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Quarantine, Contact Tracing, and Testing:
Implications of an Augmented SEIR Model
WP 21-08
Andreas Hornstein
Federal Reserve Bank of Richmond
Quarantine, Contact Tracing, and Testing:
Implications of an Augmented SEIR Model ∗
Andreas Hornstein
Federal Reserve Bank of Richmond
andreas.hornstein@rich.frb.org
First Version March 25, 2020
This Version March 31, 2021
Abstract
I incorporate quarantine, contact tracing, and random testing in the basic SEIR
model of infectious disease diffusion. A version of the model that is calibrated to known
characteristics of the spread of COVID-19 is used to estimate the transmission rate of
COVID-19 in the United States in 2020. The transmission rate is then decomposed into
a part that reflects observable changes in employment and social contacts and a residual
component that reflects disease properties and all other factors that affect the spread of
the disease. I then construct counterfactuals for an alternative employment path that
avoids the sharp employment decline in the second quarter of 2020 but also results in
higher cumulative deaths due to a higher contact rate. For the simulations, a modest
permanent increase of quarantine effectiveness counteracts the increase in deaths, and
the introduction of contact tracing and random testing further reduces deaths, although
at a diminishing rate. Using a conservative assumption on the statistical value of life,
the value of improved health outcomes from the alternative policies far outweighs the
economic gains in terms of increased output and the potential fiscal costs of these
policies.
∗
I would like to thank Alex Wolman and Zhilan Feng for helpful comments and Elaine Wissuchek and Zach
Edwards for research assistance. Any opinions expressed are mine and do not reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System. Andreas Hornstein: andreas.hornstein@rich.frb.org
1
1
Introduction
From March to April 2020, after a number of states introduced stay-at-home orders in
response to the emerging COVID-19 pandemic, employment in the United States declined
precipitously, by about 15 percent. Upon relaxing the stay-at-home orders, employment
recovered to within 6 percent of pre-pandemic levels in the late summer of 2020. The
decline in work-related contacts and social contacts associated with these stay-at-home orders
arguably helped contain the pandemic and the number of disease-related deaths but came
at the cost of lost output. In this paper, I argue that a more targeted approach, such as
improved quarantine, contact tracing, and random testing, could have attained similar or
better disease-related outcomes for an alternative employment path that avoided the sharp
and deep decline. Furthermore, the benefits of this alternative approach, in terms of the
value of lives saved, far outweigh its potential costs.
Indiscriminate social distancing limits the spread of COVID-19 because it reduces contact
rates for all individuals, whether they are infectious or not. Quarantine is less disruptive
in that it only removes known infectious individuals. Clearly, infectious individuals who
display symptoms can be quarantined, but a feature of COVID-19 is that even individuals
who do not display symptoms can be infectious. Contact tracing can work backwards from
newly identified symptomatic individuals to identify individuals who may have been infected
but do not yet display any symptoms. Random testing of the population is another way to
identify individuals who are infectious but asymptomatic.
I study the effects of quarantine, contact tracing, and random testing in a modified
susceptible-exposed-infected-recovered (SEIR) model where I differentiate between asymptomatic and symptomatic infected individuals. The modified SEIR model is calibrated to
the known characteristics of COVID-19 and is then used to infer the transmission rate of
COVID-19 from data on daily deaths. Information on employment and social mobility indices is then used to separate out the impact of work contacts on the transmission rate. This
procedure suggests that the shift toward work at home—as documented in Bick, Blandin
and Mertens (2021)—had a larger impact on overall work contacts than the reduced employment in April and May 2020. I then assume an alternative employment path that avoids the
sharp decline and rebound of employment, thus avoiding the output losses associated with
the employment reductions.
The alternative employment path implies a higher overall transmission rate and more fatalities. We consider various combinations of improved quarantine measures, contact tracing,
and random testing to counteract the increased contact rate from the alternative employment
path. The simulations suggest that a moderate and permanent increase of the quarantine
2
rate much better contains cumulative deaths over the second half of the year, even though it
is initially not as effective as the steep but temporary actual employment reductions in the
first part of the year. In particular, in the simulations, a moderate permanent increase of the
quarantine rate prevents the surge of infections and deaths at the end of the year. Adding
contact tracing and random testing yields additional reductions of cumulative deaths but
not of the same magnitude as the initial increase of quarantine effectiveness.
We provide some rough estimates of the gains from avoided output losses and deaths and
the potential fiscal costs of the simulated alternative policies. For a conservative estimate
of the statistical value of a life ($7 million), the gains from improved health outcomes, in
the range of $3.5 trillion to 5 trillion, far outweigh gains from avoided output loss and the
potential fiscal costs of the program, about $450 billion and $60 billion, respectively.
One can have well-founded reservations on the use of the kind of model described here
for policy analysis, and Jewell, Lewnard and Jewell (2020) provide an extensive list of these
reservations. On the other hand, short of running actual “experiments” on an economy,
models like the one described here provide some guidance on possible outcomes for these
policy interventions.
1.1
Related work in epidemiology
We work with an augmented version of the standard SEIR model of disease diffusion with
Poisson arrival rates for health-state changes and implied exponential distributions for stage
duration. Most epidemiological work on quarantine and contact tracing models these interventions as setting aside a fraction of newly infected individuals and gradually moving them
to a quarantine state, similar to the transition between health states. The effectiveness of
these interventions is then determined by the share and speed parameters, see for example
Wearing, Rohani and Keeling (2005) or Feng (2007). Lipsitch et al. (2003) use a similar
approach to study the issue of contact tracing in the context of the SARS epidemic.
Compared with this epidemiological work, the approach I take is more direct: some
infected individuals can be identified, and they are immediately quarantined, but only a
fraction of quarantined individuals can be excluded from the infectious pool. This “targeted
quarantine” model makes the interpretation of its effectiveness more transparent.
1.2
Related work by economists using SIR-type models
There has been an outburst of work in economics on COVID-19, much of it appearing in the
CEPR online publication Covid Economics, Vetted and Real-Time Papers.1
1
https://cepr.org/content/covid-economics-vetted-and-real-time-papers-0#block-block-9.
3
Baqaee, Farhi, Mina and Stock (2020) are closest in spirit to this paper in that they study
the implications of alternative paths for employment and non-pharmaceutical interventions,
including quarantine, for output losses and fatalities. Their environment contains more demographic and economic detail; they differentiate by age groups and industries, but their
approach to quarantine follows the standard epidemiological literature, and they interpret
contact tracing and testing as changes in the rate at which infectious individuals are quarantined. Baqaee et al. (2020) also find non-pharmaceutical interventions to be preferable to
employment reductions.
Berger, Herkenhoff, Huang and Mongey (2021) and Piguillem and Shi (2020) study random testing in a SEIR-type model of targeted quarantine. Berger et al. (2021) study the
relative effectiveness of tests and argue that the frequent use of cheap low-sensitivity tests
dominates the infrequent use of expensive perfect tests. Neither of these papers considers
contact tracing.
My approach to infer the diffusion of COVID-19 from data on daily deaths is based on
Fernandez-Villaverde and Jones (2020) and Atkeson, Kopecky and Zha (2020). In a simple
SEIR model, there is a linear progression from being susceptible to possible death. Given this
structure, one can work backwards and infer current infected individuals from future deaths
in a parameterized version of the model. My approach to extract the effect of employment
changes on the transmission rate is related to Baqaee et al. (2020). My approach to estimate
the potential cost and benefits of alternative policies uses the value of a statistical life (VSL)
and is related to Greenstone and Nigam (2020) and Cutler and Summers (2020). Greenstone
and Nigam (2020) use the VSL to calculate the benefits from reduced mortality rates for
various pandemic scenarios in the large-scale pandemic model of Ferguson et al. (2020).
Cutler and Summers (2020) estimate the costs of COVID-19 from reduced output, increased
mortality, and negative health outcomes for survivors.
Finally, this paper is a substantially revised version of Hornstein (2020), which considered
a more stylized version of social distancing as a constant but temporary reduction in contact
rates in the face of a constant transmission rate. In terms of health outcomes, this paper
focuses on fatalities, whereas the pandemic model in Hornstein (2020) was intended as a
stylized version of Ferguson et al. (2020) and also considered the implications for hospital
stays and the use of intensive care units.
4
Figure 1: The SIR Model
γ
S
2
I
R
The basic SIR model
Define the stock of susceptible population S, infected and infectious population I, and recovered population R. Total population is
N = S + I + R.
(1)
Individuals transition sequentially between the states determined by Poisson processes with
given arrival rates. Assume that the disease transmission rate for a given encounter is α,
that the recovery rate from the disease is γ, and that recovered individuals are immune to
the disease. See Figure 1 for a graphic representation.
Total disease transmission, M , following from meetings between the susceptible and
infected population is then,
M=
S
α
N
I,
(2)
where the term in brackets is the new infections caused per infected individual. The dynamics
are described by the differential equations
IS
,
N
IS
I˙ = α
− γI,
N
Ṙ = γI.
Ṡ = −α
(3)
(4)
(5)
The infectious group grows at the rate
α
S
Iˆ =
− 1 γ.
γN
Assume that the initial value for the population share of susceptible individuals is essentially
one, S (0) ≈ N . Then, the number of infected people is initially increasing if
α
R0 = > 1.
γ
5
(6)
The ratio R0 is called the basic reproduction number because at time zero it is approximately
the average number of new infections caused by an infected individual before that individual
recovers,
Z ∞
S(τ )
α
α
τ γe−γτ dτ ≈ = R0 ,
N
γ
0
where the first term in the integral is the average number of infections over a time interval
τ , and the second term is the probability of staying infectious for that time.
3
An extended SEIR model
I now extend the basic SIR model to explore the relative merits of various policy measures,
such as social distancing, quarantine, contact tracing, and random testing in a unified framework. Susceptible individuals are exposed to the infection but are not immediately infectious.
Exposed individuals become infectious, but they initially do not show any symptoms. After
some time, asymptomatic infected individuals do show symptoms of the disease and can be
quarantined. Symptomatic infected individuals either recover over time and become immune,
or they die.
Introducing the exposed state into the SIR model is standard, and we do it because there
is evidence for a non-negligible latency period for the case of COVID-19. Adding an exposed
state changes the dynamics of the model, e.g., it tends to change peak infection rates, but
it usually does not affect terminal outcomes much. Getting the disease dynamics right is
important, since in the next section we will account for the spread of the disease through
the lens of the model.
We make a distinction between asymptomatic and symptomatic infectious individuals
because we want to study the role of quarantine and contact tracing as alternative policy
tools to generalized social distancing measures. Quarantine can only apply to identified
individuals, that is, symptomatic individuals. But there is reason to believe that with
COVID-19 a large share of all infected are asymptomatic. In this case, it might be useful
to identify and subsequently quarantine asymptomatic individuals through contact tracing
and random testing.
Figure 2 provides a graphic representation of the model. Susceptible individuals, S,
become exposed at the rate αI ∗ /N , and exposed individuals, E, become infectious without
symptoms, IA , at the rate φ. Asymptomatic individuals recover at rate γA , and they become
symptomatic at rate β. Symptomatic individuals, IS , recover or die at rates γS and δ,
6
Figure 2: The Extended SEIR Model
γA
γS
αI ∗
φ
𝑁
S
β
E
D
β
qA
qE
φ
ET
δ
IS
γA
IAT
respectively.2 The flow terms, qE and qA , and the stocks, ET and IAT , refer to the exposed
and asymptomatic individuals who have been identified through tracing and/or random
testing as discussed below.
The effects of policy interventions, such as social distancing and quarantining known
infected individuals, are modeled through their impact on the flow of new infections. As in
the basic SIR model, the flow of new infections is proportional to the product of susceptible
individuals and infectious individuals, but quarantine can reduce the number of infectious
individuals who can meet the susceptible population. We assume that symptomatic individuals are always known and that tracing and random testing can identify some exposed
and asymptomatic individuals. Let εi denote the effectiveness of quarantine for the known
infectious population groups, i ∈ {S, AT }, and also assume that asymptomatic infected are
less infectious than symptomatic infected. The relative infectiousness of asymptomatic individuals is σ ≤ 1. The effective pool of infectious individuals who meet the susceptible
population and the inflow of newly infected individuals, I ∗ and M , respectively, are3
I ∗ = σ [IA + (1 − εAT )IAT ] + (1 − εS )IS ,
(7)
M = αSI ∗ /N.
(8)
2
Total deaths are small enough such that the implicit assumption of a constant population is not too
distorting.
3
This version of targeted quarantine deviates from the standard model used in the epidemiological literature in the sense that identified people are added instantaneously to the quarantine pool, but some infections
seep out of that pool. The epidemiological literature I am aware of assumes that infected individuals join
the quarantine pool gradually following a Poisson process, but then quarantine is perfect. For example, Feng
(2007).
7
The following system of differential equations provides the formal representation of the
process dynamics.
Ṡ = −M,
(9)
Ė = M − φE − qE ,
I˙A = φE − (β + γA ) IA − qA ,
(10)
ĖT = qE − φET ,
I˙AT = qA + φET − (β + γA ) IAT ,
I˙S = β(IA + IAT ) − (γS + δ)IS ,
(12)
(11)
(13)
(14)
Ṙ = γA (IA + IAT ) + γS IS ,
(15)
Ḋ = δIS .
(16)
Since the cumulative number of deaths remains sufficiently small for our analysis, we ignore
the impact of deaths on the surviving population in equation (8) to keep things simple.
Social distancing is assumed to directly reduce the rate at which individuals, infectious
and susceptible, contact each other. We assume that the transmission rate can be split
in two components: the contact rate for an individual, c, that captures social distancing
interventions and individual responses to the spread of the disease, and a residual term,
ψ, that captures all other disease-related features of transmission and non-pharmaceutical
interventions. Then the transmission rate is
α = (cψ)2 .
(17)
Social distancing is thus potentially a very effective way to contain the spread of the
disease since a reduction of contact rates applies to all individuals, infectious and noninfectious. Therefore, a reduction of contact rates implies a squared reduction of transmission
rates. Social distancing is also “easy” to implement since all individuals are supposed to
reduce their contact rates, that is, no particular information is required. This indiscriminate
reduction of contact rates also makes social distancing very disruptive for the economy.
Quarantine methods, on the other hand, target individuals who are infectious, that is,
they require information on an individual’s health status. As long as the health status is
observable, that is, for symptomatic individuals, it is relatively straightforward to implement,
though not without cost. The problem with a disease like COVID-19 is that a large share of
infectious individuals—current estimates are around 50 percent—may never show symptoms.
Thus, even if one were to quarantine all symptomatic individuals, the pool of infectious
individuals would be reduced by only half. On the other hand, quarantine is somewhat more
efficient than that since symptomatic individuals are more infectious than asymptomatic
8
individuals. Contact tracing and random testing are attempts to reduce the pool of infectious
individuals by quarantining asymptomatic individuals.
Tracing of asymptomatic infected individuals is modeled as follows. We assume that
at the time an asymptomatic individual becomes symptomatic, a fraction εT of the people
who this individual has infected and who are still in the exposed or asymptomatic state are
traced.4 The inflow into the pool of identified exposed and asymptomatic individuals is then
qT E = εT RAT E βIA and qT A = εT RAT A βIA .
(18)
In appendix A.2 we show that
S
σβ
,
N (β + γA ) (β + γA + φ)
S
σβφ
=α
.
N 2 (β + γA )2 (β + γA + φ)
RAT E = α
(19)
RAT A
(20)
Testing is modeled as follows. Let f N be the flow rate at which not yet identified infected
people are randomly tested. Assume that asymptomatic infected can be identified through
tests but not merely exposed individuals. Also assume that recovered individuals are not
tested. Then the share of identified asymptomatic in a random test is5
pF =
IA
.
S + E + IA
(21)
The inflow of newly identified exposed and asymptomatic individuals through random testing
is
qF E = εT RAT E pF f N and qF A = (1 + εT RAT A ) pF f N,
(22)
where we allow for the possibility that previous contacts of newly identified asymptomatic
individuals are then also traced. Total inflows to the stock of identified exposed and asymptomatic individuals are
qE = qT E + qF E and qAT = qT A + qF A .
4
(23)
Accounting for the pandemic
I now use a quantitative version of the extended SEIR model to account for the spread of
the pandemic in the United States. First, I calibrate the parameters of the SEIR model
4
We essentially assume that tracing does not require time but is instantaneous. It is straightforward to
introduce a time delay for the recovery of tracked individuals. Again, we model the effectiveness of tracing
not through the rate at which potentially traceable individuals enter the quarantine pool, but through the
size of the captured pool, see footnote 3.
5
This potentially overstates the effectiveness of random testing with incomplete quarantine to the extent
that the infectious pool also contains symptomatic individuals.
9
based on available evidence on how COVID-19 spreads in the population. Then, I use the
parametric model to infer the path of the transmission rate from observations of smoothed
daily death rates.
4.1
Calibration
I select the model parameters based on three surveys of the epidemiological profile of COVID19: the fact sheet posted by the Robert Koch Institut (RKI), and the summary of evidence
in Ferguson et al. (2020) (F) and in Bar-On, Flamholz, Phillips and Milo (2020) (B).6
References to the surveys cover either the surveys’ summary of the literature or particular
studies cited in the surveys.
• The latency period from infection to becoming infectious, TE→A , is about 3 days, (B).
• The incubation period from infection to the onset of symptoms, TE→S , is about 6 days,
(RKI) and (F).
• There is a wide range of estimates for the prevalence of asymptomatic infections. (RKI)
references studies that put the share of asymptomatic infections in a range from 20
percent to 90 percent.7 (F) argues that 40 percent to 50 percent of infected are never
identified, mainly because they are asymptomatic. We set the probability of never
becoming symptomatic, pA→R , at 40 percent.
• The (RKI) suggests that contagiousness for light to moderate disease progression declines after 10 days, (F) puts it at 6.5 days, and (B) at 3.5 days. We set the average
duration of infectiousness, TAS , at 8 days.
• The infection fatality rate (IFR) is not well estimated, (B) puts in a range between 0.3
percent and 1.3 percent. We assume it to be pAD = 0.01.
• Asymptomatic infections are between 30 percent (F) and 40 percent (RKI) less infectious than symptomatic infections. We set σ = 0.6.
• As for the effectiveness of quarantine, (F) assumes that two-thirds of symptomatic
individuals self-isolate after one day. Since our quarantine does not involve any time
delay, we assume that the baseline quarantine effectiveness, εS , is 50 percent.
6
The RKI COVID-19 fact sheet is available at
https://www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Steckbrief.html.
7
The lower bound is based on a meta study of at-risk population groups, and the upper bound is based
on serological studies that test for the presence of coronavirus antibodies in population samples.
10
In appendix A.1, we derive the average latency period, incubation period, duration of
infectiousness, recovery probability from an asymptomatic infection, and IFR:
1
φ
1
1
= +
φ β + γA
γA
=
γA + β
β
1
1+
=
β + γA
γS + δ
β
δ
=
.
β + γA δ + γS
TE→A =
(24)
TE→S
(25)
pA→R
TAS
pE→D
(26)
(27)
(28)
We infer the parameters (φ, γA , β) from (TE→A , TE→S , pA→R ) using equations (24), (25),
and (26). The remaining parameters (δ, γS ) follow from (TAS , pE→D ) using equations (27)
and (28). Table 1 summarizes the implied parameter values and observations used in their
derivation.
Table 1: Baseline Calibration
Observation
TE
= 3 days
TE→S = 6 days
pA→R = 40%
TAS
= 8 days
pE→D = 1%
Parameter
φ
= 0.33
γA = 0.13
β
= 0.20
γS = 0.12
δ
= 0.0020
Finally, we note that we have not used two pieces of information from (RKI) in our
calibration of the disease process. First, asymptomatic individuals cause about 45 percent
of all infections. Second, 95 percent of individuals that eventually become symptomatic do
so within 10-14 days. We can calculate these statistics for our calibration and find that they
are quite close to the (RKI) statistics, 35 percent and 98 percent, respectively.
4.2
Inferred transmission rates
We now use the calibrated model to infer the transmission rate in the United States from the
observed daily deaths for the period from March 2020 to December 2020. For this purpose,
we assume that there is no contact tracing or random testing, that is, only symptomatic
individuals are quarantined. From the flowchart of the model, Figure 2, we can see that in
this case, there is a singular flow from exposure to death. This means that we can work
11
Figure 3: Daily Deaths
3500
Actual
Moving Average
3000
Thousands
2500
2000
1500
1000
500
0
50
100
150
200
250
300
350
Days since January 1, 2020
Note: The black line represents reported daily deaths, and the red line represents a Gaussian moving average
of daily deaths with a symmetric forty-one-day window. The shaded area denotes the period after cumulative
daily deaths exceed twenty-five.
our way backwards from daily deaths to initial infections that resulted in these deaths.8
This is the approach Fernandez-Villaverde and Jones (2020) and Atkeson et al. (2020) use
to estimate transmission rates and effective reproduction numbers.
In the online-only Technical Appendix to this paper, Hornstein (2021), we describe the
algorithm that uses daily data on deaths to recover the stocks of preceding susceptible,
exposed, and (a)symptomatic infectious individuals and the transmission rate. Suffice it to
say that the stock of exposed individuals becomes a function of up to fourth order changes of
daily deaths. This means that unless the path for daily deaths is very smooth, the inferred
transmission rate will be exceedingly volatile and can be negative.
8
Note that such a simple mechanical approach is not possible if daily deaths were the outcome of more
than one distinct progression of the disease.
12
Actual daily deaths are extremely volatile—the black line in Figure 3—with a clear
weekend seasonal in reported daily deaths. We therefore smooth the daily death data using
a Gaussian moving average which filters out high frequency movements, the red line in Figure
3.9
We infer the transmission rate through the model’s interpretation of smoothed observed
daily deaths. The transmission rate is determined by the meetings of susceptible and infectious individuals, relative to the total size of the population. In the United States, as
in most other countries, reported COVID-19-related deaths are concentrated in the adult
population.10 Furthermore, there is some evidence that younger children are less likely to
transmit the coronavirus, e.g., Lewis (2020). Thus, an argument could be made to normalize
the transmission rate with respect to the working age population. Qualitatively, the results
do not depend on the normalization, but since the working age population is much smaller
than the total population, the transmission rate inferred from the working age population
measure is noticeably higher.11 We follow the literature and use total population.
In Figure 4, we plot the log-level of the square root of the transmission rate, the black
line.12 The transmission rate declines rapidly in the first month after the first COVID-19related deaths have been observed.13 In particular, the transmission rate declines before
California issues the first state wide stay-at-home order on March 19. The transmission rate
precedes the path of daily deaths bottoming out in May, increasing again in the summer,
declining in early fall, and then increasing in late fall and winter.
Associated with inferred transmission rates from daily deaths in Figure 4, are the time
paths for the population shares of susceptible, exposed, (a)symptomatic individuals, cumulative deaths, and also the effective reproduction rate, the black lines in Figure 5. At its first
peak in early April, about 0.5 percent of the population are infectious, Panels C&D, but
that share declines relatively fast before it increases again in mid-summer and reaches a new
higher peak in December. Given the calibration, at any point in time, somewhat less than
half of infectious individuals are asymptomatic, Panel C, and about one-fourth of infected
individuals are not yet infectious, Panel B. Associated with the transmission rate is the
effective reproduction number in Panel F. The reproduction number is initially quite high,
9
To be precise, we use the MATLAB routine smoothdata with the gaussian option and a symmetric
thirty-one-day window. Essentially, the Gaussian moving average is a moving average with weights that
correspond to a normal probability density function.
10
As of January 2021, more than 99.8 percent of all deaths are among those 17 years and older. See
https://covid.cdc.gov/covid-data-tracker/#demographics. Accessed 1/6/2021.
11
The Census estimates U.S. working age population, that is, those aged 16 and older, to be 263.5 million,
as opposed to a total population of 329.5 million.
12
We plot the transmission rate for the period once cumulative deaths exceed twenty-five. Prior to that,
the inferred transmission rate tends to be very large and volatile, even using smoothed daily deaths.
13
A point emphasized by Atkeson et al. (2020).
13
Figure 4: Transmission Rate
0.6
0.5ln( )
ln(c)
ln( )
0.4
0.2
Log Levels
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
100
150
200
250
300
350
Days since January 1, 2020
Note: The solid black line represents the inferred transmission rate, 0.5 ln α, the solid red line represents
the aggregate measured contact rate, ln c, and the solid blue line represents the residual component, ln ψ.
All rates are in log levels. The red and black dashed lines represent the alternative measured contact and
transmission rates based on the alternative employment rate. The shaded area denotes the period for which
the hypothetical alternative employment rate deviates from the actual employment rate. The left-hand side
boundary of the shaded area is the seventy-ninth day of the year, March 19, and the right-hand side boundary
is the 278 day of the year, October 4.
14
above four, but rapidly declines to values below two in April and then fluctuates between 0.5
and two for the rest of the year, being elevated toward the end of the year when infections
and deaths are increasing. At the end of the year, cumulative deaths make up about 0.095
percent of the population.
5
Experiments
Various measures affect the transmission rate of COVID-19 and the way the pandemic
spreads: from simple non-pharmaceutical interventions, such as hand washing and wearing
face masks, to extended social distancing measures, such as scaling back certain economic
activities. This section uses the previously derived framework to focus on the effects of the
sharp reduction in employment in spring 2020 and possible alternatives to this approach. For
this purpose, we first construct a simple model that maps employment and other social interactions into an aggregate contact rate. We then remove the aggregate contact rate from the
inferred transmission rate and obtain a residual transmission rate. By construction, the residual transmission rate captures all other social distancing measures and non-pharmaceutical
interventions in addition to disease-related characteristics that are not captured by the aggregate contact rate. We then replace the observed employment decline with an alternative
employment path that converges gradually to the levels of fall 2020 employment and avoids
the sharp decline and recovery in the spring and summer. Obviously, the increased contact
rates from this alternative employment path would have increased fatalities. Then, we ask
what alternative quarantine policy could have contained fatalities. Finally, we ask if contact
tracing or random testing could have improved on this outcome.
While the construction of our counterfactual experiments is straightforward, there is an
obvious qualification to the results we obtain. We are combining the counterfactual paths
for work contacts and COVID-19 containment policies with the residual transmission rate
path, but the latter incorporates behavioral responses to the actual path of fatalities. To
the extent that our experiments yield different paths for fatalities, one would expect that
residual contact rates change. In particular, the endogenous response of contact rates would
dampen the impact on fatalities in the counterfactual.
5.1
Measured contact rates
How much of the observed variation in transmission rates can we attribute to measured
variations of contact rates, in particular, contact rates related to employment? To answer
this question, we construct an index that aggregates information on employment and mobility
indices. Our approach is a simplified version of Baqaee et al. (2020).
15
Figure 5: Policy Interventions
A. Susceptible
90
80
0.5
0
100
200
100
300
Days since January 1, 2020
300
D. Symptomatic
1
Percent
Percent
0.5
200
Days since January 1, 2020
C. Asymptomatic
1
0
0.5
0
100
200
300
100
200
300
Days since January 1, 2020
Days since January 1, 2020
E. Deaths
F. Reff
0.2
Percent
B. Exposed
1
Percent
Percent
100
4
0.1
2
0
0
100
200
300
100
Days since January 1, 2020
200
300
Days since January 1, 2020
Note: The black lines represent the baseline outcome using the inferred transmission rate. The red lines
represent the outcome for the alternative transmission rate and no change in quarantine effectiveness, S =
0.5. The blue lines represent the outcome for the alternative transmission rate and increased quarantine
effectiveness, S = 0.7. The shaded area denotes the period for which the alternative transmission rate
deviates from the inferred transmission rate, see notes to Figure 4.
16
We first construct work-related contacts from monthly payroll employment data and the
Bick et al. (2021) survey of workplace locations. Based on survey data that are patterned
on the monthly CPS, Bick et al. (2021) argue that the share of the U.S. workforce that is
working from home increased from about 8 percent in February 2020 to about 30 percent in
May 2020 and then declined to about 20 percent in the fall.14 We define work contacts as the
product of payroll employment and the share of those working away from home. Essentially,
we assume that different contact rates apply to those working at home and those working
away from home.
The employment series and the share of those working away from home are plotted in
Figure 6.a.15 Despite the large decline of employment early in the pandemic, about 15
percent at its trough in April, the 25 percent decline of those working away from home
dominates the decline in work contacts. Work contacts decline persistently over the year by
20 percent, two-thirds due to the decline in the share of those working away from home.
We combine the work contact index with a social contact index related to non-work
activities away from home by aggregating Google mobility indices for shopping and recreation
activities.16 Social contacts started to increase in early March but then dropped off sharply,
preceding the drop-off in work contacts by about a week, Figure 6.b. These social contacts
recovered somewhat faster than work contacts in the summer, but they also did not return to
their pre-pandemic levels, and they declined in the fall when daily deaths started to increase
again.
Aggregate contacts are defined as the sum of contacts from three activities, ‘Home’ (H),
‘Work’ (W), and ‘Social’ (S),
ct =
X
xit ci ,
(29)
i∈{H,S,W }
with fixed weights, ci , and the normalized social and work indices as activity measures
relative to pre-pandemic levels, xit . Home contacts are fixed at one, xH
t = 1.
We obtain the contact weights from a survey on social contacts in Europe.17 Mossong et
al. (2008), Figure 2, display the average distribution of contacts for the pool of all countries.
14
Parker, Horowitz and Minkin (2020) report a similar pattern in Pew survey data.
We interpolate the two monthly series using piece-wise cubic splines and normalize each series at one on
February 15, 2020.
16
We average the daily Google mobility indices starting February 15 for Retail and Recreation locations,
Grocery and Pharmacy locations, and Transit Stations. We then calculate a seven-day moving average of
the average Google index and normalize the series to one on February 15.
17
The survey covers eight European countries in 2005 and 2006. The purpose of the study was to “determine patterns of person-to-person contact relevant to controlling pathogens spread by respiratory or closecontact routes,” Mossong, Hens, Jit, Beutels, Auranen, Mikolajczyk, Massari, Salmaso, Tomba, Wallinga,
Heijne, Sadkowska-Todys, Rosinska and Edmunds (2008).
15
17
Roughly, the allocation of contacts among activities is: ‘Other’ 15 percent, ‘Leisure’ 15
percent, ‘School’ 12 percent, ‘Home’ 25 percent, ‘Work’ 20 percent, ‘Multiple’ 8 percent, and
‘Transport’ 5 percent. We drop ‘School’ contacts and combine all other groups except ‘Home’
and ‘Work’ into ‘Social’, now 35 percent. That leaves us with 25 percent H, 20 percent W,
and 35 percent S, and we re-normalize the weights so that they add up to one.
The aggregate contact index is displayed as the red line in Figure 6.b. Since constant
home contacts receive a weight of one-third in our weighting scheme, aggregate contacts do
not fall as much as either work or social contacts.
We now split the transmission rate into a part that is due to the measured contact rates
and an unexplained residual term, ψ,
0.5 ln α = ln c + ln ψ,
(30)
the red and blue lines in Figure 4, respectively. According to this procedure, the impact
of reduced social contacts and employment on the transmission rate is limited: changes in
these contacts account for about one-third of the decline in the overall transmission rate. We
will now use our calibrated model to study what could have happened if social distancing
policies had been less restrictive and had been accompanied by other non-pharmaceutical
interventions such as efficient quarantine, contact tracing, or random testing.
5.2
Alternative employment path
We take the sharp decline of employment in March and April, followed by a partial recovery
by October, as the main employment impact of stay-at-home orders. Taking the partial
recovery by the fall as given, we essentially assume that many of the employment losses in
the hospitality and recreation sector were unavoidable without a vaccine.
The alternative employment path we propose has employment gradually converge to its
fall 2020 values, starting in mid-March, the dashed purple line in Figure 6.a.18 Since social
contacts are correlated with market activity, we assume that social contacts follow a similar
path, with the same beginning and end points, the dashed cyan line in Figure 6.b. Combining
the time paths for work and social contacts yields the aggregate contact index, the dashed red
line in Figure 6.b. Combining the alternative contact index with the residual transmission
rate we now have the transmission rate, associated with the alternative employment path,
the dashed red and black lines, respectively, in Figure 4.
Avoiding the sharp decline in employment doubles the eventual fatalities in the calibrated
model. The red lines in Figure 5 plot the time paths for the population shares of susceptible,
18
The alternative employment path is a cubic spline that connects March 19, when California issued the
first statewide stay-at-home order, with October 4, when employment has apparently stabilized.
18
Figure 6: Measured Contact Rates
1.05
Employment
Work Away from Home
1
Work Index, xW
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
50
100
150
200
250
300
350
Days since January 1, 2020
(a) Work Contacts
1.1
Social, xS
Work, xW
Aggregate, c
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
50
100
150
200
250
300
350
Days since January 1, 2020
(b) Aggregate Contacts
Note: In Panel (a), the purple line is total nonfarm payroll employment, the green line is the fraction of
those employed that are working away from home, and the blue line is employment away from home, also
in Panel (b). In Panel (b), the magenta line is social contacts not at home and not at work, and the red line
is an index of all contacts. All series are normalized to one on February 15, 2020. The construction of the
contacts is described in the text. The solid lines represent observed variations in contacts, and the dashed
lines represent hypothetical time paths for contacts as described in the text. The shaded area highlights the
time period for which the hypothetical contacts deviate from actual contacts.
19
exposed, and (a)symptomatic individuals, and cumulative deaths for the transmission rates
implied by the alternative employment path. The number of infectious individuals on the
alternative path more than triples relative to the baseline path and so do cumulative deaths
during the time the alternative path deviates from the baseline path. Because there are so
many more infections on the alternative path, the population share of susceptible individuals
declines to about 80 percent, as opposed to staying above 90 percent for the baseline path.
Fewer susceptible individuals then means lower effective reproduction rates at the end of
the year and fewer additional infections and deaths. Nevertheless, cumulative deaths at the
end of the year make up about 0.19 percent of the population, twice the number actually
realized.
5.3
Alternative policies: quarantine, contact tracing, testing
We now consider three alternative policies that might be put in place for the alternative
employment path: more efficient quarantine, contact tracing, and finally random testing.
Increased quarantine effectiveness keeps cumulative fatalities low in the longer run, although
in the short run, fatalities increase faster than in the baseline case. Contact tracing reduces fatalities somewhat, but the reductions, even from a perfectly effective contact tracing
scheme, are limited when combined with a more effective quarantine program. Finally, the
reductions of fatalities from even large-scale random testing programs are limited.19
5.3.1
Quarantine
Quarantine is assumed to be 50 percent effective in our baseline calibration, that is, half
of symptomatic individuals are assumed to (self)quarantine. We now consider the case
when quarantine effectiveness is permanently increased to 70 percent for the alternative
employment path.
A more efficient permanent quarantine regime does not prevent the increase of fatalities
relative to the baseline case in the short run, but it substantially reduces cumulative fatalities
in the long run. The blue lines in Figure 5 plot the time paths for the population shares
of susceptible, exposed, and (a)symptomatic individuals, and cumulative deaths for the
transmission rates implied by increased quarantine for the alternative employment path.
With the higher quarantine effectiveness, the number of infectious individuals now only
doubles relative to the baseline path for the first thirty days, and so do cumulative deaths.
But after seventy days, the number of infectious individuals is less than in the baseline case,
and cumulative deaths level out. In fact, with the higher permanent quarantine rate, the
19
For each policy parameter, we consider a permanent increase from the baseline calibration value to its
alternative value. We assume that the increase is spread out over forty days, starting March 19, 2020.
20
Table 2: Effectiveness of Quarantine, Tracing, and Random Testing
(1)
(2)
0.000
0.187
0.101
0.063
0.044
0.033
0.026
0.250
0.181
0.097
0.060
0.043
0.032
0.025
εQ
0.500
0.600
0.700
0.800
0.900
1.000
(3)
εT
0.500
0.176
0.094
0.058
0.041
0.031
0.024
(4)
(5)
(6)
0.750
0.171
0.090
0.057
0.040
0.030
0.024
1.000
0.166
0.087
0.055
0.039
0.029
0.023
0.001
0.153
0.071
0.042
0.029
0.022
0.017
(7)
f
0.010
0.148
0.069
0.041
0.028
0.021
0.016
(8)
0.100
0.110
0.054
0.034
0.024
0.018
0.014
Note: Cumulative deaths at end of sample, December 31, 2020, as a percent of population for given alternative transmission path and variations of quarantine effectiveness,
εQ , tracing effectiveness, εT , and testing rate, f . Rows represent different values of the
common quarantine effectiveness, εQ = εS = εAT . Columns 1 through 5 represent variations of the tracing effectiveness parameter in the absence of random testing, f = 0.
Columns 6 through 8 represent variations of the testing rate for a fixed tracing effectiveness, εT = 0.5. The testing rate f denotes the daily fraction of population that is
randomly selected for testing.
pool of infected individuals becomes so small that infections are not increasing in the late
fall as the transmission rate and the effective reproduction number is increasing again.
Increasing quarantine effectiveness permanently from 50 percent to 70 percent reduces
cumulative deaths on the alternative employment path at the end of the year by two-thirds,
Table 2, column 1. This represents about two-thirds of the cumulative deaths on the baseline
path, and most of that reduction is attained at the end of the year when increased quarantine
effectiveness contains the spike in transmission rates. But even a more limited permanent increase of quarantine effectiveness to 60 percent contains cumulative deaths on the alternative
employment path at the same level as the baseline path. Increasing quarantine effectiveness
beyond 70 percent further reduces cumulative deaths at a diminishing rate.
5.3.2
Contact tracing
Contact tracing, as defined in equations (19) and (20), is assumed to instantaneously identify
and quarantine a fraction of the individuals who a newly symptomatic individual has infected.
It follows that contact tracing can only be effective as long as quarantine is effective.
Our model suggests that the reductions of cumulative deaths from contact tracing are
limited, even though about half of infectious individuals are asymptomatic. We display
end-of-year cumulative deaths for various combinations of quarantine and contact tracing
21
efficiencies in Table 2, columns 1 through 5.20 As we can see, the most that contact tracing
can achieve, if it is perfectly efficient, is to reduce cumulative deaths by about 10 percent
for any level of quarantine effectiveness considered. Contact tracing that quarantines half of
the identified infected individuals reduces cumulative deaths by about 5 percent.
In our environment, even perfect contact tracing captures only a small fraction of the
asymptomatic infectious individuals. Figure 7 replicates Figure 5 for the alternative employment path with more effective quarantine, εS = 0.7, without and with perfect contact
tracing, εT = 0 and 1. As we can see, only a small fraction of asymptomatic individuals,
maybe a tenth, is identified and quarantined.
The reason why contact tracing has limited effects can be understood as follows. From
equation (20) the maximal measure of asymptomatic individuals that can be traced from
newly symptomatic individuals is
RAT A βIA = α
S
σβ
βIA .
N (β + γA ) (β + γA + φ)
(31)
Compare this inflow into the pool of traced asymptomatic individuals with the inflow into
the pool of asymptomatic individuals from new infections
α
S
S ∗
I ≈ α [σ + (1 − εQ )] IA ,
N
N
(32)
where the approximation is based on about equal numbers of asymptomatic and symptomatic
individuals.21 From our calibration, the scale factors in equations (31) and (32) are 0.3 and
1.3, respectively. Thus, inflows into the pool of traced asymptomatic individuals are less
than one-fourth of the inflows into the pool of asymptomatic individuals. Which explains
why contact tracing, even in the best of cases, cannot track that many infected individuals.22
5.3.3
Testing
Early on in the pandemic, daily tests covered less than 0.05 percent of the population, but
at the end of 2020, daily tests covered about 0.7 percent of the population. Within the
framework of this paper, this ten-fold increase may attain most of the gains that could be
obtained from random testing.23
20
We assume that the quarantine effectiveness for traced asymptomatic and exposed individuals is the
same as for symptomatic individuals.
21
We have also ignored the time delay coming from the exposed state.
22
In appendix A.4, we consider alternative calibrations that increase the share and the relative infectiousness of asymptomatic infectious individuals. Neither alternative calibration increases the effectiveness of
contact tracing.
23
We should note that throughout the pandemic, tests were predominately used for the confirmation of
symptomatic cases or suspected infections, not random testing.
22
Figure 7: Contact Tracing
A. Susceptible
B. Exposed
0.6
Percent
Percent
100
90
0.4
0.2
80
0
100
150
200
250
300
100
350
200
250
300
350
D. Symptomatic
1
0.4
Percent
Percent
C. Asymptomatic
0.2
0.5
0
0
100
150
200
250
300
350
100
Days since January 1, 2020
150
200
250
300
350
Days since January 1, 2020
E. Deaths
F. Reff
4
Percent
0.2
Percent
150
Days since January 1, 2020
Days since January 1, 2020
0.1
0
2
0
100
150
200
250
300
350
100
Days since January 1, 2020
150
200
250
300
350
Days since January 1, 2020
Note: The blue lines represent the outcome for the alternative transmission rate and increased quarantine
effectiveness, εS = 0.70, but no contact tracing. The red lines represent the outcomes when perfectly effective
contact tracing is added, εT = 1.0. The dashed lines in panels B and C represent the traced and quarantined
asymptomatic individuals, and the solid lines represent the unidentified and not quarantined individuals.
23
In Table 2, columns 6 through 8, we display end-of-year cumulative deaths for various
combinations of quarantine effectiveness and testing rate, given the alternative employment
path and a 50 percent tracing effectiveness. Relative to no random tests, Table 2, column
5, randomly testing 0.1 percent of the population every day reduces cumulative deaths by
one-fourth to one-third when quarantine effectiveness exceeds 70 percent. Increasing the
daily testing rate to 1 percent of the population does not reduce cumulative deaths much
more, but increasing the daily testing rate to 10 percent of the population cuts cumulative
deaths in half relative to no testing.
5.4
The costs and benefits of alternative policies
Cutler and Summers (2020) have calculated the net-cost of COVID-19 from impaired health
and lost lives and output at $16.1 trillion, about 90 percent of 2019 GDP. We use their approach to calculate the benefits and costs of the alternative policies described in the previous
section. We estimate the gains from increasing quarantine effectiveness to 70 percent, combined with a 50 percent effective contact tracing scheme and a daily testing of 1 percent of
the population to be $4 trillion, about 18 percent of 2019 GDP. Increasing the effectiveness
of quarantine and contact tracing to 100 percent, if it was feasible, would result in additional
net gains of $3 trillion, another 14 percent of GDP. Almost all of the gains are associated
with reductions in loss of life and improved health outcomes of survivors.
We estimate the GDP gains from the alternative employment path assuming that labor
productivity on the alternative path is the same as on the actual path. We calculate the
GDP gain as roughly half a trillion dollars, about 2.5 percent of 2019 GDP. Our procedure
is likely to overstate the potential GDP gains from the alternative employment path since
labor productivity increased substantially in the 2020 downturn: employment losses were
concentrated in low productivity sectors, such as hospitality and leisure. As we now show,
this is unlikely to matter much since benefits from improved health outcomes far outweigh
the gains from increased GDP.
Greenstone and Nigam (2020) and Cutler and Summers (2020) use the VSL to evaluate
the benefits from reduced mortality rates. The VSL represents a person’s willingness to pay
for a reduced probability of death and is a concept used by U.S. federal agencies to evaluate
the benefits of policies. For example, the EPA uses an inflation adjusted 2020 VSL for an
adult U.S. citizen of $11.5 million (Greenstone and Nigam 2020, p. 14). I follow Cutler and
Summers (2020), who on the one hand, use a more conservative estimate of $7 million for
the VSL, but on the other hand assume that for every death there are seven survivors who
suffered severe infections, the negative health effects of which are equivalent to a 35 percent
24
reduction of their VSL. The net effect of these assumptions is that a death is valued at $24
million.
Finally, the assumptions on the cost of quarantine, contact tracing, and random testing
are as follows. For quarantine, we assume that an identified infected individual is quarantined
for two weeks and gets reimbursed for lost income and expenses. The average weekly income
in the BLS Covered Employment and Wages for the first quarter of 2020 was $1,200, and
we assume that a quarantined individual receives a weekly per diem of $1,000. Thus, the
total cost per quarantined individual is $4,400.24 Cutler and Summers (2020) assume that a
contact tracing program would cost $25 billion, which we treat as a fixed cost, independent
of how many cases will be traced.25 We assume that the unit cost of a random test is $100.26
Table 3 restates the impact on cumulative deaths from alternative policies in Table 2
in terms of the benefits from improved health outcomes and the cost of attaining them.
The first thing to notice is that the output gains from the alternative employment path are
roughly one-tenth the value of the improved health outcomes for quarantine effectiveness at
70 percent or higher. Even if we did not consider permanent health damages to survivors
of severe infections and scaled back the value of a death by two-thirds, improved health
outcomes would still be twice as valuable than the gains from increased output. Second, the
costs of attaining these outcomes are less than one-fifth of the increased output, and they
are decreasing as quarantine effectiveness increases since the number of infected individuals
who need to be quarantined declines. Third, even though reductions of cumulative deaths
from contact tracing and random testing on top of effective quarantine are limited, their
value is still of a similar magnitude as the gains from increased output.
6
Conclusion
I have studied the effectiveness of alternative policies to contain the spread of a pandemic
in a SEIR model that is calibrated to the characteristics of COVID-19. In particular, the
model allows for asymptomatic infectious individuals, an approach that suggests the use of
contact tracing, conditional on having an effective quarantine policy in place. When the
model’s disease transmission is matched to the 2020 U.S. experience, I find that permanent
24
This cost does not include any resources used to verify that individuals indeed stay quarantined. On the
other hand, we assume that all quarantined individuals receive the payment and not just the ones in excess
of the baseline quarantine rate.
25
This cost corresponds to scaled versions of the cost of contact tracing programs in South Korea and
Taiwan.
26
On October 15, 2020, the Centers for Medicare and Medicaid services announced that Medicare would
reimburse high throughput COVID-19 tests at $100, https://www.cms.gov/newsroom/press-releases/cmschanges-medicare-payment-support-faster-covid-19-diagnostic-testing.
25
Table 3: Costs and Benefits of Alternative Policies
(1)
(2)
0.000
0.250
εQ
(3)
εT
0.500
(4)
(5)
(6)
0.750
1.000
0.001
(7)
f
0.010
(8)
0.100
A. GDP gain from alternative employment path, $ 0.509 Trillion
0.500
0.600
0.700
0.800
0.900
1.000
B. Benefit from improved health outcome, Trillions of
-6.560 -6.093 -5.652 -5.236 -4.843 -3.850 -3.384
0.290 0.594 0.872 1.127 1.362 2.668 2.863
3.332 3.505 3.662 3.807 3.941 4.986 5.068
4.790 4.906 5.013 5.112 5.203 6.004 6.051
5.657 5.744 5.824 5.899 5.967 6.595 6.627
6.237 6.306 6.369 6.427 6.481 6.985 7.008
$s
-0.409
4.049
5.616
6.388
6.860
7.178
0.500
0.600
0.700
0.800
0.900
1.000
C. Cost of improved health outcome, Trillions of $s
0.082 0.081 0.079 0.078 0.081 0.161
0.052 0.051 0.050 0.049 0.048 0.130
0.038 0.037 0.036 0.036 0.036 0.118
0.030 0.030 0.030 0.029 0.031 0.113
0.026 0.025 0.025 0.025 0.027 0.109
0.022 0.022 0.022 0.022 0.025 0.107
0.974
0.949
0.940
0.936
0.933
0.931
0.083
0.053
0.038
0.031
0.026
0.023
Note: This table displays the benefits and costs associated with the alternative policies
considered in Table 2. Panel A displays the GDP gains from the alternative employment
path, Panel B displays the benefits from reduced cumulative deaths and other COVID-19related health outcomes, and Panel C displays the costs associated with the quarantine,
tracing, and testing programs as described in the text.
26
improvements of quarantine effectiveness, together with contact tracing and possibly random
testing, can substantially lower cumulative deaths. These policies contain fatalities, even for
an alternative employment path that avoids the sharp drop in March and April 2020.
Using standard VSL assumptions, we show that the gains from reduced fatalities could
be up to one third of GDP, an order of magnitude larger than the gains from improved
employment outcomes and the potential fiscal costs of the programs. We have not tried to
find an optimal policy, but the results from our more limited analysis suggest that if such an
analysis were to use standard VSL for fatalities, the reduction of fatalities rather than the
containment of output losses would likely be the primary objective.
We should qualify the statement on the potential gains from these policies for at least
two reasons. First, apart from the proposed changes in employment paths, we assume that
any other behavioral responses to the pandemic remain unchanged. For example, if in our
simulations, alternative policies reduce cumulative deaths, individuals are assumed not to
let down their guard and return to their previous pattern of social interaction. Second, we
have not considered the potentially large losses due to the disruptions of schooling. For
example, Fernald, Li and Ochse (2021) argue that reduced educational attainment due to
the pandemic can lower the level of GDP by up to 0.5 percentage points for the next sixty
years. If we use a one percent discount rate, that would mean a 15 percent reduction in the
present value of GDP, which is large but still only half of the potential benefits from reduced
fatalities.
27
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29
A
Appendix
Detailed derivations for the statistics defined in this appendix can be found in the online-only
Technical Appendix to this paper, Hornstein (2021).
A.1
Calibration
Here we define certain statistics that are used in the calibration of the model.
The probability that an exposed individual eventually becomes symptomatic is
Z ∞ Z τ
−φs −β(τ −s) −γ (τ −s)
A
φe
βe
e
ds dτ,
pE→S =
0
0
where the first term is the probability that the individual becomes asymptomatic infectious
at s and then becomes symptomatic at τ , without recovering. This can be solved as
pE→S =
β
.
γA + β
(A.1)
The average time for an exposed individual to become symptomatic, conditional on eventually becoming symptomatic, is
(Z
Z
∞
τ
τ
TE→S =
0
0
−φs −β(τ −s) −γ (τ −s) )
φe
βe
e A
ds dτ,
pE→S
where the term in curly brackets is the probability of becoming symptomatic at τ , conditional
on eventually becoming symptomatic. This can be solved as
TE→S =
1
1
+
.
φ γA + β
(25)
Note that this can be rewritten as
TE→S =
1
β 1
+
= TE + pA→S TA→S = TE + TA ,
φ γA + β β
where the first term is the average time spent being latent, and the second term is the
probability of becoming symptomatic times the average time it takes to become symptomatic
from asymptomatic.
The probabilities that an infected individual becomes symptomatic or recovers before
becoming symptomatic are
β
γA + β
γA
.
=
γA + β
pA→S =
pA→R
30
(A.2)
(26)
The average duration of infectiousness, that is, the average time spent in states A and S
is
∞
Z
TAS =
−βτ
−γA τ
+ τ+
τ γA e
e
0
1
γS + δ
−γ τ −βτ
e A βe
dτ,
where the first term in the integral represents being asymptomatic for duration τ , followed by
a recovery, and the second term of the integral represents being asymptomatic for duration
τ , followed by being symptomatic with an average duration 1/ (γS + δ). This expression
simplifies to
TAS =
1
β
1
+
= TA + pA→S TS ,
β + γA β + γA γS + δ
(27)
which is the average duration of being asymptomatic plus the average duration of being
symptomatic times the probability of becoming symptomatic.
The infection fatality rate (IFR), that is, the probability of dying when symptomatic is
Z ∞
δ
pS→D =
δe−δτ e−γS τ dτ =
.
δ + γS
0
The probability of dying, conditional on being infected, is equal to the probability of making the transition from asymptomatic to symptomatic times the probability of dying when
symptomatic
β
δ
.
(28)
pE→D = pA→S pS→D =
δ + γS β + γA
A.2
Reproduction rate with quarantine and contact tracing
A symptomatic individual who is not quarantined infects on average RS = α (S/N ) RS
individuals until he recovers or dies, with
R ∞ −γ τ
τ (e s ) δe−δτ + (γS e−γs τ ) e−δτ dτ =
0
RS =
1
.
γS + δ
An asymptomatic individual who is not quarantined infects on average RA = α (S/N ) RA
until she recovers or becomes symptomatic, with
RA =
R∞
0
στ (e−γA τ ) βe−βτ + (γA e−γA τ ) e−βτ dτ = σ
1
.
γA + β
An asymptomatic individual who is quarantined with rates εQA and εQS infects on average
RAS = α (S/N ) RAS until he recovers or becomes symptomatic, with
RAS (εQA , εQS ) = (1 − εQA ) RA + (1 − εQS )
31
β
RS .
γA + β
Let R̄AS (εT , εQA , εQS ) denote the expected infection factor for a latent individual (the individual of interest) before knowing whether the individual will be traced
R̄AS (εT , εQA , εQS ) = εT RAS (εQA , εQS ) + (1 − εT ) RAS (0, εQS ) .
Consider an individual who is latent, traced with efficiency εT , and quarantined with
rates εQA and εQS . The individual of interest may have been infected by (1) a symptomatic
individual that was not quarantined, there are (1 − εQS ) IS of them; (2) an asymptomatic
individual that was traced, but not quarantined, there are (1 − εQA ) IAT of them; and (3) an
asymptomatic individual that was not yet traced, there are IA of them. We want to calculate
the average of new infections coming from this individual until she recovers or dies.
• Case 1 and 2: Since infectious individuals are only traced at the time they become
symptomatic, the individual of interest will never be traced. Therefore, the expected
number of new infections coming from the infected individual is
RAS (0, εQS ) = α (S/N ) RAS (0, εQS );
• Case 3: The expected number of new infections coming from the infected individual
when the infecting individual was asymptomatic is RE = α (S/N ) RE , with RE =
Z ∞
γA e−(γA +β)τ0 [RAS (0, εQS )] dτ0
Z0 ∞
−(γ +β)τ0 −φτ0
+
βe A
e
R̄AS (εT , εQA , εQS ) dτ0
0
Z τ0
Z ∞
−(γ +β)τ0
−φs −(γ +β)(τ0 −s)
+
βe A
φe
e A
σ (τ0 − s) + R̄AS (εT , εQA , εQS ) ds dτ0
0
0
Z τ0
Z τ0 −s
Z ∞
−(γ +β)τ0
−φs
−(γA +β)t
A
+
βe
φe
γA e
[σt] dt ds dτ0
0
0
0
Z τ0
Z τ0 −s
Z ∞
−φs
−(γ +β)t
−(γ +β)τ0
A
A
βe
[σt + (1 − εQS ) RS ] dt ds ,
βe
φe
+
0
0
0
where the five components of case 3 are
• Case 3.1: The infecting individual recovers at τ0 , and the infected individual is not
traced;
• Case 3.2 through 3.5: The infecting individual becomes symptomatic at τ0 , and
– Case 3.2: The infected individual never became asymptomatic;
– Case 3.3: The infected individual became asymptomatic at s and stayed so until
τ0 ;
32
– Case 3.4: The infected individual became asymptomatic at s and recovered at
s + t;
– Case 3.5: The infected individual became asymptomatic at s and symptomatic
at s + t.
The probabilities for the components of case 3 are
γA
,
γA + β
β
=
,
γA + β + φ
βφ
=
,
2 (γA + β + φ) (γA + β)
1
1 βφγA
,
=
2 ·
2 (γA + β) γA + β + φ
1 φβ 2
1
.
=
2 ·
2 (γA + β) (γA + β + φ)
pE1 =
pE2
pE3
pE4
pE5
The expected infections for the components of case 3 are
γA
RAS (0, εQS ) ,
γA + β
β
=
R̄AS (εT , εQA , εQS ) ,
γA + β + φ
σ
+ R̄AS (εT , εQA , εQS ) ,
=pE3
2 (γA + β)
βφγA σ
=
,
4 (γA + β + φ) (γA + β)3
β
= [RE4 + pE4 (1 − εQS ) RS ] ,
γA
RE1 =
RE2
RE3
RE4
RE5
and
RE =
5
X
RE,i .
i=1
A newly infected individual then infects on average R = α (S/N ) R individuals where
R=
A.3
[(1 − εQA ) IAT + (1 − εQS ) IS ] RAS (0, εQS ) + IA RE
.
(1 − εQA ) IAT + (1 − εQS ) IS + IA
Traceable individuals
We consider an asymptomatic infectious individual who is quarantined once he becomes
symptomatic. For this case, we calculate the average number of exposed and infectious
asymptomatic individuals that this individual has created.
33
By the time an asymptomatic individual becomes symptomatic, on average that individual has infected α (t) S(t)/N (t)RAT other individuals, where
Z ∞
(στ ) βe−βτ e−γA τ dτ = σ
RAT =
0
β
.
(β + γA )2
The average number of individuals that the infectious individual has infected and that are
not yet infectious at the time the individual becomes symptomatic is α (t) S(t)/N (t)RAT E ,
where
Z
∞
σpEE (τ ) βe−(β+γA )τ dτ,
RAT E =
0
and the term in brackets is the probability that the infectious individual has been asymptomatic for duration τ , and pEE (τ ) denotes the fraction of individuals that were infected
by the infectious individual over the interval τ and that have not yet become infectious at
the time the individual becomes symptomatic. The probability that an individual that was
infected time s ago and has not yet become infectious is e−φs . Thus
Z τ
1
e−φs ds =
pEE (τ ) =
1 − e−φτ
φ
0
and
RAT E = σ
β
.
(β + γA ) (β + γA + φ)
(19)
The average number of individuals who an infectious individual has infected and that are
infectious but asymptomatic at the time the individual becomes symptomatic is α (t) S(t)/N (t)RAT A ,
where
Z ∞ Z τ
RAT A =
σ
pEA (s) ds βe−(β+γA )τ dτ
0
0
and pEA (s) denotes the fraction of individuals that were infected time s ago, have become
infectious in the meantime but have not yet recovered or become symptomatic. Thus
Z s
φ
1 − e−(φ−γA −β)s
pEA (s) =
φe−φv e−(γA +β)(s−v) dv =
φ − γA − β
0
and
RAT A = σ
φβ
.
2 (β + γA ) (β + γA + φ)
2
(20)
The average number of individuals that the infectious individual has infected and that
have become symptomatic or recovered at the time the individual becomes symptomatic is
α (t) S(t)/N (t)RAT R , where
Z ∞ Z τ
RAT R =
σ
pER (s) dτ βe−(β+γA )τ dτ
0
0
34
and pER (s) denotes the fraction of individuals that were infected time s ago and that have
recovered or become symptomatic,
Z
pER (s) =
s
φe−φv 1 − e−(γA +β)(s−v) dv
0
and
RAT R =
A.4
σβφ
= RAT A .
2 (β + γA )2 (β + γA + φ)
Robustness
In section 5.3.2, we have shown that the effectiveness of contact tracing turns out to be
limited even though the environment is consistent with a large share of asymptomatic infected
individuals. The environment was calibrated to match what is known about the spread of
COVID-19, but the uncertainty about how the disease spreads is large. We now consider
some alternative selections of parameter values and how they affect the effectiveness of
contact tracing. One would expect that contact tracing becomes relatively more effective if
there are relatively more asymptomatic infectious individuals or if asymptomatic individuals
become relatively more infectious. The following two experiments suggest that neither of
the two alternative calibrations increase the efficiency of contact tracing.
Table A1 reports results from alternative calibrations that increase the probability that
asymptomatic infectious recover before they show symptoms. The top panel replicates the
information from Table 2, columns 1 through 5, with baseline calibration pA→R = 0.4. The
next two panels increase the recovery probability to 60 and 80 percent. For each of the two
alternative calibrations we re-estimate the implied transmission rate. As we can see from
comparing the upper left-hand side cells of each panel, εQ = 0.5 and εT = 0.0, the alternative
employment path yields only slightly different cumulative deaths at the end of year for the
different calibrations. Notice that without contact tracing, increasing quarantine efficiency
has less of an impact when the share of asymptomatic infectious is larger, moving down column 1 of each panel in Table A1. Also, for any given quarantine efficiency, increasing contact
tracing efficiency has a smaller impact on end-of-year cumulative deaths as the recovery probability increases. Essentially, once an infectious individual becomes symptomatic, fewer of
the individuals who he infected are still around to be traced if asymptomatic individuals are
recovering faster.
Table A2 reports results for alternative calibrations that increase the relative infectiousness of asymptomatic individuals. The top panel replicates the information from Table 2,
columns 1 through 5, with baseline calibration σ = 0.6. The next two panels increase the
relative infectiousness to 80 and 100 percent. We again re-estimate the implied transmission rate for each of the two alternative calibrations. Again, the alternative employment
35
Table A1: Share of Asymptomatic
(1)
εQ
0.000
0.500
0.600
0.700
0.800
0.900
1.000
0.187
0.101
0.063
0.044
0.033
0.026
0.500
0.600
0.700
0.800
0.900
1.000
0.190
0.107
0.069
0.050
0.039
0.032
0.500
0.600
0.700
0.800
0.900
1.000
0.189
0.125
0.088
0.068
0.056
0.048
(2)
(3)
(4)
εT
0.250 0.500 0.750
pA→R =0.40
0.181 0.176 0.171
0.097 0.094 0.090
0.060 0.058 0.057
0.043 0.041 0.040
0.032 0.031 0.030
0.025 0.024 0.024
pA→R =0.60
0.187 0.184 0.182
0.106 0.104 0.102
0.068 0.067 0.066
0.050 0.049 0.048
0.039 0.038 0.038
0.031 0.031 0.030
pA→R =0.80
0.189 0.188 0.188
0.125 0.124 0.124
0.088 0.088 0.087
0.068 0.068 0.068
0.056 0.056 0.056
0.048 0.047 0.047
(5)
1.000
0.166
0.087
0.055
0.039
0.029
0.023
0.179
0.101
0.065
0.048
0.037
0.030
0.187
0.123
0.087
0.067
0.055
0.047
Note: Cumulative deaths at end of sample, December 31, 2020, as a percent of population
for given recovery probability of asymptomatic individuals, pA→R , its implied alternative
transmission path, and variations of quarantine efficiency, εQ , and tracing efficiency, εT ,
and no testing, f = 0. See also Table 2.
36
Table A2: Infectiousness of Asymptomatic
(1)
(2)
0.000
0.250
0.500
0.600
0.700
0.800
0.900
1.000
0.187
0.101
0.063
0.044
0.033
0.026
0.181
0.097
0.060
0.043
0.032
0.025
0.500
0.600
0.700
0.800
0.900
1.000
0.188
0.109
0.070
0.050
0.039
0.031
0.180
0.103
0.067
0.048
0.037
0.029
0.500
0.600
0.700
0.800
0.900
1.000
0.188
0.115
0.076
0.056
0.043
0.035
0.178
0.108
0.072
0.053
0.041
0.033
εQ
(3)
εT
0.500
σ=0.60
0.176
0.094
0.058
0.041
0.031
0.024
σ=0.80
0.173
0.098
0.064
0.046
0.035
0.028
σ=1.00
0.170
0.102
0.068
0.050
0.039
0.031
(4)
(5)
0.750
1.000
0.171
0.090
0.057
0.040
0.030
0.024
0.166
0.087
0.055
0.039
0.029
0.023
0.166
0.094
0.061
0.044
0.034
0.027
0.159
0.090
0.059
0.042
0.033
0.026
0.162
0.096
0.064
0.047
0.037
0.030
0.154
0.091
0.061
0.045
0.035
0.028
Note: Cumulative deaths at end of sample, December 31, 2020, as a percent of population
for given relative infectiousness of asymptomatic individuals, σ, its implied alternative
transmission path, and variations of quarantine efficiency, εQ , and tracing efficiency, εT ,
and no testing, f = 0. See also Table 2.
path yields slightly different end-of-year cumulative deaths for each calibration. Notice that
without contact tracing, increasing quarantine efficiency has less of an impact when the
asymptomatic are more infectious since asymptomatic individuals are not quarantined and
are relatively more infectious, moving down column 1 of each panel in Table A2. On the
other hand, for any given quarantine efficiency, increasing contact tracing efficiency now has
a larger impact on end-of-year cumulative deaths as the relative infectiousness increases.
37
@FedFRASER