Post on 03-Jun-2020
Estimating and Simulating a
SIRD Model of COVID-19 for
Many Countries, States, and Cities
Jesus Fernandez-Villaverde and Chad Jones
May 29, 2020
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xkcd: Everyone’s an Epidemiologist
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Outline
• Basic model
◦ Social distancing via a time-varying R0t
• 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?
Our dashboard contains 25 pages of results
for each of 100 cities, states, and countries
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Basic Model
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Notation
• Number of people who are (stocks):
St = Susceptible
It = Infectious
Rt = Resolving
Dt = Dead
Ct = ReCovered
• Constant population size is N
St + It + Rt + Dt + Ct = N
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SIRD Model: Overview
• Susceptible get infected at rate βtIt/N
New infections = βtIt/N · St
• Infectiousness resolve at Poisson rate γ, so the average number
of days that a person is infectious is 1/γ so γ = .2 ⇒ 5 days
• Post-infectious cases then resolve at Poisson rate θ. E.g.
θ = .1 ⇒ 10 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 = −βtStIt/N︸ ︷︷ ︸
new infections
∆It+1 = βtStIt/N︸ ︷︷ ︸
new infections
− γIt︸︷︷︸
resolving infectious
∆Rt+1 = γIt︸︷︷︸
resolving infectious
− θRt︸︷︷︸
cases that resolve
∆Dt+1 = δθRt︸ ︷︷ ︸
die
∆Ct+1 = (1 − δ)θRt︸ ︷︷ ︸
reCovered
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)
• Continuous time, constant β
• Let st ≡ St/N = fraction susceptible
• If R0st > 1, the disease spreads, otherwise declines
• Initial exponential growth rate of infections is β − γ = γ (R0 − 1)
• As t → ∞, the total fraction of people ever infected, e∗, solves
(assuming s0 ≈ 1)
e∗ = −1
R0
log(1 − e∗)
Long run is pinned down by R0 (and death rate),
γ and θ affect 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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Recovering βt and R0t, I
• Recovering βt, a latent variable, from the data is straightforward.
• Dt+1: stock of people who have died as of the end of date t + 1.
• ∆Dt+1 ≡ dt+1: number of people who died on date t + 1.
• After some manipulations, we can “invert” the model and get:
βt =N
St
(
γ +1
θ∆∆dt+3 +∆dt+2
1
θ∆dt+2 + dt+1
)
and:
St+1 = St
(
1 − βt1
δγN
(1
θ∆dt+2 + dt+1
))
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Recovering βt and R0t, II
• With these two equations, a time series for dt, and an initial
condition S0/N ≈ 1, we iterate forward in time and recover βt and
St+1.
• We are using future deaths over the subsequent 3 days to tell us
about βt today.
• While this means our estimates will be three days late (if we have
death data for 30 days, we can only solve for β for the first 27
days), we can still generate an informative estimate of βt.
• More general point about SIRD models: state-space
representation that we can exploit.
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Recovering βt and R0t, III
• We can also recover the basic reproduction number:
R0t = βt × 1/γ
and the effective reproduction number:
Ret = R0t · St/N
• Now we can simulate the model forward using the most recent
value of βT and gauge where a region is headed in terms of the
infection and current behavior.
• And we can correlate the βt with other observables to evaluate
the effectiveness of certain government policies such as
mandated lock downs.
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An endogenous R0t when simulating future outcomes
• Individuals react endogenously to risk
• Indeed, much of the reaction is not even government-mandated.
• We could solve a complex dynamic programming problem.
• Instead, Cochrane (2020) has suggested:
R0t = Constant · e−αdt
where dt is daily deaths per million people.
• We estimate α from our data on R0t (more below)
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Estimates and Simulations
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Parameters assumed fixed and homogeneous
• γ = 0.2: average duration is 5 days (or γ = 0.1)
• θ = 0.1: average duration post-infectious is 10 days.
◦ average case takes 10+5 = 15 days to resolve.
◦ Long tail for exponential distribution
• α = 0.05: estimate αi for each location i.
◦ Tremendous heterogeneity across locations
◦ We report results with α = 0 and α = .05.
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Mortality rate (IFR): δ = 1.0%
• Case fatality rates not helpful: no good measure of how many
people are infected.
• Evidence from seroepidemiological national survey in Spain:
◦ Stratified random sample of 61,000 people
◦ δ in Spain is between 1% and 1.1%.
• Correction by demographics to other countries
◦ Most countries cluster around 1%.
◦ U.S.: 0.76% without correcting for life expectancy and 1.05%
correcting by it.
• New York City data suggests death rates of around 0.8%-1%.
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Estimation based entirely on death data
• Johns Hopkins University CSSE 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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New York City: Estimates of R0t = βt/γ
Mar 14 Mar 21 Mar 28 Apr 04 Apr 11 Apr 18 Apr 25 May 02 May 09
2020
0
0.5
1
1.5
2
2.5
3
3.5R
0(t
)New York City (only)
= 0.010 =0.10 =0.20
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New York City: Daily Deaths and HP Filter
Feb Mar Apr May
2020
0
10
20
30
40
50
60
70
80
90
New York City (only): Daily deaths, d
= 0.010 =0.10 =0.20
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New York City: Change in Smoothed Daily Deaths
Feb Mar Apr May
2020
-4
-3
-2
-1
0
1
2
3
4
5
6
New York City (only): Delta d
= 0.010 =0.10 =0.20
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New York City: Change in (Change in Smoothed Daily Deaths)
Feb Mar Apr May
2020
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
New York City (only): Delta (Delta d)
= 0.010 =0.10 =0.20
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New York City: Percent of the Population Currently Infectious
Mar 14 Mar 21 Mar 28 Apr 04 Apr 11 Apr 18 Apr 25 May 02 May 09
2020
0
1
2
3
4
5
6P
erce
nt
curr
entl
y i
nfe
ctio
us,
I/N
(p
erce
nt)
New York City (only)
Peak I/N = 5.67% Final I/N = 0.43% = 0.010 =0.10 =0.20
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Estimates of R0t = βt/γ
Mar Apr May
2020
0
0.5
1
1.5
2
2.5
3R
0(t
) = 0.010 =0.10 =0.20
New York City (only)Lombardy, Italy
Atlanta
Stockholm, Sweden
SF Bay Area
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Percent of the Population Currently Infectious
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Dashboard Table (link)
Total (pm) — R0 — R0 · S/N % Infectious Total (pm)
Deaths, t initial today today peak today Deaths, t+30
NYC (only) 2482 2.71 0.77 0.57 5.67% 0.43% 2650
Lombardy, Italy 2050 2.51 0.92 0.72 3.50% 0.32% 2236
Madrid, Spain 1782 2.58 0.20 0.15 3.97% 0.19% 1841
Stockholm, SWE 1499 2.61 1.17 0.97 2.44% 0.73% 2027
Boston 1198 2.12 0.72 0.62 2.63% 0.65% 1568
Paris, France 1003 2.39 0.20 0.01 1.99% 0.17% 1052
Philadelphia 885 2.46 0.88 0.78 1.68% 0.72% 1291
Spain 786 2.41 0.53 0.49 1.59% 0.12% 844
Chicago 738 2.17 0.93 0.84 1.10% 1.01% 1144
D.C. 723 1.99 0.94 0.85 1.28% 0.78% 1105
Italy 702 2.22 1.01 0.93 1.07% 0.15% 808
United Kingdom 679 2.37 0.96 0.88 1.16% 0.29% 845
France 567 2.17 1.15 1.07 1.26% 0.17% 682
United States 362 2.02 0.91 0.87 0.52% 0.24% 478
California 110 1.45 1.04 1.02 0.16% 0.13% 174
Brazil 102 1.26 1.13 1.10 0.28% 0.28% 240
SF Bay Area 77 1.26 0.98 0.97 0.12% 0.04% 97
Mexico 54 1.31 1.12 1.10 0.15% 0.15% 128
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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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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 = 0.8%)
◦ Red = lowest parameter value (e.g. δ = 0.3%)
◦ Green = highest parameter value (e.g. δ = 1.0%)
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Guide to Graphs (continued)
• R0 in subtitle:
◦ 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 (7 days): Daily Deaths per Million People
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
New York City (plus)
R0=2.6/0.5/0.9 = 0.010 =0.05 =0.1 %Infect=22/23/23
DATA THROUGH 19-MAY-2020
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New York City (7 days): Cumulative Deaths per Million (Future)
Mar Apr May Jun Jul Aug Sep
2020
0
500
1000
1500
2000
2500
3000
3500
Cum
ula
tive
dea
ths
per
mil
lion p
eople
New York City (plus)
R0=2.6/0.5/0.9 = 0.010 =0.05 =0.1 %Infect=22/23/23
DATA THROUGH 19-MAY-2020
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Lombardy (7 days): Daily Deaths per Million People
Mar Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
Dai
ly d
eath
s per
mil
lion p
eople
Lombardy, Italy
R0=2.5/1.1/1.3 = 0.010 =0.05 =0.1 %Infect=22/24/27
DATA THROUGH 19-MAY-2020
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Italy (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
2
4
6
8
10
12
14
16
18
20
Dai
ly d
eath
s per
mil
lion p
eople
Italy
R0=2.2/1.1/1.1 = 0.010 =0.05 =0.1 %Infect= 8/ 9/11
DATA THROUGH 19-MAY-2020
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Madrid (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
Dai
ly d
eath
s per
mil
lion p
eopleMadrid, Spain
R0=2.6/0.2/0.3 = 0.010 =0.05 =0.1 %Infect=18/18/18
DATA THROUGH 19-MAY-2020
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Spain (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
5
10
15
20
25
30
Dai
ly d
eath
s per
mil
lion p
eople
Spain
R0=2.4/0.6/0.7 = 0.010 =0.05 =0.1 %Infect= 8/ 9/ 9
DATA THROUGH 19-MAY-2020
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Paris (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
5
10
15
20
25
30
35
Dai
ly d
eath
s per
mil
lion p
eopleParis, France
R0=2.4/0.3/0.3 = 0.010 =0.05 =0.1 %Infect=11/11/11
DATA THROUGH 19-MAY-2020
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California (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep Oct
2020
0
1
2
3
4
5
6
Dai
ly d
eath
s p
er m
illi
on
peo
ple
California
R0=1.5/1.0/1.0 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 5
DATA THROUGH 19-MAY-2020
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California (7 days): Cumulative Deaths per Million (Future)
Mar Apr May Jun Jul Aug Sep Oct
2020
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
California
R0=1.5/1.0/1.0 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 5
New York City
Italy
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U.K. (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
United Kingdom
R0=2.4/1.1/1.2 = 0.010 =0.05 =0.1 %Infect= 8/10/14
DATA THROUGH 19-MAY-2020
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U.K. (7 days): Cumulative Deaths per Million (Future)
Mar Apr May Jun Jul Aug Sep
2020
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
United Kingdom
R0=2.4/1.1/1.2 = 0.010 =0.05 =0.1 %Infect= 8/10/14
New York City
Italy
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United States (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
1
2
3
4
5
6
7
8
9
10
Dai
ly d
eath
s per
mil
lion p
eopleUnited States
R0=2.0/1.0/1.1 = 0.010 =0.05 =0.1 %Infect= 5/ 6/ 8
DATA THROUGH 19-MAY-2020
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United States (7 days): Cumulative Deaths per Million (Future)
Mar Apr May Jun Jul Aug Sep
2020
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
United States
R0=2.0/1.0/1.1 = 0.010 =0.05 =0.1 %Infect= 5/ 6/ 8
New York City
Italy
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U.S. excl. NYC (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep Oct
2020
0
1
2
3
4
5
6
7
8
Dai
ly d
eath
s p
er m
illi
on
peo
ple
U.S. excluding NYC
R0=1.8/1.0/1.1 = 0.010 =0.05 =0.1 %Infect= 4/ 5/ 8
DATA THROUGH 19-MAY-2020
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U.S. excl. NYC (7 days): Cumulative Deaths per Million (Future)
Mar Apr May Jun Jul Aug Sep Oct
2020
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Cum
ula
tive
dea
ths
per
mil
lion p
eople
U.S. excluding NYC
R0=1.8/1.0/1.1 = 0.010 =0.05 =0.1 %Infect= 4/ 5/ 8
New York City
Italy
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Stockholm (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
Dai
ly d
eath
s per
mil
lion p
eople
Stockholm, Sweden
R0=2.6/1.5/1.6 = 0.010 =0.05 =0.1 %Infect=18/25/39
DATA THROUGH 19-MAY-2020
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France (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
France
R0=2.2/1.2/1.1 = 0.010 =0.05 =0.1 %Infect= 6/ 8/12
DATA THROUGH 19-MAY-2020
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Washington (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
1
2
3
4
5
6
7
8
9
Dai
ly d
eath
s per
mil
lion p
eopleWashington
R0=1.6/0.3/0.4 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 2
DATA THROUGH 19-MAY-2020
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Louisiana (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
5
10
15
20
25
Dai
ly d
eath
s per
mil
lion p
eople
Louisiana
R0=2.2/1.0/1.1 = 0.010 =0.05 =0.1 %Infect= 8/10/13
DATA THROUGH 19-MAY-2020
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Florida (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep Oct
2020
0
0.5
1
1.5
2
2.5
3
3.5
4
Dai
ly d
eath
s p
er m
illi
on
peo
ple
Florida
R0=1.6/0.9/1.0 = 0.010 =0.05 =0.1 %Infect= 2/ 2/ 3
DATA THROUGH 19-MAY-2020
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Detriot (Wayne County, 7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep
2020
0
10
20
30
40
50
60
70
Dai
ly d
eath
s per
mil
lion p
eople
Detroit
R0=2.4/0.8/1.3 = 0.010 =0.05 =0.1 %Infect=18/19/20
DATA THROUGH 19-MAY-2020
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Boston (7 days): Daily Deaths per Million People
Apr May Jun Jul Aug Sep Oct
2020
0
10
20
30
40
50
60
70
Dai
ly d
eath
s per
mil
lion p
eople
Boston+Middlesex
R0=2.1/1.2/1.5 = 0.010 =0.05 =0.1 %Infect=15/20/32
DATA THROUGH 19-MAY-2020
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District of Columbia (7 days): Daily Deaths per Million People
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=2.0/1.1/1.3 = 0.010 =0.05 =0.1 %Infect=10/15/26
DATA THROUGH 19-MAY-2020
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Reopening and Herd Immunity
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Percent Ever Infected would be very informative
— Percent Ever Infected (today) —δ = 0.5% δ = 1.0% δ = 1.2%
New York City (plus) 44 22 19
Lombardy, Italy 43 22 19
Madrid, Spain 36 18 15
Detroit 36 18 15
Stockholm, Sweden 36 18 15
Boston+Middlesex 29 15 12
Paris, France 21 11 9
Philadelphia 23 12 10
Spain 17 8 7
Chicago 21 11 9
District of Columbia 20 10 8
United Kingdom 16 8 7
France 13 6 5
United States 9 5 4
New York excluding NYC 8 4 3
Miami 7 3 3
Ecuador 6 3 3
Los Angeles 5 3 2
Minnesota 5 3 2
Germany 3 1 1
California 3 2 1
Brazil 4 2 2
Mexico 2 1 1
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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)
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Herd Immunity and Opening the Economy? δ = 1.0%
Percent R0(t+30) PercentSusceptible with no way back
R0 R0(t) t+30 outbreak to normal
New York City (plus) 2.6 0.4 77.5 1.3 41.5
Lombardy, Italy 2.5 0.9 77.5 1.3 23.4
Madrid, Spain 2.6 0.2 81.5 1.2 43.2
Detroit 2.4 0.5 81.6 1.2 37.6
New Jersey 2.6 1.1 78.3 1.3 11.4
Stockholm, Sweden 2.6 1.2 78.3 1.3 7.2
Boston+Middlesex 2.1 0.7 84.9 1.2 32.9
Paris, France 2.4 0.2 89.4 1.1 42.0
Philadelphia 2.5 0.9 87.2 1.1 17.0
Spain 2.4 0.5 91.5 1.1 29.8
Chicago 2.2 0.9 87.0 1.1 18.0
District of Columbia 2.0 0.9 87.9 1.1 19.0
United Kingdom 2.4 1.0 91.0 1.1 10.0
United States 2.0 0.9 94.7 1.1 13.1
New York excluding NYC 2.0 1.1 92.8 1.1 -2.3
U.S. excluding NYC 1.8 0.9 95.6 1.0 11.8
Ecuador 1.5 0.8 95.7 1.0 30.8
Florida 1.6 0.9 98.0 1.0 15.3
Germany 1.7 0.2 98.6 1.0 55.8
Brazil 1.3 1.1 95.0 1.1 -54.7
SF Bay Area 1.3 1.0 98.8 1.0 10.3
Mexico 1.3 1.1 97.3 1.0 -45.8
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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 adjust to daily deaths via α
◦ Each daily death reduces R0 by 5 percent
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Spain: Re-Opening
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Italy: Re-Opening
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New York City: Re-Opening
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New York excluding NYC: Re-Opening
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California: Re-Opening
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Sweden: Re-Opening
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Detroit: Re-Opening
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Massachusetts: Re-Opening
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Washington, DC: Re-Opening
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Conclusions
Speculations based on model, we are not epidemiologists
• Time-varying βt (or R0t) needed to capture social distancing, by
individuals or via policy
• Is the death rate 1.0% or ??? How many people currently
infectious? Random sampling!
• “One size fits all” will not work for re-opening
◦ Susceptible rates are heterogeneous
Our dashboard contains 25 pages of results
for each of 100 cities, states, and countries
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