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Federal Reserve Bank of Chicago
A Model of Economic Activity
in San Francisco During the
1918 Influenza Epidemic
François R. Velde
February 15, 2022
WP 2022-04
https://doi.org/10.21033/wp-2022-04
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
*
A Model of Economic Activity in San Francisco During the
1918 Influenza Epidemic
François R. Velde∗
Federal Reserve Bank of Chicago
February 15, 2022
Abstract
I jointly use daily data on deaths and public transportation ridership in San Francisco in 1918–19 to estimate a model in which agents choose their level of economic
activity based on perceived infection risk, modeled as a function of current and lagged
infections or deaths. Agents’ choices in turn affect the dynamics of the epidemic by
reducing contacts in an otherwise standard SEIR model. Non-pharmaceutical interventions restrict agents’ activity either as a tax or a bound. I estimate the parameters
by maximum likelihood and use the best-fitting model to compute counterfactuals. San
Francisco’s intervention reduced deaths by a few percent only, and it was away from
the Pareto frontier: an earlier and milder intervention would have done better. The
behavioral feedback narrows the room for intervention compared to a model with unresponsive agents, and ill-timed interventions can worsen outcomes. Masks also had
an effect on transmission rates.
Keywords: 1918 influenza epidemic, San Francisco, public transportation, non-pharmaceutical interventions, SIR macro model, policy evaluation, counterfactuals (JEL
H12, I18, I19, N12, R40).
∗
I thank my colleagues at the Chicago Fed, particularly Gadi Barlevy, François Gourio, and Martì
Mestieri; as well as participants in seminars at Manchester University and Università di Roma La
Sapienza. The views expressed do not necessarily represent those of the Federal Reserve System or
the Federal Reserve bank of Chicago.
Introduction
The Covid-19 pandemic has sparked interest in earlier pandemics, in particular the 1918–
19 influenza pandemic (Beach, Clay, and Saavedra 2021). During that event, almost all
large US cities (and many small ones as well) used non-pharmaceutical interventions
(NPIs) to contain the disease. The nature of the restrictions was broadly similar: closings
of places of amusement, public gatherings, schools and churches. The timing and duration of the interventions varied, and in some places restrictions were reimposed during
the second wave.
This variation across a relatively homogenous set has provided ways to evaluate NPIs.
An older epidemiological literature (Markel et al. 2007; Bootsma and Ferguson 2007) evaluates the impact on the dynamics of epidemics. With the Covid-19 pandemic, economists
have used the same data to make arguments about the tradeoffs (or absence thereof) between epidemic containment and foregone economic activity (Correia, Luck, and Verner
2020).
I also want to understand the choices that were and could have been made in 1918.
The contribution is twofold: (1) I use mobility data in addition to deaths data, and (2) I
add optimizing agents to a SEIR model in order to make inferences from the mobility and
death data jointly, taking mobility data as a proxy for economic activity as in FernándezVillaverde and Jones (2020). Korolev (2020) has pointed out identification problems in
the basic SIR model and called for using additional data such as mobility. The structural
approach allows me to estimate the parameters of the model and run counterfactuals. This
paper centers on the city of San Francisco because I have found data at the daily frequency
for both deaths and mobility. The SEIR model is standard (Bootsma and Ferguson 2007)
and the economic component taken from Eichenbaum, Rebelo, and Trabandt (2021).
The ridership data and anecdotal evidence show that the use of public transportation
began to decline before the imposition of restrictions. If individuals were limiting interactions even in the absence of public policy, how much would the transmission rate of
the disease have been reduced anyway? Empirically, the crucial element is the behavioral
feedback, or endogenous social distancing. Epidemiologists have long recognized that
the rate at which agents infect each other need not be constant as the epidemic explodes,
and during the Covid-19 pandemic economists have made the point repeatedly (Atkeson
2020, e.g.,) and devised various techniques to estimate this rate as a time-varying parameter (see Arias et al. 2021, and the literature cited therein). I derive the rate from agents’
perceptions of risk, which nest rational expectations or full information (Toxvaerd 2021,
as in), but may depend on lags of infections or deaths, rather than depend on current infections only (Jones, Philippon, and Venkateswaran 2021, as in). This behavioral feedback
through agents’ precautionary reductions in mobility and economic activity, not present
in Acemoğlu et al. (2020) and Glover et al. (2020, e.g.,) is important, not only to match the
data, but also to understand the counterfactuals and evaluate the policies.
The model does a good job of matching the mobility data and rejects rational expectations. Parameter estimates indicate strong persistence in the feedback: current and lagged
1
deaths affect current behavior. The counterfactual implies that the reduction in deaths
achieved by the interventions in San Francisco was not negligible but relatively modest:
a few percent of total deaths, or two hundred lives. Moreover the intervention was well
away from the Pareto frontier: authorities could have saved more lives with an earlier,
shorter, and milder intervention. The reason is that the behavioral channel moderates the
epidemic considerably, compared to an epidemic of similar characteristics but with completely unresponsive agents. There is less overshooting than in the SIR model with constant transmission, but the herd immunity level puts the same upper bound on reduction
in deaths: hence, the room for maneuver is limited. Interestingly, the Pareto frontier has
a sharp kink: most reduction in deaths could be achieved with very mild interventions,
and further gains were very limited and required much longer interventions.
Finally, the model allows me to estimate the impact of masks separately from other
restrictions, because during San Francisco’s second wave masks were required again but
there were no closings. I find that masks in the second wave may have been as effective
on their own as masks and closings during the first wave.
1
The epidemic in San Francisco
The purpose of this section is to provide some context for the data and model.1 The focus
is on the development of the epidemic as seen by contemporaries, and the reaction of the
city authorities. As in many other places in the US, measures were taken at the city level,
either by the Board of Health of the city and county, or by the Board of Supervisors. Dr.
William C. Hassler, as head of the Health Department or Health Officer from 1915 to his
death in 1931, was the most prominent official and led the city’s response.
The influenza pandemic of 1918 (aside from a little-noticed herald wave in the spring
of 1918) arrived in the US in late August 1918 and hit the East Coast first. It spread quickly
to other parts of the country, and by mid-September authorities in California were well
aware of the danger, but the timing of its arrival on the West Coast was uncertain and they
initially thought they had time. As of September 20, George Ebright, the president of the
State Board of Health, thought that there was no evidence of influenza in California, but
that it would arrive within two to three weeks, following the lines of railroad travel. In the
meantime health officials focused on educating the public and making recommendations.
Hassler described the state of knowledge about the disease, its propagation from person
to person through nose and mouth, the rapid progress and frequently fatal pneumonia
that ensued, and the susceptibility of “the most robust members of the community.” He
recommended avoiding public places and contacts with infected persons. There was no
talk of restrictions or closings yet (the first such measures were taken on September 20 in
Milford, MA).
When the first case was reported in the city on September 24, Hassler was optimistic
that the epidemic would be contained, but within a few days he admitted that the disease
1 The main source of this section is contemporary news articles from the San Francisco Examiner, the San
Francisco Chronicle, and the Recorder; see also Crosby (2003, 91–116).
2
was well distributed and a serious epidemic was to be feared, and on October 4 he conceded that “the spread of the disease here now will be very rapid [. . .] an exceedingly large
roll of cases can be expected.” From the end of September large cities on the East coast
resorted to closings of places of amusement and schools, and the US Surgeon General Rupert Blue stated on October 4 that “the only way to stop the spread of Spanish influenza
is to close churches, schools, theatres and public institutions in every community where
the epidemic has developed.”
Yet in San Francisco authorities remained ambivalent. While nearby communities
adopted such measures (Redwood City on October 10, San Jose the next day), the city’s
Board of Health decided on October 10 that there was no need to close schools or theaters, and Hassler opined that reports from eastern cities suggested that closing theaters
did little to check the epidemic. As late as October 19, the State Board of Health stated:
The closing of the public schools is a measure that the State Board of Health
does not favor, provided that the pupils are inspected daily by teacher or nurse,
those who show signs of illness being immediately sent home. The State Board
of Health has not issued any general order nor made any drastic moves in
prohibiting public meetings or in closing places of amusement. The Board
believes that it is much better to supply citizens with full information as to the
ways in which the disease may be contracted.
These ways included acts of personal hygiene, walking to work and avoiding crowded
places, self-isolation in case of symptoms.
Case numbers continued to climb: by October 13 hospitals were overwhelmed, the
San Francisco hospital was devoted wholly to caring for influenza cases and hospitals
were “requested not to take in any cases which do not absolutely demand hospital care.”
Notably, attendance at places of amusement was already declining: “all the theatres are
feeling the effects of the campaign against Spanish influenza, and even on O’Farrell Street,
though the audiences are still large, there has been a falling off in patronage” (Examiner,
Oct 14, p. 11). Finally, the city Board of Health held a special meeting on the evening of
October 17, attended by the mayor, various officials of the US military, US Public Health
Service, the Red Cross, and representatives of the entertainment industry, and voted to
close “all places of amusement, including theaters, moving-picture theaters, concert halls,
dance halls and dances in all cabarets, cafes and hotels, and all form of entertainment
in any or all of them”. The order also applied to lodge and fraternal meetings, public
amusement places (penny arcades, merry-go-rounds), private dances, halls, social gatherings, Sunday school classes, church services, public and private schools and kindergartens. Any public meeting required a permit. Representatives of entertainment houses
were strongly in favor because they were already suffering financially and hoped that the
disease would be eradicated quickly (Examiner, Oct 18).
Hassler strongly believed in the effectiveness of masks and urged their use on the public street. Dr. Woods Hutchinson, a health writer and strong promoter of masks, argued
that “if one half or even one third of the population would wear masks the number of
3
attacks and deaths due to Spanish influenza would be cut from one-half to two-thirds”
(Recorder, Oct 23, p. 8) and Hassler said that they gave “90% immunity” (Examiner, Oct. 22,
p. 13). At Hassler’s urging the Health Board made masks compulsory for clerks, druggists, tellers, hotel employees, barbers on October 19. Then, on October 24, the Board
of Supervisors passed a penal ordinance making the wearing of masks mandatory in all
places except the home, under penalty of a fine between $5 and $100, up to ten days imprisonment, or both (Municipal Record, 1918, 352). Newspapers reported enforcement
actions against “mask slackers” (250 arrested on November 4), but Hassler claimed that
the ordinance was widely abided. The State Board of Health, which ultimately concluded
that masks had been useless, agreed that “contrary to expectation, the masks were worn
cheerfully and universally” (Kellogg 1919a, 39).2
Reported cases peaked in late October. Hassler attributed this to masks but expected
that the city would remain masked for two months to avoid a second wave (Chronicle,
Nov 6, p. 9) and quashed early talk of reopening theaters. Although “fully aware of the
losses incurred under the present rigid regime,” he wanted closings to continue until December: “I can almost deplore the necessity for printing daily the number of new cases,
for I fear it will tend to lull the public into a false sense of security, as was the case in Los
Angeles, where , with all the amusement houses permitted to remain open, the disease
passed beyond the control of the authorities” (Chronicle, Nov 9, p. 9). But after four weeks
of closure, “been the maximum time for the closing of amusement places in other cities,”
the entertainment sector increased pressure on the Board of Health and on the mayor.
The closing order was lifted: theaters were allowed to reopen on the afternoon of November 16,3 churches on November 17, and schools on November 18 (although the Board of
Education deferred the reopening for a week). After waiting for a few days to see the impact of reopening, Hassler proclaimed that “Spanish influenza has been eradicated from
San Francisco” and announced that the mask mandate would be lifted on November 21
(Examiner, Nov 18, p. 3).
The epidemic dropped from headlines and newspapers ceased to report cases and
deaths for a while, but in early December new cases were reported again. At first Hassler
denied any indication of a flare-up (Chronicle, Dec 4, p. 8) but the next day he required
store clerks to wear masks again and warned that a reinstatement of the ban would be
necessary if cases continued to rise. Within a few days he urged the mayor to consider
reinstating the mask mandate. This time he met with serious opposition: at a conference
on December 9 with Hassler and the Board of Supervisors, the entertainment and retail
sectors disputed the case numbers and pushed back against masks, despite Hassler’s prediction that the second wave would be worse than the first “as has been the case in many
other cities.” After being widely feted, Hassler became the object of strong feelings, to
the point that a bomb was mailed to him (Examiner, December 18, p. 1). The Supervisors
2 Another
measure that Hassler thought important, despite widespread skepticism, was the administration
of a vaccine developed by Dr, Timothy J. Leary of Tufts Medical College, sent from Boston with great fanfare
and administered for free, but with disappointing take-up (Crosby 2003, 100).
3 The Mission and North Beach areas remained closed a little longer, and large dancing remained banned
until November 30.
4
postponed a decision on masks for several days, took it up again on December 17 for 4
and a half hours with no conclusion, and finally rejected the mask mandate by a 9–7 vote
on December 19. Cases continued to grow, albeit more slowly than in October. Hassler
judged the situation “bad” on December 25 and “alarming” on Dec 28, but was still reluctant to go back to the Supervisors. The Board of Health stated on January 2 that it
had done all it could and it was up to the legislative body of the city to act. At its urging
the Board of Education required masks in public schools on January 4. Finally, enough
supervisors were willing to change their minds: the Health committee approved a mask
ordinance just as cases passed 600 (compared with a peak of over 2000 on October 25)
and on January 11 the ordinance was announced, although it would not take effect until
January 17. Immediately cases began a precipitous drop, which Hassler attributed to the
masks. Within a week cases had fallen by a factor of ten, hospitals were relieved, and
Hassler declared the situation “bright beyond our highest expectations.” Late January
the pressure to repeal the ordinance increased, as compliance fell, and at a special meeting held on a Sunday morning, February 1, the Board of Health relented and the mayor
promptly nullified the ordinance at 11am.
Hassler’s views apparently shifted from the first to the second wave. He was initially
optimistic that the city would be spared, and only slowly came to recommend restrictive
measures. He proclaimed the disease eradicated after the first wave, and dismissed the
first signs of the second wave, but then strongly pushed for another mask mandate. It’s
unclear what model or data he had in mind, although he explicitly referred to the experience of other cities during the 1918 pandemic, and also the pandemic of 1889–91 which
was the main reference point at the time (California State Board of Health 1918).
In summary, San Francisco authorities saw the influenza epidemic arrive by September 24, and within a week or two had the examples of other cities’ closings, but no NPI
was undertaken until October 18, when schools, churches, and places of amusement were
closed. On October 24 a mask mandate was also put in place and enforced, and seems to
have been widely obeyed. The closing of places of amusement lasted until November
16 (this is the end date I will use in the estimation) while schools remained closed until
November 24 and the mask mandate was lifted on November 21. Health officials would
have wanted to extend the closings but strong pressure led the city authorities to lift the
ban. When the second wave arrived in early December Hassler urged a reinstatement of
the mask mandate but this time met with opposition and did not succeed until January
17, at which point the wave dropped off quickly and the mandate was lifted on February
1. There were no closings during the second wave.
2
The Data
This section describes the two data series on mortality and streetcar ridership.
5
Mortality data
As mentioned in the previous section cases and deaths were reported regularly at news
conferences by Dr. Hassler and relayed in the local newspapers. Although they tell us
what information was available to contemporaries (see section 3 below), these series are
not good measures of actual cases and deaths, as was understood even then (Kellogg
1919a).
I have collected all deaths in San Francisco on a daily basis from September 1, 1918 to
April 15, 1919 using ancestry.com.4 The data includes first and last names, year of birth,
and day of death. They appear to have been manually transcribed from indices of death
registration records and are not free of error, but aggregated at the monthly frequency
the totals differ by no more than 3% from the official monthly totals for San Francisco
(Bureau of the Census 1913–23). I also collected daily deaths for the same months in the
years 1912 to 1916 in order to establish a baseline of daily expected deaths (adjusting for
population growth, Department of Commerce, Bureau of the Census 1923). This gives
me excess deaths on a daily basis, which I assume to be entirely due to the epidemic.
140
12000
daily excess deaths
inferred infections
120
10000
100
8000
80
6000
60
4000
40
2000
20
0
Sep 1918
0
Oct 1918
Nov 1918
Dec 1918
Jan 1919
Feb 1919
Mar 1919
Figure 1: Excess deaths in San Francisco, daily (left axis) and estimated infections based
on a deconvolution of deaths (right axis). The yellow area indicates the period when
places of amusement were closed. Sources: ancestry.com.
Using the delay from exposure to death5 it is possible to deconvolute the daily death
series and construct an estimated infection series (Goldstein et al. 2009). The result is
shown in Figure 1 and will be used later.
4 Ideally one would want to collect all deaths ascribed to pneumonia and influenza in death certificates, but
they are not available digitally for California in that time period.
5 The delay from exposure to symptoms is taken from Ferguson et al. (2005) and the delay from symptoms
to death is from Keeton and Cushman (1918, chart 2), based on 603 deaths observed in Cook County Hospital.
6
Ridership data
The ridership data pertains to the San Francisco Municipal Railways (the “Muni”), one of
three street railway systems in San Francisco (O’Shaughnessy 1921). The largest system
was the United Railroads, which operated 257 miles; the California Street Cable Railroad
Company operated 10.5 miles of cable car. The Muni, city-owned as its name indicates,
began operating a line on Geary Street in 1912 and by June 1918 it operated 57.4 miles.6
Figure 2 shows the network in 1920, which covered the central business district (Market
Street) and connected to the main residential areas of the city. In 1920 the Muni accounted
for around 22% of the total ridership, which was on average 511,060 per day for a city
population of 511,300 (O’Shaughnessy 1921, 13, 122). The Muni ridership data is thus
likely to be a good measure of the use of public transportation.
I collected weekly reports of daily receipts of the Muni from 1917 to 1922 in local newspapers.7 Newspapers reported gross revenues, but passenger revenues represent 99.1% of
gross revenues. Revenues are also an excellent measurement of (paying) passengers: the
fare remained unchanged at 5 cents throughout the period, the average fare per revenue
passenger was 4.92 cents (Wilcox 1921, 185), and the correlation of monthly passengers to
monthly gross revenues is 0.998 for 1917–19 (San Francisco Municipal Railway 1915–22).
Figure 3 shows the data (smoothed using a 7-day centered moving average) and two
methods of deriving an index of ridership. The first method is to regress the data over
the full sample (June 1918 to December 1922) on a linear time trend, week-of-year, and
day-of-week dummies.8 The second method is to take the ratio of a day’s revenues to the
revenues 364 days later, and then scale the resulting series by the average annual growth
rate estimated in the previous regression. The two methods show little difference; I use
the first in the estimation.
Is ridership a good proxy for economic activity?
One goal of this paper is to use the ridership data to validate the epidemiological model,
and in particular to assess the nature and importance of any behavioral feedback. To do
this I will use in the next section a simple model in which agents decrease their economic
activity in response to the epidemic as well as to restrictions imposed by authorities. But
is ridership a good proxy?
The impact of the epidemic on electric railway ridership was widely noted. The journal
of the American Electric Railway Association noted that nationwide the epidemic “caused
an immense falling off in the number of passengers carried” (AERA, 1918, 7(6):575). In
San Francisco, the Examiner reported (28 Nov 1918, p. 6):
6 Mileage
increased by 2.4mi on Feb 21 1919 and 3.0mi on Apr 12, 1919.
7 The newspaper are the San Francisco Examiner, San Francisco Chronicle, and Daily Recorder.
It’s not clear why
newspapers published these reports, but they begin in August 1917, during a strike that affected the private
competitors from August 12 to November 23, 1917 (Commercial and Financial Chronicle, 106(2744):119). They
become regular in October 1917 but data is occasionally missing (from Jan 6 to Feb 9 and Feb 17 to Mar 2, 1918;
weeks ending Apr 13, Jul 13, 1918, May 3, 1919, Dec 16, 1920; from Dec 24 to Jan 15, 1921; weeks ending Jul 9,
Aug 6, 1921; Jan 7, Aug 5 1922).
8 I only use dummies for Saturday and Sunday because they are the only ones significant. I interact these
day-of-week dummies with calendar year dummies.
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Figure 2: Map of the San Francisco Municipal network in 1920 (San Francisco Municipal
Railway 1915–22, 1920, 15).
7500
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s.a. and detrended
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Dec 1918
Jan 1919
Feb 1919
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Mar 1919
Figure 3: Daily revenues on the San Francisco Municipal Railways, 7-day moving average
(right axis), and two indices (left axis): one detrends and seasonally adjusts, the other
takes the ratio of t to t + 365 and adjusts for 1-year average growth. The yellow area indicates the period when places of amusement were closed. Sources: San Francisco Examiner,
San Francisco Chronicle, Daily Recorder (various issues).
8
During the epidemic all street railway earnings were cut down enormously.
The United Railroads suffered in proportion to the Municipal Railway lines.
The deficit has amounted to more than $1,000 a day on the Municipal lines
ever since the ban has been raised. Traffic has been abnormally light since the
influenza epidemic broke out, whereas it should be the heaviest during the
rainy season.
Fewer riders does not translate one to one into less economic activity. Not all workers
and shoppers used public transportation, and those that usually did could choose to walk
during the epidemic, albeit at a cost in terms of time. But workers who usually rode the
Muni and stayed home would reduce measured ridership.
The maximum decrease in ridership in Figure 3 is about 30%. The only quantitative
evidence I have found for the effect of the epidemic on activity is a statement from the
entertainment industry on November 15 that $400,000 was “weekly diverted from the
normal channels.” This translates into about $1 per person or $4 per household per week,
compared with GDP per capita per week of $12.50 or per household of $60, this means
a reduction of 8%. Using median hourly wages of $0.50 for men, $0.30 for women, with
only half of women in the labor force compared to men, and a 50-hour workweek would
give a higher number of 12%. It seems likely, then, that the 30% figure is an over-estimate.
Beyond the historian’s defense that this is all the data that I have, mobility data arguably provides an upper bound on the reduction in economic activity. Moreover, if one
assumes a proportional (though less than one-for-one) relation between the two, mobility data allows for ordinal comparisons in outcomes: a counterfactual with more or less
ridership corresponds to a counterfactual with more or less economic activity. This will
be important when thinking about the Pareto frontier in section 5.
3
Model
The model, adapted from Eichenbaum, Rebelo, and Trabandt (2021), has an economic
and an epidemiological side.
The epidemiological component
The epidemiological side is based on the classic compartmental model (Bootsma and Ferguson 2007). The population is partitioned into St susceptibles, E exposed, I infected,
and R removed:
St + Et + It + Rt = 1.
The categories of interest evolve according to the following laws of motion:
Ṡt = −λt It St
(1)
Ėt = λt St It − αEt
(2)
I˙t = αEt − νIt
(3)
(4)
Ṙt = νIt
9
In addition, deaths evolve according to
Z ∞
Dt = µ
f (s)λ(t − s)S(t − s)I(t − s)ds
0
where the delay function f (s) is the distribution of time from exposure to death.
The literature has long recognized that the force of infection λt need not be constant
over time, and can depend on current and past values of the epidemic’s variables. Several
ad-hoc forms have been used.
Let f (t) be some function of the epidemic’s history {Ss , Es , Is , Ds }s≤t satisfying f (0) =
0. One common specification is that λt is a Hill function of f (t):
λt
1
=
.
λ0
1 + f (t)
(5)
Yu et al. (2017) compare the Hill function with alternatives such as e−κf (t) and (1−f (t))κ .
Bootsma and Ferguson (2007) use the Hill function and assume that f (t) a function of
RT
observed current and lagged deaths over a window of length Tm : f (t) = 0 m D(t − s)ds
while Yu et al. (2017) use exponential weighting of lagged deaths: f˙(t) = Dt − τ f (t).
The economic component of the model will provide a foundation for the functional
form, and link λt to measurements of economic activity.
The economic component
The agent’s preferences over consumption c and hours worked n
X c1−σ
n1+ϵ
−v
βt
1−σ
1+ϵ
t
are maximized subject to the budget constraint wt nt = ct . The parameters σ and ϵ are the
inverses of the intertemporal elasticity of substitution and of the Frisch elasticity. A linear
technology converts hours worked into consumption good at a constant rate, so that in
equilibrium the wage rate is constant.
The force of infection λt becomes endogenous in the following way. Eichenbaum, Rebelo, and Trabandt (2021) assume that incidence (new infections per period) λt St It arises
from interactions between the susceptible and the infected, and these interactions are chosen by agents taking account the contagion risk. I also want to allow for economic activity
to be affected directly through the labor supply, so I assume that a fraction ϕ of the infected are asymptomatic, and make the same economic choices as the susceptible, while
the fraction 1 − ϕ is out of the labor force.
Specifically, new infections are modeled as
− (St+1 − St ) = λt St It = π1 (St cst )(ϕIt cit ) + π2 (St nst )(ϕIt nit ) + π3 St It .
(6)
The first two terms on the right-hand side measure infections of susceptibles during the
purchase of consumption and the production of goods respectively, while the third term
measures infections that are not affected by economic choices and would occur even if the
economy shut down completely.
10
A susceptible individual understands that her probability of infection depends on her
consumption and labor decisions. The true probability derived from (6)
[ϕ(π1 w2 + π2 )nit n + π3 ]It
(7)
is an affine function of her choice n, given the prevalence It and the choices nit and cit = wnit
of the infected. Rational expectations would require that the agent use (7). I will be more
flexible and only require that the agent models sees the mapping from n to the probability
of not being infected as an affine function at −bt n in which at and bt depend on the current
and lagged epidemic variables. This assumption nests rational expectations as we shall
see in the next section.
To simplify I will assume that agents maximize
max
nt ,ct
s.t.
(at − bt nt )W +
c1−σ
n1+ϵ
t
−v t
1−σ
1+ϵ
(8)
ct = wnt
where W is the continuation value, or value of life and w is the (exogenous) wage. The
first-order condition
w1−σ = bt W nσt + vnϵ+σ
≡ F (nt , bt )
t
(9)
maps bt , the marginal riskiness of working, into the choice of work nt . The function F (·, b)
is increasing from F (0, b) = 0 and has a unique solution n(b), interior for low enough w.
1
1−σ
Denote n∗ = v − ϵ+σ w ϵ+σ the solution to F (n∗ , 0) = 0, and denote c∗ = wn∗ the corresponding choice of consumption. The allocation (c∗ , n∗ ) represents the economy before
the epidemic, at t = 0.
1
Eichenbaum, Rebelo, and Trabandt (2021) assume σ = ϵ = 1, in which case n∗ = v − 2
and the agent’s choice of hours worked is given by
q
n
1 + αt2 − αt
=
n∗
1
αt =
W n∗ bt .
2
(10a)
(10b)
Assuming that the infected not out of the labor force behave like the susceptibles, the
factor λt in (6) and (1) becomes
λt
= π1 cst cit ϕ + π2 nst nit ϕ + π3
n 2
= (π1 w2 + π2 )ϕn∗ 2
+ π3
n∗
n 2
= (λ0 − π3 ) ∗ + π3
n
with λ0 = (π1 w2 + π2 )ϕn∗ 2 + π3 . If we further assume π3 = 0 this reduces to
n 2
λt
t
=
.
λ0
n∗
The force of infection is the square of agents’s choices of economic activity.
Aggregate economic activity yt is the sum of individual activities. The recovered
choose n∗ , while the susceptible, exposed, and asymptomatic infected choose nt . The
11
joint behavior of the epidemic and economic activity (relative to the pre-epidemic level
y ∗ = n∗ ) is described by
yt
y∗
λt
λ0
nt
n∗
αt
= Rt + (St + Et + ϕIt )
=
n
t
(11a)
n∗
n 2
t
(11b)
n∗
p
=
1 + αt 2 − αt
=
(11c)
1
W n∗ bt .
2
(11d)
At this point I haven’t yet specified the form taken by bt , which is the marginal reduction in infection risk from reducing economic activity.
Rational expectations equilibrium
Under rational expectations the probability perceived by the agent must equal the true
probability (7) when nit = nt satisfies (10a):
p
bt = ϕ(π1 w2 + π2 )nit It = n∗ ϕ(π1 w2 + π2 )( 1 + αt 2 − αt )It .
Substituting bt =
2
W n∗ αt
from (10b) leads to
αt
n
n∗
λt
λ0
=
=
=
κI
1
√ t
2 1 + κIt
1
√
1 + κIt
1
.
1 + κIt
(12a)
(12b)
(12c)
with κ = W λ0 .
Note that (12c coincides with the Hill function assumed by epidemiologists, with
f (t) = It and the κ parameter has a structural interpretation as the value of life.
Under rational expectations the dynamics of an epidemic with and without behavioral
feedback are easy to compare in a phase diagram, shown in Figure 4.9 The process starts
at S = 1, when the whole population is susceptible. Since S cannot increase the path can
be read from right to left as a time series.
When there is no behavioral channel the vertical line at S ∗ =
ν
λ0
= 1/R0 (herd im-
munity) separates two regions with I increasing to the right and decreasing to the left.
The epidemic starts at S = 0, peaks at S = S ∗ , and ends at S∞ . The best outcome would
be a path that ends at (S ∗ , 0), the herd immunity level. At that point, by (2) and (3)
Ė + I˙ < 0 even if λt = λ0 , in the absence of any behavioral reaction. Without intervention
the epidemic overshoots herd immunity.
With the behavioral channel the red dotted line separates the two regions, hence the
epidemic peaks earlier (for a higher S), but there is only one wave. Hence behavioral
feedback, of itself, is not sufficient to generate multiple waves; nor does it change the herd
immunity level or the best outcome.
9 For
simplicity I consider here a simpler SIR model in which the compartment E is eliminated, and (2–3)
are replaced by I˙t = λSt It − νIt .
12
I = κ1 ( SS∗ − 1)
I
1
I = κ1 ( ŜS∗ − 1)
0
S∗ =
ν
λ0
Ŝ ∗ = z1 S ∗
1
S
Figure 4: Phase diagram with and without behavioral channel.
One way to generate waves is to introduce exogenous time variation in λ, that is, lockdowns. The dashed black lines outline the phase diagram when transmissibility has been
reduced by a factor z. If z is large enough (z > 1/S ∗ ) the vertical line is shifted to the right
of the current position (St , It ) and infections begin decreasing immediately upon imposition, no matter what the current value of I is. But if restrictions are removed before St has
fallen below S ∗ , infections rise again and produce a second wave (multiple lockdowns
can generate multiple waves).
In terms of optimal policy, a well-timed lockdown can bring path of the epidemic to
end at (S, I) = (S ∗ , 0) at which point the epidemic cannot restart. One such policy is to let
the epidemic progress and, as soon as S reaches S ∗ , impose an absolute lockdown (z = 0)
that halts new infections: I˙ immediately, S remains constant at S ∗ and I decreases to 0.
This eliminates the epidemic’s overshoot.
Ad hoc behavioral models
The assumption of rational expectations raises two questions: What information did agents
have in real time? Did behavior seem to respond to current infections only?
On the first question, contemporary newspapers suggest that there was a strong demand for information about the epidemic. The city’s Health Commissioner, William C.
Hassler, held briefings at least once a day on the state of the epidemic and relayed numbers of cases and deaths. But the numbers were problematic. Determining whether a
death was due to influenza or opportunistic pneumonia was not obvious. Influenza had
not been made a reportable disease until the beginning of the epidemic (September 27),
13
so physicians were not in the habit of reporting it. In the midst of an epidemic, many were
too busy to provide information in a timely manner, and sometimes filed their reports very
late and by mail. It was often unclear if cases were dated by the date on which the report
was sent or on which it was recorded by the city, and Hassler himself was increasingly
unclear on this point.
Such as they are, the numbers of cases and deaths were reported in newspapers and
may have been used by people as a measure of the epidemic. I have collected these reports;
in addition, Chowell, Nishiura, and Bettencourt (2006) provide a shorter but apparently
official series of cases.10
The top panel of Figure 5 shows that reported cases do not line up well with the infections that can be inferred from excess deaths and the known delay between infection and
deaths. In addition, the difference in scale suggests that only about 20% of actual cases
were reported. The bottom panel shows that death reports were delayed by a few days
but otherwise close to the truth, and that reported cases match the time pattern (but not
the scale) of deaths.
Thus, although rational expectations would impose that bt is a function of current
infections It only, I allow bt to depend on deaths, possibly with lags or cumulation.
The model I estimate consists of (11a–11b), rewritten as
yt
y∗
λt
λ0
p
= Rt + (St + Et + ϕIt ) K(t)
(13a)
= K(t)
(13b)
and several variants, namely:
1
1+κf
(t)
(1 − f (t))κ
K(t) =
e−κf (t)
1 + 2α 2 − 2α √1 + α 2 , α = 1 κf (t)
t
t
t
t
2
(Hill)
(Power)
(Exp)
(model)
and
Z
f (t) =
Tm
(default)
Xt−s ds
0
(Alt)
f˙(t) = Xt − τ f (t)
and Xt = Mt (default) or Xt = It (Inf).11
The different functional forms K(t) as functions of f (t) coincide to the first order:
Hill :1 − κx + κ2 x2 − κ3 x3 + O(x4 )
1
1
Power :1 − κx + κ(κ − 1)x2 − κ(κ − 1)(κ − 2)x3 + O(x4 )
2
6
1 2 2 1 3 3
Exp :1 − κx + κ x − κ x + O(x4 )
2
6
1 2 2 1 3 3
model :1 − κx + κ x − κ x + O(x5 )
2
8
10 They
cite a Japanese publication from the 1920s, whose ultimate source is unclear.
are normalized per 105 and infections per 108 .
11 Deaths
14
2500
12000
reported cases (newspapers)
reported cases (Chowell e.a. 2006)
inferred infections
10000
2000
8000
6000
infections
cases
1500
1000
4000
500
0
2000
Oct 1918
Nov 1918
Dec 1918
0
Feb 1919
Jan 1919
2500
120
reported cases (newspapers)
reported cases (Chowell e.a. 2006)
excess deaths
reported deaths
100
2000
80
60
deaths
cases
1500
1000
40
500
0
Sep 1918
20
Oct 1918
Nov 1918
Dec 1918
Jan 1919
Feb 1919
Mar 1919
Apr 1919
0
May 1919
Figure 5: Reported cases from newspapers and Chowell, Nishiura, and Bettencourt (2006)
(left axis) compared to inferred infections from Figure 1 in the top figure, to reported
deaths from newspapers and excess deaths from Figure 1 in the bottom figure.
15
Note that a Hill function with f (t) =
R Tm
0
I(t − s)ds nests the rational expectation
model for small Tm = 1 (τ = 1 for the Alt model).
Introducing a lockdown
There are various ways one can introduce a lockdown, or restrictions on economic activity.
Bootsma and Ferguson (2007) assume that the effect is multiplicative, that is, lockdowns
act as a tax (Eichenbaum, Rebelo, and Trabandt 2021). Alternatively, they may act as an
upper bound on n. With a lockdown of intensity 0 ≤ pct ≤ 1, (10a) becomes either of
p
(1 − pct )( 1 + α2 − αt )
(default)
nt
t
=
p
n∗
min{1 − pct , 1 + αt2 − αt } (Min)
and K(t) is replaced in (13) by either of
(1 − pct )2 K(t)
K̃(t) =
min{(1 − pct )2 , K(t)}.
A suite of models
I will consider a suite of models. The naming convention will indicate the choices for:
1. the argument of the behavioral function bt : deaths (the default) or infections (Inf)
2. the functional form for memory: default or modified (Alt)
3. the functional form for λt : Hill, Exp, Power, or that implied by (13) (model)
4. the way in which lockdowns interact with the dynamics: multiplicative (default) or
Min.
4
Estimation
The estimation procedure can be summarized as follows. Given parameter values (including the epidemic’s starting date) I solve the deterministic model and compute predicted
for deaths and activity.
I assume that the each day’s deaths follow a Poisson process whose parameter is the
model’s predicted value for that day, while measured economic activity follows a Gaussian process whose mean is the model’s predicted activity for that day and whose variance
is σ.12 The data consists of daily deaths D and economic activity y (proxied by the index
of ridership).
The log likelihood is proportional to
L(D, y, θ) ∝
X
log(D̂t (θ))Dt − D̂t (θ) −
t
12 This
yt − ŷt (θ) log(σ)
−
2σ
2
treats the noise term in activity as measurement error. An alternative would be to model the transmission rate as stochastic and let the daily fluctuations in ridership affect the epidemic.
16
model
(LL)
Hill
9211.3
HillAlt
9208.7
HillMin
9233.8
HillAltMin
9221.9
HillAltInf
9186.5
HillAltInfMin
9187.1
Power
9203.2
PowerAlt
9203.9
PowerAltMin
9226.1
Exp
9203.2
ExpAlt
9203.9
ExpAltMin
9226.1
model
9203.9
modelAlt
9204.6
modelMin
9235.3
modelAltMin
9225.9
µ
(%)
R0
κ
τ
(%)
1.22
[1.17,1.27]
1.17
[1.13,1.21]
1.22
[1.18,1.27]
1.15
[1.1,1.19]
1.10
[1.06,1.14]
1.08
[1.04,1.12]
1.22
[1.17,1.27]
1.18
[1.13,1.22]
1.15
[1.1,1.19]
1.22
[1.17,1.27]
1.18
[1.13,1.22]
1.15
[1.1,1.19]
1.22
[1.18,1.27]
1.18
[1.13,1.22]
1.22
[1.17,1.26]
1.15
[1.1,1.19]
2.25
[2.24,2.27]
2.42
[2.41,2.43]
2.39
[2.38,2.4]
2.66
[2.65,2.67]
2.86
[2.83,2.88]
2.99
[2.97,3.01]
2.20
[2.19,2.22]
2.36
[2.34,2.37]
2.65
[2.64,2.67]
2.20
[2.19,2.22]
2.36
[2.34,2.37]
2.65
[2.64,2.67]
2.20
[2.19,2.22]
2.36
[2.34,2.37]
2.39
[2.37,2.4]
2.65
[2.64,2.67]
320.1
[284,357]
638.5
[563,719]
369.3
[333,406]
660.8
[586,741]
213.3
[187,241]
223.9
[196,253]
259.7
[231,289]
515.0
[456,577]
533.7
[477,594]
259.8
[232,289]
515.3
[456,577]
534.0
[478,593]
261.3
[233,291]
519.2
[460,582]
312.0
[285,340]
537.9
[481,598]
1
[0.93,1.10]
0.76
[0.69,0.83]
0.4
[0.36,0.44]
0.39
[0.34,0.44]
1
[0.96,1.15]
0.73
[0.67,0.80]
1
[0.96,1.15]
0.73
[0.67,0.80]
1
[0.96,1.14]
0.73
[0.67,0.80]
Tm
pc
deaths
(%)
17
[16.2,18.3]
0.41
[0.39,0.43]
0.42
[0.40,0.43]
0.61
[0.61,0.62]
0.66
[0.65,0.67]
0.46
[0.44,0.49]
0.70
[0.69,0.71]
0.40
[0.38,0.42]
0.39
[0.37,0.42]
0.66
[0.65,0.66]
0.40
[0.38,0.42]
0.39
[0.37,0.42]
0.66
[0.65,0.66]
0.40
[0.38,0.42]
0.39
[0.37,0.42]
0.61
[0.60,0.62]
0.66
[0.65,0.66]
5
[3,7]
2
[-1,4]
9
[7,10]
7
[5,9]
-2
[-4,1]
-2
[-4,2]
1
[-2,4]
-4
[-6,-1]
3
[-0,5]
1
[-2,4]
-4
[-6,-1]
3
[-0,5]
1
[-1,4]
-4
[-6,-1]
5
[3,7]
3
[1,6]
22
[21.3,23.1]
16
[15.3,17.5]
16
[15.3,17.5]
17
[15.4,17.6]
21
[20.5,22.4]
Table 1: Median estimates without the ridership data (95% confidence intervals between
brackets).
I calibrate the mean latency period to 1/α = 1.5 days and the mean infectious period to
1/ν = 1.8 days as in Bootsma and Ferguson (2007). I set the fraction of asymptomatics
ϕ at 0.7. I estimate the rest, namely σ (the variance of ridership data) and θ = (µ,R0 ,κ,
Tm ,pc ,d0 ), where R0 = λ/ν and d0 is the first day of the epidemic. The mappings Dt (θ)
and yt (θ) is clearly nonlinear: for a given θ I simulate the epidemic with a seed infection
of 1 at d0 and a time scale of 0.1 day.13 I use MCMC methods with a flat prior to estimate
the vector of parameters.14
Tables 1 and 2 report the results with and without the ridership data.
There is little difference between Tables 1 to 2: the additional ridership data does not
change the parameter estimates. This may be disappointing (the ridership data contain no
information) or comforting (the model’s behavioral channel is validated by observations
on mobility). I tend to the latter view. Validating the behavioral channel postulated by
epidemiologists with actual mobility data has important consequences for optimal policy
as we shall see.
The models that use infections rather than deaths as input to the behavioral function
do poorly. In addition, rational expectations (which imposes a dependence on current
infections only, hence τ close to 1) is rejected.
The models that assume that lockdowns are an upper bound (Min) rather than a tax
do systematically better. The functional forms Power, Exp, and Hill do reasonably well,
but the form model does at least as well, and it has the additional advantage that it maps
into an economic interpretation.
13 The seed makes some difference for the parameter estimates, notably R and the starting date which is
0
later for a larger seed. A time unit smaller than 0.1 does not alter the estimates.
14 The chains are 500,000 long with a 20% burn-in.
17
model
(LL)
Hill
9581.6
HillAlt
9580.2
HillMin
9603.9
HillAltMin
9585.8
HillAltInf
9552.4
HillAltInfMin
9548.4
Power
9572.7
PowerAlt
9575.8
PowerAltMin
9591.4
Exp
9572.8
ExpAlt
9575.8
ExpAltMin
9591.4
model
9573.6
modelAlt
9576.5
modelMin
9606.5
modelAltMin
9591.1
µ
(%)
R0
κ
τ
(%)
1.21
[1.17,1.26]
1.16
[1.12,1.21]
1.21
[1.17,1.26]
1.15
[1.1,1.19]
1.09
[1.05,1.13]
1.09
[1.05,1.12]
1.21
[1.17,1.26]
1.17
[1.13,1.21]
1.15
[1.11,1.19]
1.21
[1.17,1.26]
1.17
[1.13,1.21]
1.15
[1.11,1.19]
1.21
[1.17,1.26]
1.17
[1.13,1.21]
1.21
[1.16,1.25]
1.15
[1.11,1.19]
2.26
[2.24,2.27]
2.42
[2.41,2.44]
2.39
[2.38,2.4]
2.59
[2.58,2.61]
2.78
[2.76,2.8]
2.81
[2.79,2.83]
2.21
[2.19,2.22]
2.36
[2.34,2.37]
2.59
[2.57,2.6]
2.21
[2.19,2.22]
2.36
[2.34,2.37]
2.59
[2.57,2.6]
2.21
[2.19,2.22]
2.36
[2.35,2.37]
2.39
[2.38,2.4]
2.59
[2.57,2.6]
312.4
[277,349]
635.0
[558,716]
355.3
[320,392]
630.7
[556,710]
202.7
[177,229]
208.0
[179,238]
253.9
[226,283]
513.6
[454,577]
522.3
[465,582]
253.9
[225,284]
512.9
[453,577]
521.8
[465,582]
256.3
[228,286]
517.9
[457,583]
302.7
[276,331]
524.9
[468,586]
1
[0.96,1.13]
0.81
[0.74,0.88]
0.44
[0.39,0.48]
0.45
[0.39,0.50]
1.1
[0.98,1.18]
0.78
[0.72,0.85]
1.1
[0.98,1.18]
0.78
[0.72,0.85]
1.1
[0.98,1.17]
0.78
[0.72,0.85]
Tm
pc
deaths
(%)
17
[16.0,18.1]
0.41
[0.39,0.43]
0.41
[0.39,0.43]
0.61
[0.60,0.62]
0.65
[0.64,0.66]
0.45
[0.43,0.47]
0.67
[0.67,0.68]
0.40
[0.38,0.42]
0.39
[0.37,0.41]
0.64
[0.64,0.65]
0.40
[0.38,0.42]
0.39
[0.37,0.41]
0.64
[0.64,0.65]
0.40
[0.38,0.42]
0.39
[0.37,0.41]
0.60
[0.60,0.61]
0.64
[0.64,0.65]
6
[4,8]
2
[-1,4]
9
[8,10]
6
[4,8]
-2
[-4,1]
-3
[-5,0]
1
[-1,4]
-4
[-6,-1]
1
[-2,4]
1
[-1,4]
-4
[-6,-1]
1
[-2,4]
2
[-1,4]
-3
[-6,-1]
6
[4,7]
2
[-1,5]
22
[21.1,22.9]
16
[15.1,17.3]
16
[15.1,17.3]
16
[15.2,17.4]
21
[20.4,22.3]
Table 2: Median estimates with the ridership data (95% confidence intervals between
brackets).
The best model is (modelMin), both without and with ridership data. The values of µ
and R0 are fairly consistent across models, and with the epidemiological literature (Mills,
Robins, and Lipsitch 2004; Biggerstaff et al. 2014). The magnitudes of κ can be gauged as
follows. The level of ft (in deaths per 100,000) that halves the transmission rate is N/κ
√
for Hill, N/ 2κ for model, N (1 − 21/κ ) for Power, N log(2)/κ for Exp (the population of
San Francisco can be estimated at 494,350 in September 1918 Department of Commerce,
Bureau of the Census 1923). For the estimates of κ these values are all in the range of 1300–
RT
1800. By comparison the peak of 0 m D(t − s)ds (for Tm = 20) was 1680 on November 7.
This suggests a fairly strong, but (given the value of Tm ) sluggish response. The values
of pc indicate a substantial reduction in activity (in the Min model, the upper bound is
√
1 − pc ∼ 60%).
A final remark on the results: the last column of both tables gives the reduction in
deaths, compared to a counterfactual of no intervention. In terms of magnitudes, a reduction of 6% corresponds to about 200 lives saved during the six months of the epidemic.
The numbers are relatively small and even negative for some models, for reasons that will
become apparent in the next section.
Figures 6 and 7 compare the simulations of the best-fitting model with the data for
deaths and ridership (Figures 12 to 13 in the Appendix show the same graphs for all
models).
In Figure 6 the model’s fit can be compared with the data. The counterfactual assumes
no restrictions and results in a single wave that peaks at 230 deaths per day instead of 110.
Figure 7 plots predicted ridership, and also the ridership at each point in time that would
result if restrictions were lifted. Comparing the two lines shows when the restrictions
18
7=0.012, R0=1.7, 5=302.7, T m=21.4, p c =0.60
250
actual
predicted
counterfactual
200
excess deaths
150
100
50
0
Oct 1918
Nov 1918
Dec 1918
Jan 1919
Feb 1919
Mar 1919
LL = 9606.5, start date: 16-Sep-1918 (modelMin, mean)
reduction in deaths: 6%
Apr 1919
Figure 6: Deaths, actual (estimated excess deaths), fitted, and counterfactual (with no
lockdown), best model. The yellow area indicates the period when places of amusement
were closed.
were binding: most of the time, but not always. The restrictions were mostly binding in
the early part of the restriction period (the yellow area in the Figure), when cumulated
deaths hadn’t risen enough to prompt an equal-sized reduction in activity through the
behavioral channel.
5
Optimal policy
The previous section showed that the reduction in deaths was small. The phase diagram,
constructed with the estimated parameters and shown in Figure 8 shows why.15
The red line shows the path traced by the counterfactual without lockdown. The blue
line shows the simulated path with lockdown. The light grey lines are not properly speaking tracing the phase diagram, which is difficult to do with the lags incorporated in the
behavioral function. Instead they trace paths without lockdowns and starting from values
of S < 1. The dark blue line represents the points along those paths where Ė + I˙ = 0,
the equivalent of the vertical line at 1/R0 in Figure 4. Finally, the outer green line plots a
counterfactual with no behavioral channel.
If one knew the parameters but ignored the behavioral channel, one would take the
green line to represent the result of no intervention: the overshoot is massive and interven15 The simulations done for this section were computed for 1200 days, to make sure that the epidemic is not
merely postponed beyond the simulation horizon.
19
1.4
ridership
predicted ridership
predicted ridership without pc
1.3
1.2
index of ridership
1.1
1
0.9
0.8
0.7
0.6
Oct 1918
Nov 1918
Dec 1918
Jan 1919
modelMin (mean)
Feb 1919
Mar 1919
Figure 7: Economic activity (as proxied by ridership), actual, predicted, and predicted if
restrictions were lifted. The yellow area indicates the period when places of amusement
were closed.
tion could reduce deaths considerably. In reality there is little margin between the herd
immunity threshold and the final outcome without intervention. The behavioral channel
is to some degree a substitute for intervention.
Note that an ill-timed intervention can do worse than no intervention at all, which is
not possible in the model without lags in the behavioral function. The reason is that there
are various paths leading to the same point in the (S, I) plane. A point on the counterfactual (no-lockdown) path corresponds to a particular prior history of deaths, and hence
current value of M (t). An intervention can arrive at the same point but with a lower value
of M (t), leading to less precautionary behavior and higher transmission rates going forward. This is the reason why some models estimate negative death reductions in Tables 1
and 2.
To simplify the optimal policy problem I assume that authorities can institute only one
lockdown of constant intensity. There are thus three control variables: the start date, the
end date, and the intensity of the intervention.16 By varying the control variables one can
construct the Pareto frontier tracing the trade-off between deaths and economic activity.
Figure 9 plots that frontier, with economic activity measured in annualized consumption equivalent relative to no pandemic. The no-lockdown point represents the maximum
16 In the computation the start date is constrained between 0, day of the first case, and 50. The actual intervention ran from 32 to 61. The duration is bounded above by 180. I let simulations run for 1200 days to make
sure that the epidemic does not restart: for that purpose, an infection of 1e-3 individual is the same as 0. The
longer simulation length means that some results will be slightly different from those presented in the previous
section, where simulations lasted seven months.
20
Figure 8: Phase diagram for the best-fitting model.
of deaths and minimum reduction in economic activity. The actual policy followed appears to be clearly interior to the frontier. Authorities in San Francisco (had they known
the model) could have done better on both dimensions. The very steep slopes of the
frontier indicate that, starting from no lockdown, important gains could be made at little
economic cost up to a point, after which further gains would abruptly become costly.
Arguably it might have been difficult for authorities to control the intensity of the lockdown, but the start and end dates were clear choices. The inner line in Figure 9 constrains
the intensity to be its estimated (actual) value. The results are the same.
The actual shape of the Pareto frontier effectively obviates the need to choose terms
for the tradeoff. The upper-right point is nearly a kink, suggesting that for a wide range
of relative weights on deaths and economic activity, the answer would be the same.
Underlying the Pareto frontier are three choices for the intervention: start date, end
date, and intensity. Figure 9 shows the values of these choices, with intensity on the left
axis, start and end date on the right axis. The actual values of the control variables are
also shown. Going from left to right traces the Pareto frontier from no-lockdown. The
steeply vertical part of the Pareto frontier corresponds the left part of the graph, with
short but early lockdowns of increasing but moderate intensity. The kink of the Pareto
frontier is passed when the lockdown suddenly becomes much longer. Past that point
further reductions in deaths require an increasingly stringent lockdown.
Along the Pareto frontier the relation between controls (length, intensity, earliness of
intervention) and death outcomes is monotonic, but very nonlinear. Along the whole
frontier earliness does not change much; length is nearly constant up to the kink and
then after. Only the relation between intensity and reduction in deaths seems roughly
21
680
unconstrained
constrained
690
total deaths / 100,000
700
710
720
actual
730
740
750
-1.23%
no lockdown
-1.20%
-1.17%
-1.14%
-1.12%
-1.09%
-1.06%
annual consumption equivalent (relative to no pandemic)
-1.04%
-1.01%
Figure 9: Pareto frontier between economic activity and deaths. The lockdown intensity
is fixed at its actual value in the constrained case.
linear. Away from the frontier (as the actual policy is) there is no telling how controls and
outcomes would relate—as noted earlier, interventions can actually worsen outcomes.
The next figure compares the dynamics of the epidemic under a few policies of interest.
Figure 11 plots four paths in the (S, E + I) space. One is the actual (or fitted) path, the
three others correspond to the end points and the kink of the frontier: maximum activity
(no lockdown), the kink which I label Pareto optimum, and minimum deaths on the far
left of the Pareto curve.
The maximum activity (no intervention) obviously does the worst in terms of deaths,
ending at the lowest value of S. The Pareto optimum corresponds to an early but very brief
intervention that bends the curve slightly at its start and lets the behavioral response take
over for the rest of the epidemic. The death-minimizing intervention lets infections rise as
high as the no-lockdown, but then imposes a strong and long intervention to bring down
infections definitively. The actual policy starts earlier but is milder and shorter. Once it
ends a second wave rises again, moderated by the behavioral channel, but not enough
to prevent a trailing end of deaths that extends beyond what the death-minimizing path
achieves.
Compared with the (fitted) actual policy, an earlier, shorter, and milder closing (what
I call the Pareto optimum), would have reduced incidence by 1.7%, or 8,400 cases, and
deaths (using the estimated rate of 1.2%) by 100. Compared to no lockdowns, the actual
policy avoided 10,900 cases and 130 deaths. The most stringent policy would have saved
an additional 270 lives.17 This is another way of stating that, while mitigation policies
17 This
number is slightly smaller than the 200 deaths avoided in the previous section, because the horizon
22
0.4
220
actual intensity
180
intensity of lockdown (1-p c )1/2
0.3
160
0.25
140
0.2
120
100
0.15
80
0.1
actual end
60
0.05
start and end of lockdown (relative to date of first case)
200
0.35
40
actual start
0
750
740
730
720
710
700
total deaths / 100,000
690
680
670
20
660
Figure 10: Policies that implement the Pareto frontier, indexed by total deaths. The left axis
plots the intensity of lockdown, the right axis plots the start and end dates of lockdown.
could save lives, the room for action was limited.
6
The effect of masks
The final exercise consists in estimating the effectiveness of masks. Recall that a mask
mandate was in effect from January 17 to February 1, but was not accompanied by any
other restrictions. The assumption is that masks did not affect mobility during the second
wave. I allow them to affect the transmission rate λt , with an intensity to be estimated, but
not economic activity. Table 3 presents the results. The last two columns show the overall
reduction in deaths compared to no intervention, and the reduction from the January
mask mandate alone compared to no mask mandate (the denominator is therefore deaths
from January 1919 onwards).
The parameters pc and pc2 measure the force of the first (masks and closings in November) and second (masks in January) interventions. With one additional parameters, the
models do at least as well as in Table 2. The ordering changes, so that the best fit is now
achieved by HillAlt (Hill function with exponential discounting of deaths). Estimated
death rates (µ) are higher and basic reproduction numbers (R0 ) are lower. Most models
(including the best fitting) see little difference between pc and pc2 , although the estimates
are imprecise. If so, this suggests that masks alone would have done as well in November, without economic costs. However the best model of the previous section (modelMin)
of the simulation is 1200 days in this section; as incidence drags on at a low level for a long period of time in the
model, the differences between policies narrow.
23
0.18
maximum activity
actual
Pareto optimum
mininum deaths
0.16
0.14
0.12
E+I
0.1
0.08
0.06
0.04
0.02
0
S 1 = 0.377 (max act)
0.5
0.6
S
0.399 (actual)
0.416 (Pareto opt)
0.444 (min deaths)
0.7
0.8
0.9
1
Figure 11: Paths for selected policies along the Pareto frontier.
rejects any role for masks, while another (HillMin) finds an effect of masks but the additional parameter does not increase the log-likelihood significantly.
Hassler claimed that wearing masks in early December when he first requested it
would have saved two or three hundred lives (Chronicle, 26 January 1919, p. 9). Conversely F. Holmes Smith, Health Officer of San Mateo County, expressed doubts about
the value of reimposing a mask mandate in January 1919. He pointed to the fact that in
his county, where masks had not been required, the same phenomenal decline in new
cases has been reported as in San Francisco: “I think the decline of the epidemic can be
explained by the statement that masks were required at the time when the epidemic had
reached its peak, and under normal conditions would have declined anyway” (Chronicle,
27 January 1919, p. 3). In the model Smith is partly correct: the second wave had peaked,
but the mask mandate still slowed the epidemic and reduced overshooting. However Hassler’s claim is not supported by the model: a counterfactual with a mask mandate starting
on December 5 saves only about thirty lives rather than three hundred.
Interestingly there was interest in evaluating the effectiveness of masks, using data
(but without the benefit of a model). Bases on the similarities in curves across cities that
required or didn’t require masks, the State Board of Health concluded that gauze masks,
while sound in principle, were useless because of their defective quality and the difficulty
of requiring their use in close quarters (Kellogg 1919b). My results suggest less skepticism.
24
model
(LL)
Hill
9608.2
HillAlt
9612.8
HillMin
9605.1
HillAltMin
9601.4
HillAltInf
9594.7
HillAltInfMin
9584.1
Power
9604.0
PowerAlt
9611.5
PowerAltMin
9604.4
Exp
9604.0
ExpAlt
9611.5
ExpAltMin
9604.4
model
9604.6
modelAlt
9611.9
modelMin
9606.7
modelAltMin
9604.3
µ
(%)
R0
κ
1.35
[1.29,1.41]
1.32
[1.27,1.38]
1.25
[1.2,1.3]
1.23
[1.18,1.28]
1.23
[1.18,1.28]
1.20
[1.15,1.25]
1.36
[1.3,1.42]
1.33
[1.28,1.39]
1.25
[1.2,1.3]
1.36
[1.3,1.42]
1.33
[1.28,1.39]
1.25
[1.2,1.3]
1.36
[1.31,1.42]
1.34
[1.28,1.39]
1.23
[1.18,1.27]
1.25
[1.2,1.3]
2.13
[2.12,2.14]
2.22
[2.21,2.23]
2.34
[2.33,2.35]
2.46
[2.44,2.47]
2.54
[2.52,2.56]
2.64
[2.62,2.66]
2.09
[2.07,2.1]
2.17
[2.15,2.18]
2.40
[2.38,2.41]
2.09
[2.07,2.1]
2.17
[2.16,2.18]
2.40
[2.38,2.41]
2.09
[2.08,2.1]
2.17
[2.16,2.18]
2.33
[2.32,2.35]
2.40
[2.38,2.41]
339.0
[298,382]
655.3
[571,745]
353.7
[313,395]
640.6
[560,726]
219.8
[191,249]
229.9
[198,263]
290.3
[256,326]
559.7
[491,633]
526.0
[463,595]
290.5
[256,328]
559.8
[491,633]
526.4
[464,595]
292.3
[258,329]
562.8
[494,636]
293.0
[267,320]
528.3
[465,594]
τ
(%)
1.1
[1.01,1.19]
0.84
[0.76,0.92]
0.43
[0.39,0.47]
0.44
[0.38,0.49]
1.2
[1.06,1.26]
0.84
[0.77,0.92]
1.2
[1.07,1.26]
0.84
[0.77,0.92]
1.2
[1.06,1.25]
0.84
[0.77,0.92]
Tm
pc
pc2
16
[15.0,17.0]
0.36
[0.33,0.38]
0.35
[0.33,0.38]
0.60
[0.59,0.61]
0.63
[0.62,0.64]
0.41
[0.38,0.43]
0.65
[0.65,0.66]
0.33
[0.30,0.36]
0.32
[0.28,0.35]
0.62
[0.61,0.63]
0.33
[0.30,0.36]
0.32
[0.28,0.35]
0.62
[0.61,0.63]
0.33
[0.30,0.36]
0.32
[0.29,0.35]
0.60
[0.59,0.61]
0.62
[0.61,0.63]
0.28
[0.22,0.35]
0.31
[0.25,0.38]
0.37
[0.01,0.44]
0.44
[0.38,0.50]
0.33
[0.27,0.40]
0.48
[0.42,0.54]
0.29
[0.23,0.35]
0.31
[0.25,0.37]
0.44
[0.38,0.49]
0.29
[0.23,0.35]
0.31
[0.25,0.37]
0.44
[0.38,0.49]
0.29
[0.23,0.35]
0.31
[0.25,0.37]
0.00
[0.00,0.04]
0.44
[0.38,0.49]
22
[20.5,22.6]
15
[13.8,15.9]
15
[13.8,15.8]
15
[13.9,15.9]
22
[20.6,22.5]
deaths
(%)
4
[2,6]
2
[1,3]
8
[7,10]
5
[2,7]
1
[-0,3]
0
[-1,3]
-1
[-2,1]
2
[-2,6]
-2
[-4,1]
-1
[-2,2]
2
[-2,5]
-2
[-4,1]
-0
[-1,2]
2
[-2,5]
5
[3,7]
-1
[-3,2]
32
[25,38]
33
[27,39]
12
[0,21]
24
[18,30]
33
[28,39]
31
[26,37]
34
[27,40]
34
[28,40]
25
[18,32]
34
[27,40]
34
[28,40]
25
[19,32]
34
[28,40]
34
[28,40]
0
[0,0]
25
[19,32]
Table 3: Median estimates with the ridership data and allowing for masks in January.
7
Conclusion
Even before the second wave hit, Hassler claimed that masks and other measures taken
had avoided 20,000 cases and saved 1,500 lives (Chronicle, 22 Nov, p. 4). Was he anywhere
near the truth?
Policy evaluation requires counterfactuals, which in turn require a model. The one I
use is structural. Daily observations of ridership on the San Francisco Muni, interpreted
through the lens of a simple model of activity under risk of infection, confirms that a standard SEIR model captures the dynamics of the 1918 Spanish influenza epidemic. Substantial behavioral response incorporating long lags is needed, and this has implications for
optimal policy. To a large degree the behavioral response substitutes for policy intervention, so that the room for actual improvement (in terms of deaths) is much narrower than
in a model that ignores the response. My estimate of the effect of the intervention in San
Francisco on deaths is an order of magnitude lower than Hassler’s.
The Pareto frontier between economic activity and deaths is steep, implying limited
trade-offs. Actual policy was away from the frontier, and the optimal policy would have
achieved a better outcome with a shorter intervention. Reducing deaths further was feasible, but at the cost of a six-month long intervention that seems unrealistic given the
pressures I documented in the historical section. Masks may well have been effective.
The fact that San Francisco was away from the frontier suggests that other cities may
have been. The dynamics of the SEIR model with behavioral feedback indicate that interventions can be counterproductive away from the frontier. Even along the frontier the
relation between intervention and outcome is non linear. This suggests caution when
using reduced form, linear models that rely on cross-sectional variation to evaluate the
effectiveness or long-term impact of interventions.
25
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27
Appendix
28
0
Nov 1918
Jan 1919
0
Mar 1919
actual
predicted
counterfactual
150
100
Nov 1918
Jan 1919
50
Jan 1919
250
actual
predicted
counterfactual
100
Nov 1918
Jan 1919
7=0.011, R0=2.6, 5=522.3, ==0.01, pc =0.64
actual
predicted
counterfactual
100
50
0
250
Nov 1918
Jan 1919
250
actual
predicted
counterfactual
150
100
50
Nov 1918
Jan 1919
7=0.012, R0=2.4, 5=517.9, ==0.01, pc =0.39
Nov 1918
Jan 1919
Mar 1919
actual
predicted
counterfactual
200
excess deaths
Jan 1919
100
Mar 1919
Nov 1918
Jan 1919
Mar 1919
7=0.012, R0=2.4, 5=513.6, ==0.01, pc =0.39
actual
predicted
counterfactual
150
100
0
Nov 1918
Jan 1919
Mar 1919
250
7=0.011, R0=2.6, 5=521.8, ==0.01, pc =0.64
actual
predicted
counterfactual
200
150
100
50
Nov 1918
Jan 1919
Mar 1919
0
250
actual
predicted
counterfactual
100
Nov 1918
Jan 1919
Mar 1919
LL = 9591.4, start date: 19-Sep-1918 (ExpAltMin, median)
reduction in deaths: 1%
7=0.012, R0=2.4, 5=302.7, Tm =21.4, pc =0.60
150
0
Mar 1919
LL = 9575.8, start date: 16-Sep-1918 (PowerAlt, median)
reduction in deaths: -4%
actual
predicted
counterfactual
50
LL = 9576.5, start date: 16-Sep-1918 (modelAlt, median)
reduction in deaths: -3%
Jan 1919
200
7=0.012, R0=2.4, 5=512.9, ==0.01, pc =0.39
200
150
0
Nov 1918
100
250
Nov 1918
50
LL = 9575.8, start date: 16-Sep-1918 (ExpAlt, median)
reduction in deaths: -4%
50
LL = 9573.6, start date: 13-Sep-1918 (model, median)
reduction in deaths: 2%
250
actual
predicted
counterfactual
150
0
Mar 1919
100
0
50
LL = 9572.8, start date: 13-Sep-1918 (Exp, median)
reduction in deaths: 1%
excess deaths
excess deaths
100
150
LL = 9585.8, start date: 19-Sep-1918 (HillAltMin, median)
reduction in deaths: 6%
7=0.012, R0=2.2, 5=253.9, Tm =16.2, pc =0.40
200
150
0
Mar 1919
7=0.012, R0=2.2, 5=256.3, Tm =16.3, pc =0.40
200
0
actual
predicted
counterfactual
50
LL = 9591.4, start date: 19-Sep-1918 (PowerAltMin, median)
reduction in deaths: 1%
Mar 1919
100
250
excess deaths
150
7=0.012, R0=2.2, 5=253.9, Tm =16.2, pc =0.40
200
excess deaths
200
excess deaths
250
Jan 1919
LL = 9572.7, start date: 13-Sep-1918 (Power, median)
reduction in deaths: 1%
excess deaths
250
Nov 1918
150
0
Mar 1919
actual
predicted
counterfactual
200
50
LL = 9552.4, start date: 21-Sep-1918 (HillAltInf, median) LL = 9548.4, start date: 21-Sep-1918 (HillAltInfMin, median)
reduction in deaths: -2%
reduction in deaths: -3%
7=0.011, R0=2.6, 5=630.7, ==0.01, pc =0.65
50
200
150
0
Mar 1919
100
LL = 9603.9, start date: 16-Sep-1918 (HillMin, median)
reduction in deaths: 9%
50
Nov 1918
150
0
Mar 1919
7=0.011, R0=2.8, 5=208.0, ==0.00, pc =0.67
200
excess deaths
excess deaths
250
actual
predicted
counterfactual
50
LL = 9580.2, start date: 17-Sep-1918 (HillAlt, median)
reduction in deaths: 2%
7=0.011, R0=2.8, 5=202.7, ==0.00, pc =0.45
200
0
100
50
LL = 9581.6, start date: 14-Sep-1918 (Hill, median)
reduction in deaths: 6%
250
150
250
excess deaths
50
7=0.012, R0=2.4, 5=355.3, Tm =22.0, pc =0.61
200
excess deaths
100
250
actual
predicted
counterfactual
200
excess deaths
150
7=0.012, R0=2.4, 5=635.0, ==0.01, pc =0.41
excess deaths
actual
predicted
counterfactual
200
excess deaths
250
7=0.011, R0=2.6, 5=524.9, ==0.01, pc =0.64
actual
predicted
counterfactual
200
excess deaths
7=0.012, R0=2.3, 5=312.4, Tm =17.0, pc =0.41
excess deaths
250
150
100
50
Nov 1918
Jan 1919
Mar 1919
0
Nov 1918
Jan 1919
Mar 1919
LL = 9606.5, start date: 16-Sep-1918 (modelMin, median) LL = 9591.1, start date: 19-Sep-1918 (modelAltMin, median)
reduction in deaths: 6%
reduction in deaths: 2%
Figure 12: Deaths, actual (estimated excess deaths), fitted, and counterfactual (with no
lockdown).
29
1.4
1.4
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
1.2
1.2
1.1
1.1
1.1
1.1
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
Nov 1918
Jan 1919
Hill (median)
Mar 1919
Nov 1918
Jan 1919
HillAlt (median)
Mar 1919
Nov 1918
Jan 1919
HillMin (median)
Mar 1919
0.6
1.4
1.4
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
1.2
1.2
1.1
1.1
1.1
1.1
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
Nov 1918
Jan 1919
HillAltInf (median)
Mar 1919
0.6
Nov 1918
Jan 1919
0.6
Mar 1919
HillAltInfMin (median)
Nov 1918
Jan 1919
Power (median)
Mar 1919
0.6
1.4
1.4
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
1.2
1.2
1.1
1.1
1.1
1.1
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
Nov 1918
Jan 1919
Mar 1919
0.6
PowerAltMin (median)
Nov 1918
Jan 1919
Exp (median)
0.6
Mar 1919
Nov 1918
Jan 1919
ExpAlt (median)
Mar 1919
0.6
1.4
1.4
1.4
1.4
1.3
1.3
1.3
1.3
1.2
1.2
1.2
1.2
1.1
1.1
1.1
1.1
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
Nov 1918
Jan 1919
model (median)
Mar 1919
0.6
Nov 1918
Jan 1919
modelAlt (median)
0.6
Mar 1919
Nov 1918
Jan 1919
modelMin (median)
Mar 1919
0.6
Figure 13: Economic activity (as proxied by ridership), actual and predicted.
30
Nov 1918
Nov 1918
Nov 1918
Nov 1918
Jan 1919
Mar 1919
Jan 1919
Mar 1919
Jan 1919
Mar 1919
Jan 1919
Mar 1919
HillAltMin (median)
PowerAlt (median)
ExpAltMin (median)
modelAltMin (median)
@FedFRASER