Estimating and Simulating a [-3pt] SIRD Model of COVID-19...

Estimating and Simulating a

SIRD Model of COVID-19 for

Many Countries, States, and Cities

Jesus Fernandez-Villaverde and Chad Jones

Extended results for New Hampshire

Based on data through May 28, 2020

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Outline of Slides

• Basic data from Johns Hopkins CSSE (raw and smoothed)

• Brief summary of the model

• Baseline results (δ = 1.0%, γ = 0.2, θ = 0.1)

• Simulation of re-opening – possibilities for raising R0

• Results with alternative parameter values:

◦ Lower mortality rate, δ = 0.8%

◦ Higher mortality rate, δ = 1.2%

◦ Infections last longer, γ = 0.15

◦ Cases resolve more quickly, θ = 0.2

◦ Cases resolve more slowly, θ = 0.07

• Data underlying estimates of R0(t)

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Underlying data from

Johns Hopkins CSSE

– Raw data

– Smoothed = 5 day centered moving average

– Later slides inflate deaths by 33% for “excess

deaths”

2 / 38

New Hampshire: Daily Deaths per Million People

03/23 04/06 04/20 05/04 05/18

2020

-5

0

5

10

15

20

25

30

35

40

45D

aily

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

3 / 38

New Hampshire: Daily Deaths per Million People (Smoothed)

03/21 04/04 04/18 05/02 05/16

2020

-1

0

1

2

3

4

5

6

7

8

9D

aily

dea

ths

per

mil

lio

n p

eop

le (

smo

oth

ed)

New Hampshire

4 / 38

Brief Summary of Model

• See the paper for a full exposition

• A 5-state SIRDC model with a time-varying R0

Parameter Baseline Description

δ 1.0% Mortality rate from infections (IFR)

γ 0.2 Rate at which people stop being infectious

θ 0.1 Rate at which cases (post-infection) resolve

α 0.05 Rate at which R0(t) decays with daily deaths

R0 ... Initial base reproduction rate

R0(t) ... Base reproduction rate at date t (βt/γ)

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Estimates of Time-Varying R0

– Inferred from daily deaths, and

– the change in daily deaths, and

– the change in (the change in daily deaths)

(see end of slide deck for this data)

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New Hampshire: Estimates of R0(t)

Apr 21 Apr 24 Apr 27 Apr 30 May 03 May 06 May 09 May 12 May 15 May 18

2020

0

0.5

1

1.5

2

2.5

3

3.5R

0(t

)New Hampshire

= 0.010 =0.10 =0.20

7 / 38

New Hampshire: Percent Currently Infectious

Apr 21 Apr 24 Apr 27 Apr 30 May 03 May 06 May 09 May 12 May 15 May 18

2020

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55P

erce

nt

curr

entl

y i

nfe

ctio

us,

I/N

(p

erce

nt)

New Hampshire

Peak I/N = 0.53% Final I/N = 0.23% = 0.010 =0.10 =0.20

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Notes on Intepreting Results

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Guide to Graphs

• Warning: Results are often very uncertain; this can be seen by

comparing across multiple graphs. See the original paper.

• 7 days of forecasts: Rainbow color order!

ROY-G-BIV (old to new, low to high)

◦ Black=current

◦ Red = oldest, Orange = second oldest, Yellow =third oldest...

◦ Violet (purple) = one day earlier

• For robustness graphs, same idea

◦ Black = baseline (e.g. δ = 1.0%)

◦ Red = lowest parameter value (e.g. δ = 0.8%)

◦ Green = highest parameter value (e.g. δ = 1.2%)

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How does R0 change over time?

• Inferred from death data when we have it

• For future, two approaches:

1 Alternatively, we fit this equation:

logR0(t) = a0 − α(Daily Deaths)

⇒α ≈ .05

R0 declines by 5 percent for each new daily death,

or rises by 5 percent when daily deaths decline

• Robustness: Assume R0(t) = final empirical value. Constant in

future, so no α adjustment → α = 0

11 / 38

Repeated “Forecasts” from the

past 7 days of data

– After peak, forecasts settle down.

– Before that, very noisy!

– If the region has not peaked, do not trust

– With α = .05 (see robustness section for α = 0)

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New Hampshire (7 days): Daily Deaths per Million People (α = .05)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

14

16

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.1/1.3/1.2 = 0.010 =0.1 =0.2 %Infect= 3/ 8/19

13 / 38

New Hampshire (7 days): Cumulative Deaths per Million (Future, α = .05

Apr May Jun Jul Aug Sep Oct Nov

2020

0

200

400

600

800

1000

1200

1400

1600

1800

Cum

ula

tive

dea

ths

per

mil

lion p

eople

New Hampshire

R0=2.1/1.3/1.2 = 0.010 =0.1 =0.2 %Infect= 3/ 8/19

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New Hampshire (7 days): Cumulative Deaths per Million, Log Scale (α =

15 / 38

Robustness to Mortality Rate, δ

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New Hampshire: Cumulative Deaths per Million (δ = .01/.008/.012)

Apr 21 Apr 28 May 05 May 12 May 19 May 26 Jun 02

2020

0

20

40

60

80

100

120

140

160

180

200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.1/0.5/0.6 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 2

DATA THROUGH 28-MAY-2020

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New Hampshire: Daily Deaths per Million People (δ = .01/.008/.012)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.1/0.5/0.6 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 2

DATA THROUGH 28-MAY-2020

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New Hampshire: Cumulative Deaths per Million (δ = .01/.008/.012)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

50

100

150

200

250

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.1/0.5/0.6 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 2

= 0.01 = 0.008 = 0.012DATA THROUGH 28-MAY-2020

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Reopening and Herd Immunity

– Black: assumes R0(today) remains in place forever

– Red: assumes R0(suppress)= 1/s(today)

– Green: we move 25% of the way from R0(today)

back to initial R0 = “normal”

– Purple: we move 50% of the way from R0(today)

back to initial R0 = “normal”

NOTE: Lines often cover each other up

20 / 38

New Hampshire: Re-Opening (α = .05)

(Light bars = New York City, for comparison) 21 / 38

New Hampshire: Re-Opening (α = 0)

(Light bars = New York City, for comparison) 22 / 38

Results for alternative

parameter values

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New Hampshire (7 days): Daily Deaths per Million People (α = 0)

May Jun Jul Aug Sep Oct Nov

2020

0

10

20

30

40

50

60

70

80

90

100

Dai

ly d

eath

s p

er m

illi

on

peo

ple

New Hampshire

R0=2.1/1.4/1.4 = 0.010 =0.1 =0.2 %Infect= 3/20/54

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New Hampshire (7 days): Cumulative Deaths per Million (Future, α = 0)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

1000

2000

3000

4000

5000

6000

Cum

ula

tive

dea

ths

per

mil

lion p

eople

New Hampshire

R0=2.1/1.4/1.4 = 0.010 =0.1 =0.2 %Infect= 3/20/54

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New Hampshire (7 days): Cumulative Deaths per Million, Log Scale (α =

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New Hampshire: Daily Deaths per Million People (δ = 0.8%)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.1/0.5/0.6 = 0.008 =0.1 =0.2 %Infect= 3/ 3/ 3

SOME ERRORS IN ESTIMATION...

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New Hampshire: Cumulative Deaths per Million (δ = 0.8%)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

50

100

150

200

250

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.1/0.5/0.6 = 0.008 =0.1 =0.2 %Infect= 3/ 3/ 3

SOME ERRORS IN ESTIMATION...

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New Hampshire: Daily Deaths per Million People (δ = 1.2%)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.1/0.5/0.6 = 0.012 =0.1 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

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New Hampshire: Cumulative Deaths per Million (δ = 1.2%)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

50

100

150

200

250

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.1/0.5/0.6 = 0.012 =0.1 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

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New Hampshire: Daily Deaths per Million People (γ = .2/.15)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.1/0.5/0.6 = 0.010 =0.1 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

31 / 38

New Hampshire: Cumulative Deaths per Million γ = .2/.15)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

50

100

150

200

250

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.1/0.5/0.6 = 0.010 =0.1 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

32 / 38

New Hampshire: Daily Deaths per Million People (θ = .1/.07/.2)

May Jun Jul Aug Sep Oct Nov

2020

0

2

4

6

8

10

12

Dai

ly d

eath

s per

mil

lion p

eople

New Hampshire

R0=2.3/0.2/0.3 = 0.010 =0.2 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

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New Hampshire: Cumulative Deaths per Million People (θ = .1/.07/.2)

Apr May Jun Jul Aug Sep Oct Nov

2020

0

20

40

60

80

100

120

140

160

180

200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leNew Hampshire

R0=2.3/0.2/0.3 = 0.010 =0.2 =0.2 %Infect= 2/ 2/ 2

SOME ERRORS IN ESTIMATION...

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Data Underlying Estimates

of Time-Varying R0

– Inferred from daily deaths, and

– the change in daily deaths, and

– the change in (the change in daily deaths)

35 / 38

New Hampshire: Daily Deaths, Actual and Smoothed

Feb Mar Apr May

2020

0

2

4

6

8

10

12

New Hampshire: Daily deaths, d

= 0.010 =0.10 =0.20

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New Hampshire: Change in Smoothed Daily Deaths

Feb Mar Apr May

2020

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

New Hampshire: Delta d

= 0.010 =0.10 =0.20

37 / 38

New Hampshire: Change in (Change in Smoothed Daily Deaths)

Feb Mar Apr May

2020

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

New Hampshire: Delta (Delta d)

= 0.010 =0.10 =0.20

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