Estimating and Simulating a [-3pt] SIRD Model of COVID-19 ...chadj/slides-sird.pdfEstimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities Jesu´s

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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https://web.stanford.edu/~chadj/Covid/Dashboardhtml

Basic Model

3 / 66

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

4 / 66

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

18 / 66

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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https://web.stanford.edu/~chadj/Covid/Dashboard.html

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)

28 / 66

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

29 / 66

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


30 / 66

Lombardy (7 days): Daily Deaths per Million People


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


31 / 66

Italy (7 days): Daily Deaths per Million People


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


32 / 66

Madrid (7 days): Daily Deaths per Million People


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


33 / 66

Spain (7 days): Daily Deaths per Million People


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


34 / 66

Paris (7 days): Daily Deaths per Million People


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


35 / 66

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


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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

37 / 66

U.K. (7 days): Daily Deaths per Million People


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


38 / 66

U.K. (7 days): Cumulative Deaths per Million (Future)


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

39 / 66

United States (7 days): Daily Deaths per Million People


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


40 / 66

United States (7 days): Cumulative Deaths per Million (Future)


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

41 / 66

U.S. excl. NYC (7 days): Daily Deaths per Million People


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


42 / 66

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

43 / 66

Stockholm (7 days): Daily Deaths per Million People


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


44 / 66

France (7 days): Daily Deaths per Million People


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


45 / 66

Washington (7 days): Daily Deaths per Million People


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


46 / 66

Louisiana (7 days): Daily Deaths per Million People


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


47 / 66

Florida (7 days): Daily Deaths per Million People


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


48 / 66

Detriot (Wayne County, 7 days): Daily Deaths per Million People


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


49 / 66

Boston (7 days): Daily Deaths per Million People


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


50 / 66

District of Columbia (7 days): Daily Deaths per Million People


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


51 / 66

Reopening and Herd Immunity

52 / 66

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

53 / 66

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)

54 / 66

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

55 / 66

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

56 / 66

Spain: Re-Opening

57 / 66

Italy: Re-Opening

58 / 66

New York City: Re-Opening

59 / 66

New York excluding NYC: Re-Opening

60 / 66

California: Re-Opening

61 / 66

Sweden: Re-Opening

62 / 66

Detroit: Re-Opening

63 / 66

Massachusetts: Re-Opening

64 / 66

Washington, DC: Re-Opening

65 / 66

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

66 / 66

https://web.stanford.edu/~chadj/Covid/Dashboardhtml

Documents

Estimating and Simulating a [-3pt] SIRD Model of COVID-19 ...chadj/slides-sird.pdfEstimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities Jesu´s