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Behavior and the Transmission of COVID-19
∗
By Andrew G. Atkeson, Karen Kopecky, and Tao Zha
Since the outbreak of the COVID-19 pan- the pandemic in many countries. Large un-
demic in early 2020, epidemiologists and explained factors or wedges are required to
economists have raced to develop models of account for regime shifts in the evolution of
the disease usable for forecasting the pro- daily deaths, disease transmission, and hu-
gression of the pandemic, for evaluating the man behavior that occurred in later days of
effectiveness of various interventions aimed the pandemic. Future research should aim
at mitigating the spread of the disease, at accounting for these wedges so as to de-
and for understanding the interaction of velop new models that are more useful for
the pandemic with the economy.The models analysis of this and future pandemics.
developed by economists often differ from
those developed by epidemiologists in that I. A Behavioral SIR Model
economists include equations intended to Our behavioral SIR model, which is com-
capture the impact of endogenous changes monly used in the literature, is built on the
in human behavior undertaken in response standard SIR epidemiological model that is
to the pandemic on the progression of the used to interpret the data on deaths from
pandemic itself.1 We refer to such models COVID-19 for a given region.2 At each mo-
as behavioral SIR, or BSIR, models. ment of time, the population N is divided
We show that a simple BSIR model can into four categories (states) that sum to the
match an important feature of the data: the total population: susceptible S, infected I,
growth rates of daily deaths began at high resistant R, and dead D. The transmis-
and highly dispersed levels early on in the sion rate β(t) is the rate at which infected
pandemic and then fell toward zero fairly agents spread the virus to others that they
rapidly. However, despite this remarkable encounter at date t. We use R(t) to denote
match between model and data on growth the effective reproduction number of the dis-
rates of daily deaths, our simple model can- ease at date t, the ratio of the rate at which
not match many aspects of the evolution of infected agents infect susceptible agents to
∗ Atkeson: Department of Economics, University of the recovery rate γ of infected agents. The
California, Los Angeles, NBER, and Federal Reserve effective reproduction number can fall ei-
Bank of Minneapolis, andy@atkeson.net. Kopecky: ther due to changes in the normalized trans-
Federal Reserve Bank of Atlanta and Emory Univer- mission rate, β(t)/γ, or changes in S/N .
sity, karen.kopecky@atl.frb.org. Zha: Federal Re-
serve Bank of Atlanta, Emory University, and NBER, By inverting the standard SIR model to
zmail@tzha.net. We are grateful to Hongyi Fu for interpret data on deaths, one can derive
superlative research assistance. The views expressed a simple linear relationship between the
herein are those of the authors and do not necessarily growth rate of daily deaths and the effec-
reflect the views of the Federal Reserve Banks of At-
lanta and Minneapolis, the Federal Reserve System, or
2 Details of this SIR model, as well as the BSIR
the National Bureau of Economic Research.
1 See Atkeson, Kopecky and Zha (2020) for the ex- model, are presented in Atkeson, Kopecky and Zha
tensive references to the economics literature. (2020) and Droste and Stock (2021).
12 PAPERS AND PROCEEDINGS MONTH YEAR
tive reproduction number scribing behavior as
d2 D(t)
1 dt2 (3) Y (t) = exp −κḊ(t) + ψy (t) ,
(1) R(t) = 1 + dD(t)
,
γ dt
where κ > 0 represents the semi-elasticity
of activity Y (t) with respect to daily
Our behavioral SIR model assumes that
deaths. The variable ψy (t) represents a
the transmission rate β(t) is a function of
time-varying shifter to the region-specific
human activity Y (t)
relationship between deaths and activity
(2) β(t) = β̄Y (t)α exp(ψβ (t)). that may be induced by lockdowns or
changes in behavior in response to the dis-
The parameter β̄ in equation (2) is a fixed ease.
coefficient that captures features of a re- By substituting equation (3) into equa-
gion’s population and environment deter- tion (2), we obtain a reduced-form rela-
mined prior to the pandemic that might tionship between the current level of daily
impact transmission. Factors considered in deaths and the transmission rate, given by
the literature include population density,
modes of transportation, household and de- (4) β(t) = β̄ exp(−ακḊ(t) + ψ(t)),
mographic structure, cultural norms (bow-
where ψ(t) ≡ αψy (t) + ψβ (t) is the compos-
ing, shaking hands or kissing), and temper-
ite wedge.
ature and humidity. The parameter α cap- II. Estimation
tures the elasticity of transmission with re- In Atkeson, Kopecky and Zha (2020), we
spect to activity. discuss how to recover consistent estimates
The parameter ψβ (t) represents a po- of the growth rate of daily deaths, the level
tentially time-varying wedge shifting the of daily deaths, and the level of cumulative
region-specific relationship between activity deaths from noisy reported data on daily
and transmission. This wedge may repre- deaths.3 With the estimated death growth
sent the impact of policies such as mask- and level, one can study the epidemiological
wearing, ventilation, physical distancing, dynamics implied by our BSIR model. For
redesign of workspaces, or other measures this exercise we also need to set values of
implemented after the start of the pandemic several model parameters. Specifically, we
that reduce transmission given a fixed level set ν = 0.005, a level used by the CDC for
of activity, and/or natural variation in the those aged 50-69.4 The level of the fatality
transmission of the virus over time such as rate, if held constant, does not impact our
variation driven by seasonality or mutation estimates of the evolution of the effective
of the virus. reproduction number in equation (1).
The rest of the parameters are set or es-
The behavioral component of the model timated as follows. We set γ = 0.2. We set
assumes that individuals’ decisions to en-
gage in activity in a given region at time 3 Atkeson, Kopecky and Zha (2020) study 103 regions
t, Y (t), are a declining function of the (34 states and 69 countries) with the estimation period
starting at the location-specific date when cumulative
time derivative of cumulative deaths, Ḋ(t), deaths reached 25 and ending on November 12, 2020.
which is measured by the current level of 4 See Table 1 at https://www.cdc.gov/coronavirus/
daily deaths. We specify this function de- 2019-ncov/hcp/planning-scenarios.html.VOL. VOL NO. ISSUE BEHAVIOR AND COVID-19 3
α = 2 for all locations to capture the idea
that the number of interactions between in-
dividuals goes up with the square of the
activity level of all individuals. We allow
the semi-elasticity κ to vary by location, de-
pending both on individuals’ opportunities
to reduce activity by working from home
and on their beliefs about personal trade-
offs involved in exposing themselves to virus
transmission. We normalize ψβ (0) = 0 and
the level of activity at the beginning of the
pandemic to Y (0) = 1. Given these nor-
malizations, the region-specific parameter Figure 1. : Location and sampling uncer-
β̄ determines the transmission rate of the tainty.
virus in a given region at the beginning of
the pandemic, with the basic reproduction
number of the virus in this region given by 8
10
-3 Phase Diagram for St and It
R0 = β̄/γ. 7
To solve the BSIR model, we set the ini- 6
tial fractions of susceptible, infected, recov-
ered, and dead at date t0 > 0 to their
5
Infections It
values based on estimated daily deaths in 4
the data and the SIR model as described 3
·
I=0
in Section I. For each value of κ, β̄ is set 2
such that the transmission rate at date t0 1
in the BSIR model corresponds to the rate
implied by the SIR model and the death 0
0.1 0.2 0.3 0.4
S̄ =
γ
0.5 0.6
Suscepitbles St
0.7 0.8 0.9 1
β̄
data. This is achieved by setting β̄ =
β data (t0 ) exp(ακḊdata (t)). After setting the
Figure 2. : Phase diagram for the dynamics
initial phase, for each location, we choose
implied by the standard BSIR model.
κ to minimize the distance between daily
deaths in the data as given by our Bayesian
COVID-19 fell rapidly almost everywhere
estimation procedure and daily deaths gen-
within the first 30 days after each region
erated by the BSIR model with no wedges
PT h i2 reached 25 cumulative deaths. The solid
such that t=t0 Ḋdata (t) − Ḋ(t) is mini- black line in Figure 1 shows that the me-
mized, where T indicates the terminal date. dian estimated growth rate of daily deaths
fell rapidly from an initial level of about
III. Key Findings and Challenges 12 percent to zero within the first 30 days
Our estimation yields important findings of the estimation period. During this ini-
about the COVID-19 pandemic that pose tial phase, there was a wide dispersion of
challenges to future empirical work. growth rates of daily deaths observed in
Finding I One key finding, documented most of 103 locations; then there was a
in Atkeson, Kopecky and Zha (2020), is regime shift and the dispersion remained
that the growth rate of daily deaths from in a relatively narrow range around zero4 PAPERS AND PROCEEDINGS MONTH YEAR
into mid November of 2020.5 This fact was the path of I(t) and hence the path of daily
true even with substantial second and third deaths Ḋ(t) are both single-peaked; (b) af-
waves of daily deaths observed in many lo- ter the peak of infections and daily deaths,
cations in the fall and winter of 2020. these outcomes cannot fall rapidly to a low
According to equation (1), the effec- level without a substantial decline of S(t).
tive reproduction numbers, as well as our Thus, with no wedges, the model cannot ac-
model-implied disease transmission rates, count for multiple waves of infections and
have a similar pattern to Figure 1. That daily deaths or the patterns of daily deaths
is, they fell sharply in the early phase of seen in many of the regions that were ini-
the pandemic from widely dispersed initial tially hard hit followed by a steep decline in
levels and have since remained close to 1.6 daily deaths to very low levels (Finding I).
A challenge for empirical work is how to ac- Finding III The distribution of the
count for this stark switch in regime from growth rates of daily deaths for all the re-
the early period to the remaining period of gions we study, generated by our estimated
the pandemic. BSIR model without wedges and reported
Finding II The key dynamics of the in the top panel of Figure 3, is qualitatively
BSIR model can be summarized by the similar to the data (Figure 1) in that the
phase diagram presented in Figure 2 with growth rates of daily deaths fall toward zero
S(t) on the x-axis and I(t) on the y-axis. (the effective reproduction numbers fell to-
The black curve is the locus of points (S, I) ward one) over the course of 30 days or
such that I˙ = 0. When (S, I) lies above less—a regime shift. But the dispersion in
the black curve, I˙ < 0 and Ṡ < 0. When the growth rates of daily deaths predicted
(S, I) lies below the black curve, I˙ > 0 and by the BSIR model after the initial period
Ṡ < 0. The red curve in Figure 2 shows the is substantially smaller than the dispersion
model-implied path of (S(t), I(t)). The pair in the growth rates of these deaths observed
(S(t), I(t)) starts in the lower left corner of in the data. Unlike the data, in the BSIR
the figure with S(0) very close to one and model without wedges, there are no realiza-
I(0) positive but very close to zero. I(t) tions of growth rates substantially above or
rises rapidly initially and crosses over the below zero. As a result, both the magni-
black locus of points such that I˙ = 0 and tude and the dispersion of the wedges grow
then falls slowly, remaining above that lo- larger over time (bottom panel of Figure 3).
cus until S(t) falls below S̄ where the locus The main reason for growing wedges
of points such that I˙ = 0 intersects the x- in both magnitude and dispersion is that
axis and I(t) asymptotes to zero. the relationship between the level of daily
These dynamics imply two results: (a) deaths and the logarithm of the trans-
mission rate in the standard BSIR model
5 The
(equation (4)) exists only for the initial pe-
dashed lines in Figure 1 represent the 68% and
95% posterior probability intervals, which include both
riod of the pandemic. After the initial pe-
location uncertainty and sampling uncertainty. Most of riod, however, there is little apparent rela-
the cross-sectional dispersion in growth rates in the fig- tionship. California (Figure 4), for exam-
ure, however, is driven by location uncertainty as sam- ple, illustrates a one-to-one relationship be-
pling uncertainty within a location is very small.
6 This result is robust to various extensions of the tween β(t) and daily deaths in the initial
standard SIR model and consistent with the findings by period (the circled blue line) but no one-
the IHME at the University of Washington. to-one relationship after this initial periodVOL. VOL NO. ISSUE BEHAVIOR AND COVID-19 5
100 6
and U.S. states we examine. Accounting
80 5
60 4
for such a regime shift in behavior in an eco-
40 3 nomic model is a necessary but challenging
20 2 task.
0 1
-20 0
IV. Going Forward
0 20 40 60 80 100 120 140 160 180 200
Qualitatively, the push by economists to
5 introduce theories of behavior into epidemi-
4
3
ological models of COVID-19 has been a
2 big empirical success. Even a simple BSIR
1
0
model matches the main features of the dy-
-1 namics of the growth rate of deaths ob-
-2
0 20 40 60 80 100 120 140
served in many locations around the world.
160 180 200
As our findings demonstrate, however,
Figure 3. : Top panel: outcomes from the much of the dynamics of the growth and
BSIR model without wedges. Bottom panel: level of deaths is left unexplained by the
wedges. standard BSIR model. Such a model must
-1 be augmented with very large wedges to
the transmission rate, while holding disease
-1.1 prevalence constant, to match the data on
deaths. The next immediate task is to iden-
-1.2
tify and estimate possible variations of be-
-1.3
havioral parameters such as α, ν, γ, and κ
across regions and over time, and to study
-1.4 whether these variations can account for
part of the widening wedges we find. A
-1.5 more challenging task for future research
is to assess how much of the regional and
-1.6
time variations in behavioral parameters is
-1.7
due to nonpharmaceutical policies versus
0 20 40 60 80 100 120 140
changes in voluntary behavior (e.g., wear-
ing masks and practicing social distancing)
Figure 4. : Relationship between the model- as well as versus changes in the disease it-
implied ln β(t) and the data on daily deaths self.
Ḋ(t) for California.
REFERENCES
7
(the starred red line). This pattern, which
Atkeson, Andrew, Karen Kopecky,
is suggestive of a regime shift in behavior
and Tao Zha. 2020. “Behavior and
over time in response to the deadly pan-
the Transmission of COVID-19.” Unpub-
demic, is common across all the countries
lished Manuscript.
7 Measuring Y (t) with Google mobility data, we find Droste, Michael, and James H.
a remarkably similar pattern: the one-to-one relation- Stock. 2021. “Adapting to the COVID-
ship between β(t) and Y (t) (equation (2)) or between
Y (t) and Ḋ(t) (equation (3)) breaks down after the ini-
19 Pandemic.” American Economic Re-
tial phase of the pandemic. view (2021, this issue).You can also read
