Post on 05-Aug-2020
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