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Estimating the Prevalence and Mortality of Coronavirus Disease 2019 (COVID-19) in the USA, the UK, Russia, and India

Yongbin Wang1,*, Chunjie Xu2,*, Sanqiao Yao1, Yingzheng Zhao1, Yuchun Li1, Lei Wang3, Xiangmei Zhao1

1 Department of Epidemiology and Health Statistics, School of Public Health, Xinxiang Medical University, Xinxiang, Henan Province, P.R. China; 2 Department of Occupational and Environmental Health, School of Public Health, Capital Medical University, Beijing, P.R. China; 3 Center for Musculoskeletal Surgery, Charité – Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin, Humboldt-Universität zu Berlin and Berlin Institute of Health, Berlin, Germany

Correspondence: Yongbin Wang

Department of Epidemiology and Health Statistics, School of Public Health, Xinxiang Medical University, Xinxiang 453000, Henan Province, P.R. China

Tel +86 0373 383 1646

Email [email protected] (Yongbin Wang)

*These authors contributed equally to this work

Table S1 The degrees of difference for the prevalence and mortality time series in the USA, the UK, Russia, and India

Degree of difference

USA

UK

Russia

India

ADF

p-value

ADF

p-value

ADF

p-value

ADF

p-value

Prevalence data

Actual series

1.359

0.955

0.851

0.884

4.707

0.990

4.803

0.990

First-order difference

-0.517

0.448

-0.595

0.423

-3.692

<0.001

2.413

0.990

Second-order difference

-9.660

<0.001

-10.551

<0.001

－

－

-10.054

<0.001

Mortality data

Actual series

2.932

0.990

3.846

0.990

4.695

0.990

5.255

0.990

First-order difference

-1.337

0.187

-1.364

0.159

-0.035

0.601

-0.128

0.572

Second-order difference

-9.545

<0.001

-5.684

<0.001

-10.991

<0.001

-10.694

<0.001

Abbreviation: ADF Augmented Dickey–Fuller test.

Table S2 Comparisons of possible ARIMA models developed with the prevalence and mortality of COVID-19 in these four countries.

Country

Model

MAD

MAPE

RMSE

NBIC

Stationary R2

R2

USA

Possible ARIMA models developed with the square root transformed training prevalence data

ARIMA(0,2,1)

2575.215

25.784

3963.240

16.629

0.418

1.000

ARIMA(1,2,0)

2778.711

27.850

4202.739

16.746

0.365

1.000

ARIMA(2,2,0)

2853.595

26.567

4325.661

16.863

0.419

1.000

ARIMA(3,2,0)

2788.448

26.068

4409.395

16.961

0.447

1.000

ARIMA(1,2,1)

2760.484

25.648

4223.747

16.816

0.451

1.000

Possible ARIMA models developed with the square root transformed training mortality data

ARIMA(0,2,1)

210.036

10.237

336.582

11.706

0.237

1.000

ARIMA(1,2,0)

212.551

10.311

344.3451

11.752

0.161

1.000

ARIMA(1,2,1)

210.528

10.277

340.054

11.795

0.237

1.000

ARIMA(2,2,0)

211.971

10.276

344.453

11.820

0.226

1.000

ARIMA(0,2,2)

210.826

10.270

340.463

11.797

0.237

1.000

UK


ARIMA(0,2,(1,6))

394.614

5.874

589.361

13.190

0.370

1.000

ARIMA(0,2,1)

475.356

6.483

795.262

13.417

0.194

1.000

ARIMA(0,2,2)

465.782

6.645

790.769

13.465

0.208

1.000

ARIMA(1,2,2)

455.907

6.581

776.964

13.489

0.241

1.000

ARIMA(2,2,2)

452.707

6.588

767.021

13.523

0.269

1.000

Possible ARIMA models developed with the log-transformed training mortality data

ARIMA(0,2,1)

451.222

11.931

714.520

13.215

0.391

0.994

ARIMA(1,2,0)

361.356

11.984

775.004

13.378

0.217

0.993

ARIMA(1,2,1)*

429.388

12.009

708.945

13.271

0.397

0.994

ARIMA(2,2,0)#

358.292

10.952

746.063

13.373

0.329

0.993

ARIMA(0,2,2)#

387.669

11.899

675.339

13.174

0.407

0.994

Russia

Possible ARIMA models developed with the log-transformed training prevalence data

ARIMA(1,1,1)

832.455

9.166

1898.550

15.215

0.039

0.997

ARIMA(0,1,1)

1081.249

10.369

2158.841

15.413

-0.316

0.996

ARIMA(1,1,0)

879.934

9.827

1909.207

15.168

-0.243

0.997

ARIMA(2,1,1)

1243.922

13.078

2803.403

16.053

-0.225

0.994

Possible ARIMA models developed with the log-transformed training mortality data

ARIMA(0,2,1)

19.138

10.721

30.685

6.969

0.410

0.994

ARIMA(0,2,0)

20.385

12.242

32.884

7.084

0.000

0.993

ARIMA(1,2,0)

20.319

11.752

31.538

7.000

0.184

0.994

ARIMA(2,2,0)

19.621

10.362

31.847

7.117

0.466

0.994

ARIMA(0,2,2)

18.350

10.527

32.233

7.141

0.443

0.994

India


ARIMA(2,2,1)

113.639

28.303

202.730

10.861

0.611

1.000

ARIMA(1,2,1)

119.206

29.502

205.414

10.828

0.582

1.000

ARIMA(1,2,0)

144.142

33.264

246.230

11.156

0.338

1.000

ARIMA(0,2,1)

173.786

44.036

293.548

11.423

0.000

0.999

ARIMA(1,2,2)

117.076

28.868

205.335

10.887

0.598

1.000

ARIMA(2,2,0)

116.233

27.793

212.830

10.899

0.544

1.000

Possible ARIMA models developed with the training mortality data

ARIMA(0,2,2)

6.387

30.482

8.381

4.487

0.522

1.000

ARIMA(0,2,1)

6.897

32.795

8.990

4.549

0.438

0.999

ARIMA(1,2,0)

6.771

15.787

9.453

4.571

0.366

0.999

ARIMA(1,2,1)

6.484

32.222

8.560

4.529

0.501

1.000

ARIMA(2,2,0)

6.440

30.588

8.425

4.497

0.517

1.000

Abbreviation: ARIMA Autoregressive integrated moving average method, MAD Mean absolute deviation, MAPE Mean absolute percentage error, RMSE Root mean squared error, NBIC Normalized Bayesian information criterion.

* indicates no statistical difference of the estimated parameters.

# indicates a p-value less than 0.05 under the Ljung-Box Q test.

Table S3 The candidate ETS models and their goodness of test results for the COVID-19 prevalence series in the USA

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

M,MD,N

-760.290

-706.041

1530.580

1542.100

1535.180

9.90E+07

M,M,N

-761.392

-707.143

1530.780

1540.000

1534.460

2.60E+08

M,MD,M

-759.462

-705.213

1530.920

1544.750

1536.440

9.60E+07

M,M,M

-760.573

-706.324

1531.150

1542.670

1535.740

2.50E+08

M,MD,A

-760.290

-706.041

1532.580

1546.400

1538.100

9.90E+07

M,M,A

-761.392

-707.143

1532.780

1544.310

1537.380

2.60E+08

M,A,M

-764.985

-710.736

1539.970

1551.490

1544.570

7.10E+07

M,A,N

-766.251

-712.002

1540.500

1549.720

1544.180

8.30E+07

M,AD,M

-764.985

-710.736

1541.970

1555.800

1547.490

7.10E+07

A,A,N

-766.986

-712.737

1541.970

1551.190

1545.650

6.40E+07

M,AD,N

-766.251

-712.002

1542.500

1554.020

1547.100

1.00E+100

M,A,A

-766.251

-712.002

1542.500

1554.020

1547.100

8.30E+07

A,A,A

-766.628

-712.379

1543.260

1554.780

1547.850

6.30E+07

A,A,M

-766.740

-712.491

1543.480

1555.000

1548.080

6.30E+07

A,AD,N

-766.986

-712.737

1543.970

1555.490

1548.570

6.40E+07

M,AD,A

-766.251

-712.002

1544.500

1558.330

1550.020

1.00E+100

A,AD,A

-766.628

-712.379

1545.260

1559.080

1550.770

6.30E+07

A,MD,A

-766.701

-712.452

1545.400

1559.230

1550.920

6.80E+07

A,AD,M

-766.740

-712.491

1545.480

1559.310

1551.000

1.00E+100

A,MD,M

-766.913

-712.665

1545.830

1559.650

1551.340

5.80E+07

A,MD,N

-769.312

-715.063

1548.620

1560.140

1553.220

6.40E+07

A,M,A

-781.991

-727.742

1573.980

1585.500

1578.580

1.80E+08

A,M,N

-784.092

-729.843

1576.180

1585.400

1579.860

2.10E+08

A,M,M

-783.675

-729.426

1577.350

1588.870

1581.950

1.90E+08

M,N,N

-790.107

-735.858

1584.210

1588.820

1586.050

1.80E+09

M,N,M

-789.259

-735.010

1584.520

1591.430

1587.270

1.80E+09

M,N,A

-789.424

-735.175

1584.850

1591.760

1587.610

1.60E+09

A,N,A

-849.408

-795.159

1704.820

1711.730

1707.570

1.20E+09

A,N,M

-859.652

-805.403

1725.300

1732.220

1728.060

1.30E+09

A,N,N

-892.635

-838.386

1789.270

1793.880

1791.110

1.80E+09

Abbreviation: ETS Error-trend-seasonal method, LL Log-likelihood, AIC Akaike information criterion, BIC Bayesian information criterion, HQ Hannan-Quinn criterion, AMSE Average mean square error.

Table S4 The candidate ETS models and their goodness of test results for the COVID-19 prevalence series in the UK

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

A,AD,M

-650.859

-596.610

1313.720

1327.540

1319.230

2219036.000

A,AD,N

-651.998

-597.749

1314.000

1325.520

1318.590

2193585.000

A,AD,A

-651.119

-596.870

1314.240

1328.060

1319.750

2275837.000


Table S5 The candidate ETS models and their goodness of test results for the COVID-19 prevalence series in Russia

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

A,MD,A

-687.149

-632.900

1386.300

1400.120

1391.810

1936007

A,MD,N

-692.184

-637.935

1394.370

1405.890

1398.960

2557242

A,MD,M

-694.749

-640.500

1401.500

1415.320

1407.010

2952685

A,M,N

-706.350

-652.101

1420.700

1429.920

1424.380

5433736

A,A,N

-706.562

-652.313

1421.120

1430.340

1424.800

5634054

A,M,A

-705.645

-651.396

1421.290

1432.810

1425.890

5296611

A,A,M

-705.951

-651.702

1421.900

1433.420

1426.500

5615841

A,M,M

-706.237

-651.988

1422.470

1433.990

1427.070

5464416

A,A,A

-706.562

-652.313

1423.120

1434.640

1427.720

5634061

A,AD,N

-706.562

-652.313

1423.120

1434.640

1427.720

1.00E+100

A,AD,M

-705.951

-651.702

1423.900

1437.730

1429.420

5615835

A,AD,A

-706.562

-652.313

1425.120

1438.950

1430.640

1.00E+100

A,N,A

-742.488

-688.239

1490.980

1497.890

1493.730

3.50E+07

A,N,M

-743.223

-688.974

1492.450

1499.360

1495.200

3.50E+07

A,N,N

-763.140

-708.892

1530.280

1534.890

1532.120

4.60E+07


Table S6 The candidate ETS models and their goodness of test results for the COVID-19 prevalence series in India

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

A,M,A

-560.728

-506.479

1131.460

1142.980

1136.050

180865

A,M,N

-562.179

-507.930

1132.360

1141.570

1136.030

188234

A,M,M

-562.001

-507.752

1134.000

1145.520

1138.600

186684


Table S7 The candidate ETS models and their goodness of test results for the COVID-19 mortality series in the USA

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

M,A,M

-473.070

-433.103

956.139

966.775

960.315

499399

M,A,N

-474.723

-434.756

957.447

965.955

960.787

517943

M,AD,M

-473.07

-433.103

958.139

970.902

963.150

499399

M,A,A

-474.723

-434.756

959.447

970.082

963.622

517943

M,AD,N

-474.723

-434.756

959.447

970.082

963.622

517943

M,AD,A

-474.723

-434.756

961.447

974.209

966.458

517943

A,A,N

-487.289

-447.322

982.579

991.087

985.919

468447

A,A,M

-486.996

-447.029

983.993

994.628

988.168

443598

A,A,A

-487.129

-447.162

984.258

994.894

988.434

453610

A,AD,N

-487.289

-447.322

984.579

995.214

988.754

468447

A,AD,M

-486.996

-447.029

985.993

998.755

991.004

443598

A,AD,A

-487.129

-447.162

986.258

999.021

991.269

453610

A,M,N

-491.889

-451.922

991.779

1000.290

995.119

752382

A,M,M

-493.755

-453.788

997.509

1008.140

1001.680

2428769

M,M,A

-499.414

-459.447

1008.830

1019.460

1013.000

2805910

M,N,A

-502.593

-462.626

1011.190

1017.570

1013.690

6951721

M,N,N

-520.290

-480.323

1044.580

1048.830

1046.250

7934603

M,N,M

-520.290

-480.323

1046.580

1052.960

1049.090

7934603

M,M,N

-537.515

-497.548

1083.030

1091.540

1086.370

1.80E+15

A,N,A

-539.835

-499.868

1085.670

1092.050

1088.180

5125265

A,N,M

-545.419

-505.452

1096.840

1103.220

1099.340

5411203

M,M,M

-545.998

-506.031

1102.000

1112.630

1106.170

1.50E+10

A,N,N

-575.295

-535.328

1154.590

1158.840

1156.260

7934597

A,M,A

-583.379

-543.413

1176.760

1187.390

1180.930

1.00E+07

M,MD,M

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.50E+10

M,MD,A

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

3.30E+12

A,MD,M

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

2428769

A,MD,N

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

6.90E+08

M,MD,N

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

3.30E+12

A,MD,A

1.00E+100

1.00E+100

1.00E+100

1.00E+100

1.00E+100

6.90E+08


Table S8 The candidate ETS models and their goodness of test results for the COVID-19 mortality series in the UK

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

M,A,N

-443.898

-408.444

895.796

904.038

899.006

855802

M,M,A

-442.983

-407.529

895.966

906.268

899.979

1172589

M,A,A

-443.898

-408.444

897.796

908.098

901.809

855802

M,A,M

-443.898

-408.444

897.796

908.098

901.809

855802

M,AD,N

-443.898

-408.444

897.796

908.098

901.809

1.E+100

M,AD,A

-443.898

-408.444

899.796

912.159

904.612

1.E+100

M,AD,M

-443.898

-408.444

899.796

912.159

904.612

1.E+100

M,MD,N

-449.948

-414.493

909.895

920.198

913.908

748767

M,M,N

-462.610

-427.155

933.220

941.461

936.430

1.0E+07

M,M,M

-462.610

-427.155

935.220

945.522

939.233

1.0E+07

M,N,A

-469.275

-433.820

944.549

950.731

946.957

2242613

M,MD,M

-469.275

-433.820

950.549

962.912

955.365

2242613

M,MD,A

-475.221

-439.767

962.443

974.805

967.258

835092

M,N,N

-481.194

-445.740

966.389

970.510

967.994

2328564

A,MD,N

-479.009

-443.555

968.018

978.320

972.031

500493

A,MD,A

-478.800

-443.346

969.601

981.964

974.416

489780

A,MD,M

-478.948

-443.494

969.896

982.259

974.712

496689

A,A,N

-482.619

-447.165

973.239

981.481

976.449

647994.

A,A,M

-481.678

-446.224

973.356

983.658

977.369

608757

A,A,A

-482.619

-447.165

975.239

985.541

979.252

647994

A,AD,N

-482.619

-447.165

975.239

985.541

979.252

647994

A,AD,M

-481.680

-446.225

975.359

987.722

980.175

609549

A,AD,A

-482.619

-447.165

977.239

989.602

982.054

647995

A,M,N

-488.521

-453.067

985.042

993.284

988.252

993397

A,M,M

-488.518

-453.063

987.035

997.338

991.048

1112650

A,M,A

-490.240

-454.786

990.480

1000.78

994.493

1028873

A,N,A

-497.731

-462.277

1001.46

1007.64

1003.87

1855772

A,N,M

-498.380

-462.926

1002.76

1008.94

1005.17

1874168

A,N,N

-504.743

-469.288

1013.49

1017.61

1015.09

2188041

M,N,M

-544.682

-509.228

1095.36

1101.55

1097.77

1964958


Table S9 The candidate ETS models and their goodness of test results for the COVID-19 mortality series in Russia

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

A,A,N

-192.823

-176.722

393.646

400.301

396.034

1445.800

A,A,M

-192.389

-176.288

394.778

403.096

397.763

1404.480

A,A,A

-192.823

-176.722

395.646

403.964

398.631

1445.800

A,MD,N

-192.831

-176.730

395.662

403.980

398.646

1157.690

A,AD,N

-192.839

-176.739

395.679

403.997

398.663

1452.790

A,AD,M

-192.389

-176.288

396.778

406.760

400.360

1404.460

A,AD,A

-192.823

-176.722

397.646

407.628

401.228

1445.800

A,MD,M

-192.831

-176.730

397.662

407.643

401.243

1157.690

A,MD,A

-192.831

-176.730

397.662

407.643

401.243

1157.690

A,M,A

-196.511

-180.410

403.022

411.339

406.006

1993.970

A,M,N

-197.675

-181.574

403.350

410.004

405.738

1.2E+07

M,A,N

-200.675

-184.574

409.349

416.004

411.737

2927.060

M,A,A

-200.675

-184.574

411.349

419.667

414.334

2927.020

M,AD,N

-200.675

-184.574

411.349

419.667

414.334

1.E+100

M,A,M

-200.675

-184.574

411.349

419.667

414.334

2927.360

M,AD,A

-200.675

-184.574

413.349

423.331

416.931

1.E+100

M,AD,M

-200.675

-184.574

413.350

423.331

416.931

1.E+100

M,MD,N

-206.338

-190.238

422.677

430.995

425.661

7672.630

M,MD,M

-206.338

-190.238

424.677

434.658

428.258

7672.630

M,MD,A

-206.338

-190.238

424.677

434.658

428.258

7672.630

A,N,A

-210.007

-193.906

426.015

431.005

427.805

6718.520

A,N,M

-210.288

-194.187

426.575

431.566

428.366

6753.640

M,M,A

-215.253

-199.152

440.506

448.823

443.490

1763.000

A,N,N

-221.743

-205.642

447.486

450.813

448.680

9088.170

M,N,N

-222.030

-205.929

448.060

451.387

449.253

13619.700

M,N,M

-222.030

-205.929

450.060

455.050

451.850

13619.700

M,N,A

-222.030

-205.929

450.060

455.050

451.850

13619.700

M,M,N

-274.553

-258.452

557.106

563.761

559.494

29257.200

A,M,M

-340.970

-324.869

691.940

700.258

694.925

2928109

M,M,M

-370.251

-354.150

750.502

758.820

753.486

2934296


Table S10 The candidate ETS models and their goodness of test results for the COVID-19 mortality series in India

Model

Compact LL

Likelihood

AIC

BIC

HQ

AMSE

M,M,N

-239.521

-210.573

487.041

494.846

490.034

1766.340

M,M,M

-239.444

-210.496

488.887

498.644

492.628

1623.790

M,M,A

-239.490

-210.542

488.980

498.736

492.720

1803.180


Table S11 Estimated initial parameters of the best-fitting ETS methods for the prevalence and mortality of COVID-19 in the USA, the UK, Russia, and India

Country

Initial parameters

Value

USA

Prevalence data of COVID-19

Initial level

9.846

Initial trend

1.615

Mortality data of COVID-19

Initial level

-1.518

Initial trend

3.494

Initial state 1

1.000

UK


Initial level

8.303

Initial trend

0.902

Initial state 1

1.000


Initial level

-11.055

Initial trend

-7.981

Russia


Initial level

-0.174

Initial trend

0.276

Initial state 1

0.000


Initial level

-0.169

Initial trend

2.118

India


Initial level

1.553

Initial trend

-0.137

Initial state 1

0.000


Initial level

1.502

Initial trend

1.138

Table S12 Estimated parameters of the best-fitting ETS methods developed with the whole data and their goodness of fit test results for the prevalence and mortality of COVID-19 in these four countries

Country

Parameter

Compact LL

LL

AIC

BIC

HQ

AMSE

USA

ETS(A,AD,N) model developed with the whole prevalence data

α=0.935

-924.898

-855.390

1864.447

1879.173

1864.730

8.3E+07

β=0.470

γ=0.000

δ=0.980

ETS(A,AD,N) model developed with the whole mortality data

α=1.000

-636.933

-582.684

1493.271

1507.997

1288.462

1359916

β=0.356

γ=0.000

δ=0.980

UK


α=1.000

-764.541

-695.032

1543.094

1557.820

1544.020

1.0E+100

β=0.562

γ=0.000

δ=0.980


α=1.000

-585.244

-535.872

1450.410

1465.136

1184.953

653458

β=0.183

γ=0.000

δ=0.980

Russia

ETS(M,MD,N) model developed with the whole prevalence data

α=0.153

-656.837

-587.329

1329.626

1344.352

1328.612

1.4E+07

β=0.153

γ=0.000

δ=0.980


α=0.949

-254.106

-226.210

874.080

888.806

521.903

1266.265

β=0.439

γ=0.000

δ=0.980

India


α=0.666

-673.707

-604.199

1369.245

1383.971

1362.353

1.0E+100

β=0.666

γ=0.000

δ=0.980


α=0.955

-360.843

-318.571

882.907

897.633

735.938

3109.498

β=0.451

γ=0.000

δ=0.980


Figure S1 Modeling process for time series forecasting with an ARIMA model.

Figure S2 Autocorrelation function (ACF) and partial ACF (PACF) of the differenced COVID-19 prevalence series for (A) USA, (B) UK, (C) Russia, and (D) India. Generally, the ACF offers a considerable amount of information on the order of moving average parameters, while the PACF offers a considerable amount of information on the order of autoregressive parameters. A search for the optimum p and q can be done by comparing the R2, stationary R2, and normalized Bayesian information criterion (NBIC) repeatedly.

Figure S3 Autocorrelation function (ACF) and partial ACF (PACF) of the differenced COVID-19 mortality series for (A) USA, (B) UK, (C) Russia, and (D) India. Typically, the ACF offers a considerable amount of information on the order of moving average parameters, while the PACF offers a considerable amount of information on the order of autoregressive parameters. A search for the optimum p and q can be done by comparing the R2, stationary R2, and normalized Bayesian information criterion (NBIC) repeatedly.

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