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Estimating and Simulating a
SIRD Model of COVID-19 for
Many Countries, States, and Cities
Jesus Fernandez-Villaverde and Chad Jones
Extended results for Peru
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
Peru: Daily Deaths per Million People
Apr May Jun Jul Aug Sep Oct
2020
0
5
10
15
20
25D
aily
dea
ths
per
mil
lion p
eople
Peru
3
Peru: Daily Deaths per Million People (Smoothed)
Apr May Jun Jul Aug Sep Oct
2020
0
2
4
6
8
10
12D
aily
dea
ths
per
mil
lion p
eople
(sm
ooth
ed)
Peru
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
Peru: Estimates of R0(t)
Apr May Jun Jul Aug Sep Oct Nov
2020
0
0.5
1
1.5
2
2.5
3
3.5R
0(t
)Peru
= 0.010 =0.10 =0.20
7
Peru: Percent Currently Infectious
Apr May Jun Jul Aug Sep Oct
2020
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5P
erce
nt
curr
entl
y i
nfe
ctio
us,
I/N
(per
cent)
Peru
Peak I/N = 0.47% Final I/N = 0.09% = 0.010 =0.10 =0.20
8
Peru: Growth Rate of Daily Deaths over Past Week (percent)
Apr May Jun Jul Aug Sep Oct Nov
2020
-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)
Peru
= 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
Peru (7 days): Daily Deaths per Million People (α = .05)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
14
Peru (7 days): Cumulative Deaths per Million (Future, α = .05)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
15
Peru (7 days): Cumulative Deaths per Million, Log Scale (α = .05)
Mar 2020 May 2020 Jul 2020 Sep 2020 Nov 2020 Jan 2021 Mar 2021 1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
New York City
Italy
16
Robustness to Mortality Rate, δ
17
Peru: Cumulative Deaths per Million (δ = .01/.008/.012)
Apr May Jun Jul Aug Sep Oct Nov
2020
0
200
400
600
800
1000
1200
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
lePeru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
18
Peru: Daily Deaths per Million People (δ = .01/.008/.012)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
19
Peru: Cumulative Deaths per Million (δ = .01/.008/.012)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
= 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
Peru: 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
Peru
R0(t)=0.9, R
0(suppress)=1.1, R
0(25/50)=1.2/1.4, = 0.010, =0.05
(Light bars = New York City, for comparison)22
Peru: 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
Peru
R0(t)=0.9, R
0(suppress)=1.1, R
0(25/50)=1.1/1.4, = 0.010, =0.00
(Light bars = New York City, for comparison)23
Results for alternative
parameter values
24
Peru (7 days): Daily Deaths per Million People (α = 0)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/0.9 = 0.010 =0.00 =0.1 %Infect=10/10/10
DATA THROUGH 09-OCT-2020
25
Peru (7 days): Cumulative Deaths per Million (Future, α = 0)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/0.9 = 0.010 =0.00 =0.1 %Infect=10/10/10
DATA THROUGH 09-OCT-2020
26
Peru (7 days): Cumulative Deaths per Million, Log Scale (α = 0)
Mar 2020 May 2020 Jul 2020 Sep 2020 Nov 2020 Jan 2021 Mar 2021 1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/0.9 = 0.010 =0.00 =0.1 %Infect=10/10/10
New York City
Italy
27
Peru: Daily Deaths per Million People (δ = 0.8%)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.008 =0.1 =0.2 %Infect=13/13/13
28
Peru: Cumulative Deaths per Million (δ = 0.8%)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.008 =0.1 =0.2 %Infect=13/13/13
29
Peru: Daily Deaths per Million People (δ = 1.2%)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.012 =0.1 =0.2 %Infect= 8/ 9/ 9
30
Peru: Cumulative Deaths per Million (δ = 1.2%)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.012 =0.1 =0.2 %Infect= 8/ 9/ 9
31
Peru: Daily Deaths per Million People (γ = .2/.15)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
32
Peru: Cumulative Deaths per Million γ = .2/.15)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
= 0.2 = 0.15
DATA THROUGH 09-OCT-2020
33
Peru: Daily Deaths per Million People (θ = .1/.07/.2)
May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
2
4
6
8
10
12
Dai
ly d
eath
s per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
DATA THROUGH 09-OCT-2020
34
Peru: Cumulative Deaths per Million People (θ = .1/.07/.2)
Apr 2020 May 2020 Jun 2020 Jul 2020 Aug 2020 Sep 2020 Oct 2020 Nov 2020 Dec 2020 Jan 2021 Feb 2021Mar 20210
200
400
600
800
1000
1200
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Peru
R0=1.5/0.9/1.0 = 0.010 =0.05 =0.1 %Infect=10/10/11
= 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
Peru: Daily Deaths, Actual and Smoothed
Feb Mar Apr May Jun Jul Aug Sep Oct
2020
0
2
4
6
8
10
12
Peru: Daily deaths, d
= 0.010 =0.10 =0.20
37
Peru: Change in Smoothed Daily Deaths
Feb Mar Apr May Jun Jul Aug Sep Oct
2020
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Peru: Delta d
= 0.010 =0.10 =0.20
38
Peru: Change in (Change in Smoothed Daily Deaths)
Feb Mar Apr May Jun Jul Aug Sep Oct
2020
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Peru: Delta (Delta d)
= 0.010 =0.10 =0.20
39