Estimating and Simulating a SIRD Model of COVID-19 Jes ´ us Fern´ andez-Villaverde and Chad Jones April 22, 2020 (Preliminary and incomplete) 0 / 58
Estimating and Simulating
a SIRD Model of COVID-19
Jesus Fernandez-Villaverde and Chad Jones
April 22, 2020
(Preliminary and incomplete)
0 / 58
Outline
• Basic model
◦ Social distancing via a time-varying β
• Estimation and simulation
◦ Different countries, U.S. states, and New York City
◦ Robustness to parameters
◦ “Forecasts” from each of the last 7 days
• Re-opening and herd immunity
◦ How much can we relax social distancing?
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Basic Model
2 / 58
Notation
• Number of people who are (stocks):
S = Susceptible
I = Infectious
R = Recovered
D = Dead
• Constant population size is N
St + It + Rt + Dt = N
3 / 58
SIRD Model: Overview
• Susceptible get infected at rate βIt/N
New infections = βIt/N · St
• Infections resolve at Poisson rate γ, so the average number of
days until resolution is 1/γ so γ = .2 ⇒ 5 days.
• Resolution happens in one of two ways:
◦ Death: fraction δ
◦ Recovery: fraction 1 − δ
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SIRD Model: Laws of Motion
∆St+1 = −βStIt/N︸ ︷︷ ︸
new infections
∆It+1 = βStIt/N︸ ︷︷ ︸
new infections
− γIt︸︷︷︸
resolving infections
∆Rt+1 = (1 − δ)γIt︸ ︷︷ ︸
recover
∆Dt+1 = δγIt︸︷︷︸
die
R0 = D0 = 0
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Recyled notation (terrible) R0: Initial infection rate
• Initial reproduction number R0 ≡ β/γ
R0 = β × 1/γ
# of infections
from one sick
person
# of lengthy
contacts per
day
# of days
contacts are
infectious
• R0 = expected number of infections via the first sick person
◦ R0 > 1 ⇒ disease initially grows
◦ R0 < 1 ⇒ disease dies out: infectious generate less than 1
new infection
• If 1/γ = 5, then easy to have R0 >> 1
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Basic Properties of Differential System (Hethcote 2000)
• Let st ≡ St/N = fraction susceptible
• If R0st > 1, the disease spreads, otherwise declines
• Initial exponential growth rate is β − γ
• As t → ∞, the total fraction of people ever infected, e∗, solves
(assuming s0 ≈ 1)
e∗ = −1
R0log(1 − e∗)
Long run is pinned down by R0 (and death rate),
γ affects timing
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Social Distancing
• What about a time-varying infection rate βt?
◦ Disease characteristics – fixed, homogeneous
◦ Regional factors (NYC vs Montana) – fixed, heterogeneous
◦ Social distancing – varies over time and space
• Reasons why βt may change over time
◦ Policy changes on social distancing
◦ Individuals voluntarily change behavior to protect
themselves and others
◦ Superspreaders get infected quickly but then recover and
“burn out” early
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Model of Social Distancing
• Assume two key parameters β0 and β∗
• Economy decays exponentially from β0 to β∗ at rate λ:
βt = β0e−λt + β∗(1 − e−λt)
⇒ can think about initial R0 = β0/γ
R0(t) = βt/γ and final R∗
0 = β∗/γ
• Interpretation:
◦ λ governs the rate of convergence to R∗
0
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Estimates and Simulations
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Estimation: Countries and States
• Parameters that are fixed and homogeneous
◦ γ = 0.2: average duration is 5 days (or γ = 0.1)
◦ δ = 0.003
Apr 1: 15% of mothers giving birth in NYC infected
With δ = .004, model says only 13.6% ever infected on Apr 1
• Parameters that vary across countries/states
◦ β0 and β∗
◦ λ: speed at which you move to β∗
◦ I0: initial number of infections (gets timing right)
• Objective function:
◦ Equally weighted SSR for Cumulative deaths (logs) and
Daily deaths (logs)11 / 58
Estimation based entirely on death data
• Excess death issue
◦ New York City added 3000+ deaths on April 15 ≈ 45% more
◦ The Economist ⇒ increases based on vital records
⇒ We adjust all NYC deaths before April 15 by this 45%
and non-NYC deaths upwards by 33%
• We use 5-day moving averages (centered)
◦ Otherwise, very serious “weekend effects” in which deaths
are underreported
◦ Even zero sometimes, followed by a large spike
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Guide to Graphs
• 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 = .003)
◦ Red = lowest parameter value (e.g. δ = .002)
◦ Green = highest parameter value (e.g. δ = .004)
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Guide to Graphs (continued)
• R0 in subtitle: R0 / R0(today) / R∗
0
◦ Initial / Today / Final
• “%Infect”
◦ Today / t+30 / Final
◦ This is the percent ever infected
◦ (so fraction δ will eventually be deaths)
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New York City: Cumulative Deaths per Million (δ = .003/.002/.004)
Mar 16 Mar 23 Mar 30 Apr 06 Apr 13 Apr 20 Apr 27
2020
0
200
400
600
800
1000
1200
1400
1600
1800
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leNew York City (only)
R0=3.4/1.4/0.9 = 0.003, =0.05, =0.2, %Infect=64/70/70
DATA THROUGH 21-APR-2020
δ = .003/.004 fit equally well in levels
δ = .002 cannot fit: 1700 deaths per
million people already means nearly
100% infected!
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New York City: Daily Deaths per Million People (δ = .003/.002/.004)
Mar 17 Mar 31 Apr 14 Apr 28 May 12 May 26 Jun 09
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 York City (only)
R0=3.4/1.4/0.9 = 0.003, =0.05, =0.2, %Infect=64/70/70
DATA THROUGH 21-APR-2020
δ = .003/.004 fit equally well
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New York City: Cumulative Deaths per Million (δ = .003/.002/.004)
Mar 16 Mar 30 Apr 13 Apr 27 May 11 May 25 Jun 08
2020
0
500
1000
1500
2000
2500
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leNew York City (only)
R0=3.4/1.4/0.9 = 0.003, =0.05, =0.2, %Infect=64/70/70
= 0.003
= 0.002
= 0.004
DATA THROUGH 21-APR-2020
But very different futures!
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Spain: Cumulative Deaths per Million People (γ = .2/.1)
Mar 05 Mar 12 Mar 19 Mar 26 Apr 02 Apr 09 Apr 16 Apr 23
2020
0
100
200
300
400
500
600
700
Cum
ula
tive
dea
ths
per
mil
lion p
eople
Spain
R0=4.6/1.4/0.6 = 0.003, =0.07, =0.2, %Infect=21/23/23
DATA THROUGH 21-APR-2020
γ = .2 fits slightly better in levels
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Spain: Daily Deaths per Million People (γ = .2/.1)
Mar 06 Mar 20 Apr 03 Apr 17 May 01 May 15 May 29
2020
0
5
10
15
20
25
30
Dai
ly d
eath
s per
mil
lion p
eople
Spain
R0=4.6/1.4/0.6 = 0.003, =0.07, =0.2, %Infect=21/23/23
DATA THROUGH 21-APR-2020
γ = .2 better fit for daily deaths
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Spain: Cumulative Deaths per Million (Future, γ = .2/.1)
Mar 05 Mar 19 Apr 02 Apr 16 Apr 30 May 14 May 28
2020
0
100
200
300
400
500
600
700
800
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leSpain
R0=4.6/1.4/0.6 = 0.003, =0.07, =0.2, %Infect=21/23/23
= 0.2
= 0.1
DATA THROUGH 21-APR-2020
Very different futures!
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Italy: Daily Deaths per Million People (γ = .2/.1)
Feb 24 Mar 09 Mar 23 Apr 06 Apr 20 May 04 May 18
2020
0
2
4
6
8
10
12
14
16
18
20
Dai
ly d
eath
s per
mil
lion p
eople
Italy
R0=4.4/1.8/0.8 = 0.003, =0.07, =0.2, %Infect=19/22/22
DATA THROUGH 21-APR-2020
γ = .1 gives a flatter right tail of
deaths
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Repeated “Forecasts” from the
past 7 days of data
– After peak, forecasts settle down.
– Before that, very noisy!
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Spain (7 days): Daily Deaths per Million People
Mar 06 Mar 20 Apr 03 Apr 17 May 01 May 15 May 29
2020
0
5
10
15
20
25
30
Dai
ly d
eath
s per
mil
lion p
eople
Spain
R0=4.6/0.7/0.6 = 0.003, =0.07, =0.2, %Infect=21/23/23
DATA THROUGH 21-APR-2020
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Italy (7 days): Daily Deaths per Million People
Feb 24 Mar 09 Mar 23 Apr 06 Apr 20 May 04 May 18
2020
0
2
4
6
8
10
12
14
16
18
20
Dai
ly d
eath
s per
mil
lion p
eople
Italy
R0=4.4/0.9/0.8 = 0.003, =0.07, =0.2, %Infect=19/22/22
DATA THROUGH 21-APR-2020
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New York City (7 days): Daily Deaths per Million People
Mar 17 Mar 31 Apr 14 Apr 28 May 12 May 26 Jun 09
2020
0
10
20
30
40
50
60
70
80
90
Dai
ly d
eath
s p
er m
illi
on
peo
ple
New York City (plus)
R0=3.5/1.2/0.5 = 0.003, =0.04, =0.2, %Infect=53/59/60
DATA THROUGH 21-APR-2020
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New York City (7 days): Cumulative Deaths per Million (Future)
Mar 16 Mar 30 Apr 13 Apr 27 May 11 May 25 Jun 08
2020
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leNew York City (plus)
R0=3.5/1.2/0.5 = 0.003, =0.04, =0.2, %Infect=53/59/60
DATA THROUGH 21-APR-2020
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California (7 days): Cumulative Deaths per Million
Mar 06 Mar 13 Mar 20 Mar 27 Apr 03 Apr 10 Apr 17 Apr 24
2020
0
5
10
15
20
25
30
35
40
45
50
Cum
ula
tive
dea
ths
per
mil
lion p
eople
California
R0=4.0/1.2/1.1 = 0.003, =0.08, =0.2, %Infect= 2/ 6/ 8
DATA THROUGH 21-APR-2020
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California (7 days): Daily Deaths per Million People
Mar 07 Mar 21 Apr 04 Apr 18 May 02 May 16 May 30
2020
0
1
2
3
4
5
6
7
8
9
10
Dai
ly d
eath
s per
mil
lion p
eople
California
R0=4.0/1.2/1.1 = 0.003, =0.08, =0.2, %Infect= 2/ 6/ 8
DATA THROUGH 21-APR-2020
(noisy and unreliable)
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California (7 days): Cumulative Deaths per Million (Future)
Mar 06 Mar 20 Apr 03 Apr 17 May 01 May 15 May 29
2020
0
50
100
150
200
250
300
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leCalifornia
R0=4.0/1.2/1.1 = 0.003, =0.08, =0.2, %Infect= 2/ 6/ 8
DATA THROUGH 21-APR-2020
(noisy and unreliable)
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U.K. (7 days): Cumulative Deaths per Million
Mar 07 Mar 14 Mar 21 Mar 28 Apr 04 Apr 11 Apr 18 Apr 25
2020
0
50
100
150
200
250
300
350
400
450
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leUnited Kingdom
R0=3.8/1.0/0.5 = 0.003, =0.04, =0.2, %Infect=13/20/20
DATA THROUGH 21-APR-2020
30 / 58
U.K. (7 days): Daily Deaths per Million People
Mar 08 Mar 22 Apr 05 Apr 19 May 03 May 17 May 31
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
United Kingdom
R0=3.8/1.0/0.5 = 0.003, =0.04, =0.2, %Infect=13/20/20
DATA THROUGH 21-APR-2020
(noisy and unreliable)
31 / 58
U.K. (7 days): Cumulative Deaths per Million (Future)
Mar 07 Mar 21 Apr 04 Apr 18 May 02 May 16 May 30
2020
0
100
200
300
400
500
600
700
800
Cu
mu
lati
ve
dea
ths
per
mil
lio
n p
eop
leUnited Kingdom
R0=3.8/1.0/0.5 = 0.003, =0.04, =0.2, %Infect=13/20/20
DATA THROUGH 21-APR-2020
(noisy and unreliable)
32 / 58
Sweden (7 days): Daily Deaths per Million People
Mar 14 Mar 28 Apr 11 Apr 25 May 09 May 23 Jun 06
2020
0
2
4
6
8
10
12
14
16
Dai
ly d
eath
s p
er m
illi
on
peo
ple
Sweden
R0=3.1/1.0/0.5 = 0.003, =0.05, =0.2, %Infect= 9/13/13
DATA THROUGH 21-APR-2020
(noisy and unreliable)
33 / 58
France (7 days): Daily Deaths per Million People
Feb 18 Mar 03 Mar 17 Mar 31 Apr 14 Apr 28 May 12
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
France
R0=7.4/0.9/0.5 = 0.003, =0.05, =0.2, %Infect=16/20/20
DATA THROUGH 21-APR-2020
(noisy and unreliable)
34 / 58
Cumulative Deaths per Million, Log Scale
0 10 20 30 40 50 60 0.1
0.5
1
2
4
8
16
32
64
128
256
512
1024
2048
Doubles every 10 days, g=7%Doubles e
very 5 days, g=14%
Dou
bles
eve
ry 3
day
s, g=
23%
Dou
bles
eve
ry 2
day
s, g
=35
%
Iran
Korea, South
SwedenUnited Kingdom
FranceItalySpain
United States
DATA THROUGH 21-APR-2020
DAYS SINCE 1 DEATH PER 10 MILLION PEOPLE
CUMULATIVE DEATHS PER MILLION
Italy seems a better guide to France/U.K.
35 / 58
S. Korea (7 days): Daily Deaths per Million People
Feb 23 Mar 08 Mar 22 Apr 05 Apr 19 May 03 May 17
2020
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Dai
ly d
eath
s p
er m
illi
on
peo
pleKorea, South
R0=1.5/0.8/0.5 = 0.003, =0.02, =0.2, %Infect= 0/ 0/ 0
DATA THROUGH 21-APR-2020
36 / 58
Washington (7 days): Daily Deaths per Million People
Mar 03 Mar 17 Mar 31 Apr 14 Apr 28 May 12 May 26
2020
0
1
2
3
4
5
6
Dai
ly d
eath
s p
er m
illi
on
peo
pleWashington
R0=1.8/1.0/0.5 = 0.003, =0.02, =0.2, %Infect= 4/ 6/ 6
DATA THROUGH 21-APR-2020
(noisy and unreliable)
37 / 58
Louisiana (7 days): Daily Deaths per Million People
Mar 17 Mar 31 Apr 14 Apr 28 May 12 May 26 Jun 09
2020
0
5
10
15
20
25
Dai
ly d
eath
s p
er m
illi
on
peo
ple
Louisiana
R0=3.1/0.9/0.8 = 0.003, =0.08, =0.2, %Infect=15/20/21
DATA THROUGH 21-APR-2020
38 / 58
Florida (7 days): Daily Deaths per Million People
Mar 11 Mar 25 Apr 08 Apr 22 May 06 May 20 Jun 03
2020
0
0.5
1
1.5
2
2.5
3
3.5
Dai
ly d
eath
s per
mil
lion p
eople
Florida
R0=3.9/1.0/0.5 = 0.003, =0.05, =0.2, %Infect= 2/ 3/ 3
DATA THROUGH 21-APR-2020
(noisy and unreliable)
39 / 58
Detriot (Wayne County, 7 days): Daily Deaths per Million People
Mar 21 Apr 04 Apr 18 May 02 May 16 May 30 Jun 13
2020
0
10
20
30
40
50
60
Dai
ly d
eath
s p
er m
illi
on
peo
ple
Detroit
R0=2.9/1.0/0.5 = 0.003, =0.05, =0.2, %Infect=34/41/41
DATA THROUGH 21-APR-2020
40 / 58
Massachusetts (7 days): Daily Deaths per Million People
Mar 23 Apr 06 Apr 20 May 04 May 18 Jun 01 Jun 15
2020
0
10
20
30
40
50
60
Dai
ly d
eath
s p
er m
illi
on
peo
pleMassachusetts
R0=3.4/1.6/1.6 = 0.003, =0.20, =0.2, %Infect=15/56/63
DATA THROUGH 21-APR-2020
(noisy and unreliable)
41 / 58
District of Columbia (7 days): Daily Deaths per Million People
Mar 23 Apr 06 Apr 20 May 04 May 18 Jun 01 Jun 15
2020
0
10
20
30
40
50
60
70
Dai
ly d
eath
s p
er m
illi
on
peo
ple
District of Columbia
R0=1.8/1.3/0.5 = 0.003, =0.02, =0.2, %Infect= 9/27/30
DATA THROUGH 21-APR-2020
(noisy and unreliable)
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Reopening and Herd Immunity
43 / 58
Percent Ever Infected would be very informative
— Percent Ever Infected (today) —
δ = .002 δ = .003 δ = .004
New York City (plus) 78 53 41
Detroit 51 34 26
Spain 32 21 16
Italy 28 19 14
Michigan 25 16 12
France 24 16 12
Massachusetts 23 15 12
United Kingdom 20 13 10
Sweden 13 9 6
District of Columbia 13 9 7
New York excluding NYC 8 5 4
Denmark 5 3 2
Germany 4 3 2
Florida 3 2 2
California 3 2 1
Tennessee 2 1 144 / 58
Herd Immunity
• How far can we relax social distancing?
• Let s(t) = S(t)/N = the fraction still susceptible
◦ The disease will die out as long as
R0(t)s(t) < 1
◦ That is, if the “new” R0 is smaller than 1/s(t)
◦ Today’s infected people infect fewer than 1 person on
average
• We can relax social distancing to raise R0(t) to 1/s(t)
45 / 58
Herd Immunity and Opening the Economy?
Percent R0(t+30) PercentSusceptible with no way back
R0 R0(t) t+30 outbreak to normal
New York City 3.5 1.2 40.5 2.5 55.6
Detroit 2.9 1.0 58.7 1.7 34.8
Spain 4.6 0.7 77.4 1.3 15.1
Italy 4.4 0.9 78.4 1.3 10.9
Michigan 3.4 1.0 76.3 1.3 13.3
France 7.4 0.9 79.8 1.3 5.1
Massachusetts 3.4 1.6 43.8 2.3 38.2
United Kingdom 3.8 1.0 80.5 1.2 8.4
Sweden 3.1 1.0 87.0 1.1 8.0
District of Columbia 1.8 1.3 73.4 1.4 4.2
New York excluding NYC 3.5 0.9 92.9 1.1 8.3
Washington 1.8 1.0 93.9 1.1 13.1
Denmark 2.7 0.7 96.4 1.0 16.1
Germany 4.6 0.9 95.3 1.0 3.7
Florida 3.9 1.0 96.6 1.0 2.9
California 4.0 1.2 94.1 1.1 -4.3
Korea, South 1.5 0.8 99.8 1.0 33.646 / 58
Simulations of Re-Opening
• Begin with the basic estimates shown already
• Different policies are then adopted starting around May 20
◦ 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”
• We assume these R0 values stay in place forever
◦ In practice, over course, β would likely start to fall as
mortality rises
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Spain: Re-Opening
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
80
Dai
ly d
eath
s per
mil
lion p
eople
Spain
R0(t)=0.7, R
0(suppress)=1.3, R
0(25/50)=1.7/2.6, = 0.003, =0.07
48 / 58
Italy: Re-Opening
Mar Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
80
90
Dai
ly d
eath
s per
mil
lion p
eople
Italy
R0(t)=0.9, R
0(suppress)=1.3, R
0(25/50)=1.8/2.6, = 0.003, =0.07
49 / 58
New York City: Re-Opening
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
80
90
Dai
ly d
eath
s per
mil
lion p
eople
New York City (plus)
R0(t)=1.2, R
0(suppress)=2.5, R
0(25/50)=1.8/2.3, = 0.003, =0.04
NYC has herd immunity and can
re-open significantly in a month!
(???)
50 / 58
New York excluding NYC: Re-Opening
Apr May Jun Jul Aug Sep Oct
2020
0
10
20
30
40
50
60
70
80
90
Dai
ly d
eath
s per
mil
lion p
eople
New York excluding NYC
R0(t)=0.9, R
0(suppress)=1.1, R
0(25/50)=1.5/2.2, = 0.003, =0.07
But not the rest of NY state!
51 / 58
California: Re-Opening
Apr May Jun Jul Aug Sep
2020
0
20
40
60
80
100
120
140
Dai
ly d
eath
s p
er m
illi
on
peo
ple
California
R0(t)=1.2, R
0(suppress)=1.1, R
0(25/50)=1.8/2.4, = 0.003, =0.07
52 / 58
Sweden: Re-Opening
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
Dai
ly d
eath
s per
mil
lion p
eople
Sweden
R0(t)=1.0, R
0(suppress)=1.2, R
0(25/50)=1.5/2.0, = 0.003, =0.05
53 / 58
Detroit: Re-Opening
Apr May Jun Jul Aug Sep Oct
2020
0
10
20
30
40
50
60
Dai
ly d
eath
s per
mil
lion p
eople
Detroit
R0(t)=1.0, R
0(suppress)=1.7, R
0(25/50)=1.5/2.0, = 0.003, =0.05
54 / 58
Massachusetts: Re-Opening
Apr May Jun Jul Aug Sep Oct
2020
0
5
10
15
20
25
30
35
40
45
50
Dai
ly d
eath
s per
mil
lion p
eopleMassachusetts
R0(t)=1.6, R
0(suppress)=2.7, R
0(25/50)=2.0/2.5, = 0.003, =0.20
55 / 58
Washington, DC: Re-Opening
Apr May Jun Jul Aug Sep Oct
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
District of Columbia
R0(t)=1.3, R
0(suppress)=1.4, R
0(25/50)=1.5/1.6, = 0.003, =0.02
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Conclusions
Speculations based on model, we are not epidemiologists
• Time-varying β (or R0) needed to capture social distancing, by
individuals or via policy
• New York City already has around 1700 deaths per million
people
◦ So δ > .002 is required by model
◦ 100% are not already infected and resolved
(continued...)
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Conclusions (continued)
• Random sampling in NYC would be very informative about δ and
percent ever infected (for herd immunity effects)
◦ NEJM study of 215 mothers giving birth in NYC found a 15
percent infection rate from Mar 22 to Apr 4
◦ Suggests a high current ever-infected rate (infections
doubling every 3-4 days)
◦ Model suggests δ < .004 to match this
◦ California serology testing also suggests δ ≈ .003 (but
problems?)
• “One size fits all” will not work for re-opening
◦ NYC could potentially re-open soon w/ basic precautions
◦ Rest of NY state could not
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