Estimating and Simulating a

SIRD Model of COVID-19 for

Many Countries, States, and Cities

Jesus Fernandez-Villaverde and Chad Jones

Extended results for Italy

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

Italy: Daily Deaths per Million People

Mar Apr May Jun Jul Aug Sep Oct

2020

-2

0

2

4

6

8

10

12

14

16D

aily

dea

ths

per

mil

lion p

eople

Italy

3

Italy: Daily Deaths per Million People (Smoothed)

Mar Apr May Jun Jul Aug Sep Oct

2020

0

2

4

6

8

10

12

14D

aily

dea

ths

per

mil

lion p

eople

(sm

ooth

ed)

Italy

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

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

Italy: Estimates of R0(t)

Mar Apr May Jun Jul Aug Sep Oct Nov

2020

0

0.5

1

1.5

2

2.5

3

3.5R

0(t

)Italy

= 0.010 =0.10 =0.20

7

Italy: Percent Currently Infectious

Mar Apr May Jun Jul Aug Sep Oct

2020

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8P

erce

nt

curr

entl

y i

nfe

ctio

us,

I/N

(p

erce

nt)

Italy

Peak I/N = 0.74% Final I/N = 0.02% = 0.010 =0.10 =0.20

8

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

Mar Apr May Jun Jul Aug Sep Oct Nov

2020

-10

-5

0

5

10

15

20

25G

row

th r

ate

of

dai

ly d

eath

s (p

erce

nt,

pas

t w

eek

)

Italy

= 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

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

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

Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

14

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

15

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

1

2

4

8

16

32

64

128

256

512

1024

2048

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leItaly

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

16

Robustness to Mortality Rate, δ

17

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

Mar Apr May Jun Jul Aug Sep Oct Nov

2020

0

100

200

300

400

500

600

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leItaly

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

18

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

Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

19

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

= 0.01 = 0.008 = 0.012

DATA THROUGH 09-OCT-2020

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

Italy: Re-Opening (α = .05)

Mar 2020 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

Italy

R0(t)=1.2, R

0(suppress)=1.1, R

0(25/50)=1.4/1.6, = 0.010, =0.05

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

Italy: Re-Opening (α = 0)

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

20

40

60

80

100

120

Dai

ly d

eath

s per

mil

lion p

eople

Italy

R0(t)=1.2, R

0(suppress)=1.1, R

0(25/50)=1.4/1.6, = 0.010, =0.00

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

Results for alternative

parameter values

24

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

Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.2 = 0.010 =0.00 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

25

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.2 = 0.010 =0.00 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

26

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

1

2

4

8

16

32

64

128

256

512

1024

2048

Cu

mu

lati

ve

dea

ths

per

mil

lio

n p

eop

leItaly

R0=2.0/1.2/1.2 = 0.010 =0.00 =0.1 %Infect= 6/ 6/ 7

27

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.008 =0.1 =0.2 %Infect= 8/ 8/ 9

28

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.008 =0.1 =0.2 %Infect= 8/ 8/ 9

29

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.012 =0.1 =0.2 %Infect= 5/ 5/ 6

30

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.012 =0.1 =0.2 %Infect= 5/ 5/ 6

31

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

Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

32

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

= 0.2 = 0.15

DATA THROUGH 09-OCT-2020

33

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

Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

2

4

6

8

10

12

14

Dai

ly d

eath

s p

er m

illi

on

peo

ple

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

DATA THROUGH 09-OCT-2020

34

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

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

2020

0

100

200

300

400

500

600

700

Cum

ula

tive

dea

ths

per

mil

lion p

eople

Italy

R0=2.0/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 6/ 7

= 0.1

= 0.07

= 0.2

DATA THROUGH 09-OCT-2020

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

Italy: Daily Deaths, Actual and Smoothed

Feb Mar Apr May Jun Jul Aug Sep Oct

2020

0

2

4

6

8

10

12

14

Italy: Daily deaths, d

= 0.010 =0.10 =0.20

37

Italy: Change in Smoothed Daily Deaths

Feb Mar Apr May Jun Jul Aug Sep Oct

2020

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Italy: Delta d

= 0.010 =0.10 =0.20

38

Italy: Change in (Change in Smoothed Daily Deaths)

Feb Mar Apr May Jun Jul Aug Sep Oct

2020

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Italy: Delta (Delta d)

= 0.010 =0.10 =0.20

39