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
0 / 38
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 / 38
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)
6 / 38
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
9 / 38
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
14 / 38
New Hampshire (7 days): Cumulative Deaths per Million, Log Scale (α =
15 / 38
Robustness to Mortality Rate, δ
16 / 38
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
17 / 38
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
18 / 38
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
23 / 38
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
24 / 38
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
25 / 38
New Hampshire (7 days): Cumulative Deaths per Million, Log Scale (α =
26 / 38
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...
28 / 38
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...
29 / 38
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...
30 / 38
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...
33 / 38
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...
34 / 38
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
36 / 38
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
38 / 38
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