X 0.005 0.023 0.01
    Asian 0.06 0.14 0.04 0.0435 0.045 0.38 0.06 0.035 0.038 0.08
    Black 0.03 0.06 0.1 0.4453 0.22 0.02 0.29 0.14 0.021 0.04
    LatinX X 0.39 0.16 0.113 0.09 X 0.1 X 0.142 0.13
    Multi X X 0.02 X 0.02 X X X X 0.05
    NHPI 0.01 0.039 X X X 0.1 X X 0.016 0.008
    Other 0.08 X 0.01 0.01 X 0.24 X 0.03 0.01 0.005
    White 0.65 0.37 0.67 0.4196 0.62 0.25 0.51 0.78 0.78 0.69
     | Show Table
    DownLoad: CSV

    Table 5.  Results for the CFR using the same $ R_t(n) $ prior for all populations with $ \sigma_{R(t)} = 1.5 $

    Population Prior, $ \sigma=0.05 $ Post, $ \chi^2=112 $ Prior, $ \sigma=0.05 $ Post, $ \chi^2=92 $ Prior, $ \sigma=0.03 $ Post, $ \chi^2=87 $
    Asian 0.009 0.0074 0.020 0.0093 0.001 0.0070
    Black 0.009 0.0126 0.020 0.0157 0.001 0.0084
    LatinX 0.009 0.0213 0.020 0.0246 0.001 0.0147
    Multi 0.009 0.0103 0.020 0.0104 0.001 0.0086
    Other 0.009 0.0065 0.020 0.0042 0.001 0.0065
    White 0.009 0.0229 0.020 0.0271 0.001 0.0134
     | Show Table
    DownLoad: CSV

    Table 6.  Results for the CFR using $ R_t(n) $ priors taken from piece-wise assimilation first assuming a CFR of 2% for all populations

    Population Prior, $ \sigma=0.05 $ Post, $ \chi^2=89 $ Prior, $ \sigma=0.002 $ Post, $ \chi^2=41 $
    Asian 0.020 0.0282 0.020 0.0199
    Black 0.020 0.0337 0.020 0.0199
    LatinX 0.020 0.0317 0.020 0.0200
    Multi 0.020 0.0316 0.020 0.0202
    Other 0.020 0.0299 0.020 0.0204
    White 0.020 0.0259 0.020 0.0195
     | Show Table
    DownLoad: CSV

    Table 7.  Results for the CFR assuming a CFR of 2% for only the Black Group

    Population Prior, $ \sigma=0.05 $ Post, $ \chi^2=302 $ Prior, $ \sigma=.1x $ Post, $ \chi^2=38 $ Prior, $ \sigma=0.05 $ $ \chi^2=83 $
    Asian 0.0090 0.0101 0.0090 0.0091 0.0090 0.0249
    Black 0.0200 0.0226 0.0200 0.0201 0.0200 0.0371
    LatinX 0.0090 0.0211 0.0090 0.0092 0.0090 0.0299
    Multi 0.0090 0.0097 0.0090 0.0092 0.0090 0.0285
    Other 0.0090 0.0053 0.0090 0.0093 0.0090 0.0234
    White 0.0090 0.0248 0.0090 0.0090 0.0090 0.0256
     | Show Table
    DownLoad: CSV

    Table 4.  CFR estimates coming from the run presented in Section 4.2.6

    Population Prior, $ \sigma=0.009 $ Post, $ \chi^2=31 $
    Asian 0.0900 0.0089
    Black 0.0900 0.0100
    LatinX 0.0900 0.0090
    Multi 0.0900 0.0093
    Other 0.0900 0.0094
    White 0.0900 0.0094
     | Show Table
    DownLoad: CSV

    Table 8.  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
     | Show Table
    DownLoad: CSV

    Table 9.  The $ p $-numbers indicate the fraction of sick people in an age group ending up with mild symptoms, severe symptoms (hospitalized), and fatal infection

    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
     | Show Table
    DownLoad: CSV
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