Estimating and Simulating a [-3pt] SIRD Model of COVID-19 ...chadj/Covid/CHL-ExtendedResults2.pdfJun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar

Estimating and Simulating a

SIRD Model of COVID-19 for

Many Countries, States, and Cities

Jesus Fernandez-Villaverde and Chad Jones

Extended results for Chile

Based on data through October 9, 2020

0

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)

1

Underlying data from

Johns Hopkins CSSE

– Raw data

– Smoothed = 7 day centered moving average

– No “excess deaths” correction (change as of Aug 6

run)

2

Chile: Daily Deaths per Million People

Apr May Jun Jul Aug Sep Oct

2020

0

10

20

30

40

50

60D

aily

dea

ths

per

mil

lion p

eople

Chile

3

Chile: Daily Deaths per Million People (Smoothed)

Apr May Jun Jul Aug Sep Oct

2020

0

2

4

6

8

10

12

14

16D

aily

dea

ths

per

mil

lion p

eople

(sm

ooth

ed)

Chile

4

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)

5

https://web.stanford.edu/~chadj/sird-paper.pdf

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)

6

Chile: Estimates of R0(t)

May Jun Jul Aug Sep Oct Nov

2020

0

0.5

1

1.5

2

2.5

3

3.5R

0(t

)Chile

= 0.010 =0.10 =0.20

7

Chile: Percent Currently Infectious

May Jun Jul Aug Sep Oct

2020

0

0.1

0.2

0.3

0.4

0.5

0.6P

erce

nt

curr

entl

y i

nfe

ctio

us,

I/N

(p

erce

nt)

Chile

Peak I/N = 0.56% Final I/N = 0.13% = 0.010 =0.10 =0.20

8

Chile: Growth Rate of Daily Deaths over Past Week (percent)


2020

-6

-4

-2

0

2

4

6

8

10G

row

th r

ate

of

dai

ly d

eath

s (p

erce

nt,

pas

t w

eek)

Chile

= 0.010 =0.10 =0.20

9

Notes on Intepreting Results

10

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%)

11

https://web.stanford.edu/~chadj/sird-paper.pdf

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

12

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)

13

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

Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 2021 Apr 20210

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10

DATA THROUGH 09-OCT-2020

14

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

May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 2021 Apr 20210

200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10


15

Chile (7 days): Cumulative Deaths per Million, Log Scale (α = .05)

Jan 2020 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 1

2

4

8

16

32

64

128

256

512

1024

2048

4096

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10

New York City

Italy

16

Robustness to Mortality Rate, δ

17

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


2020

0

100

200

300

400

500

600

700

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leChile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10


18

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10


19

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


200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leChile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10

= 0.01 = 0.008 = 0.012


20

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

21

Chile: Re-Opening (α = .05)

May 2020 Jul 2020 Sep 2020 Nov 2020 Jan 2021 Mar 20210

10

20

30

40

50

60

70

80

90

Dai

ly d

eath

s per

mil

lion p

eople

Chile

R0(t)=1.1, R

0(suppress)=1.1, R

0(25/50)=1.3/1.5, = 0.010, =0.05

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

Chile: Re-Opening (α = 0)

May 2020 Jul 2020 Sep 2020 Nov 2020 Jan 2021 Mar 20210

10

20

30

40

50

60

70

80

90

Dai

ly d

eath

s per

mil

lion p

eople

Chile

R0(t)=1.1, R

0(suppress)=1.1, R

0(25/50)=1.3/1.5, = 0.010, =0.00

(Light bars = New York City, for comparison)23

Results for alternative

parameter values

24

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.010 =0.00 =0.1 %Infect= 7/ 8/10


25

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


200

400

600

800

1000

1200

1400

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.010 =0.00 =0.1 %Infect= 7/ 8/10


26

Chile (7 days): Cumulative Deaths per Million, Log Scale (α = 0)

Jan 2020 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 1

2

4

8

16

32

64

128

256

512

1024

2048

4096

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.010 =0.00 =0.1 %Infect= 7/ 8/10

New York City

Italy

27

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.008 =0.1 =0.2 %Infect= 9/10/13

28

Chile: Cumulative Deaths per Million (δ = 0.8%)


200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.008 =0.1 =0.2 %Infect= 9/10/13

29

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.012 =0.1 =0.2 %Infect= 6/ 7/ 9

30

Chile: Cumulative Deaths per Million (δ = 1.2%)


200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

le

Chile

R0=1.6/1.1/1.1 = 0.012 =0.1 =0.2 %Infect= 6/ 7/ 9

31

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10


32

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


200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leChile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10

= 0.2 = 0.15


33

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


2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Chile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10


34

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


200

400

600

800

1000

1200

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leChile

R0=1.6/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 7/ 8/10

= 0.1

= 0.07

= 0.2


35

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)

36

Chile: Daily Deaths, Actual and Smoothed

Feb Mar Apr May Jun Jul Aug Sep Oct

2020

0

2

4

6

8

10

12

14

Chile: Daily deaths, d

= 0.010 =0.10 =0.20

37

Chile: Change in Smoothed Daily Deaths


2020

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Chile: Delta d

= 0.010 =0.10 =0.20

38

Chile: Change in (Change in Smoothed Daily Deaths)


2020

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Chile: Delta (Delta d)

= 0.010 =0.10 =0.20

39

Documents

Estimating and Simulating a [-3pt] SIRD Model of COVID-19 ...chadj/Covid/CHL-ExtendedResults2.pdfJun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar