Epidemics with Behavior
Satoshi Fukuda∗

Nenad Kos†

Christoph Wolf‡

February 22, 2021

Abstract
We study equilibrium distancing during epidemics. Distancing reduces the individual’s probability of getting infected but comes at a cost. It creates a singlepeaked epidemic, flattens the curve and decreases the size of the epidemic. We
examine more closely the effects of distancing on the outset, the peak and the final size of the epidemic. First, we define a behavioral basic reproduction number
and show that it is concave in the transmission rate. The infection, therefore,
spreads only if the transmission rate is in the intermediate region. Second, thepeak of the epidemic is non-monotonic in the transmission rate. A reduction in
the transmission rate can lead to an increase of the peak. On the other hand, adecrease in the cost of distancing always flattens the curve. Third, both an increase in the infection rate as well as an increase in the cost of distancing increase
the size of the epidemic. Our results have important implications on the modeling of interventions. Imposing restrictions on the infection rate has qualitatively
different effects on the trajectory of the epidemics than imposing assumptions on
the cost of distancing. The interventions that affect interactions rather than thetransmission rate should, therefore, be modeled as changes in the cost of distancing.
Keywords: epidemics; equilibrium distancing; transmission rate; interventions
JEL Classification Numbers: I12; I18; C73∗

Bocconi
University,
DepartmentofDecision
Sciences
and
IGIER.
Email:
satoshi.fukuda@unibocconi.it.
†
Bocconi
University,
Departmentof
Economics,
IGIER
and
CEPR.
Email:
nenad.kos@unibocconi.it.
‡
Bocconi University, Department of Economics and IGIER. Email: christoph.wolf@unibocconi.it.

1Introduction

The earliest account of an epidemic is the Plague of Athens—an epidemic that tore
through Athens 430 BC.1 The epidemic played an important role in the weakening ofAthens and the consequent victory of the Spartans; a detailed account is provided in
History of the Peleponnesian war by Thucydides. Outbreaks of epidemics have beena mainstay ever since, as has been their depiction in literature.2 Epidemics, however,are not merely a remnant of distant history or creative artistic representation. Not
a quarter way into the 21st century, the world has faced outbreaks of Severe AcuteRespiratory Syndrome (SARS), avian flu, swine flu, Ebola, Middle East Respiratory
Syndrome (MERS), and COVID-19.
The systematic study of epidemics and how they spread through the population gained
traction with the SIR model of Ross and Hudson (1917) and Kermack and McKendrick(1927). A disease is introduced into a population of susceptible individuals by a small
seed of infected—and infectious—persons. The disease spreads through meetings of individuals. The infected eventually recover. By studying the transmission and the recovery
rate of the disease, the model provides predictions on whether a disease will spread, what
peak prevalence it will attain and what proportion of population will be afflicted. Yet,when it comes to individuals’ behavior the model implicitly assumes that individuals engage in as many interactions at the height of the epidemic as they do when the disease isbarely present. The model of Capasso and Serio (1978) generalizes the standard model
to non-linear or time-dependent contact rates, which can be thought of as a reduced formof modeling behavior; for a more recent overview, see Funk et al. (2010) and Verelst et al.
(2016). A strong point for explicitly modeling behavior is made by Ferguson (2007).
We propose a tractable model of equilibrium distancing during an epidemic.3 Susceptible individuals non-cooperatively decide whether, and to which extent, to distanceor to engage in exposure at each point in time. Distancing comes at a quadratic cost,
but reduces the probability of getting infected. The cost of getting infected is fixed; amodel with fixed cost of infection was introduced by Chen (2012). We show that optimal
distancing is proportional to the current size of the infected population. More infected in1

While termed the Plague, it is not clear what the disease was. Examination of DNA in dental pulp
suggests it could have been typhoid fever; see Papagrigorakis et al. (2006).
2
The Plague plays an important role in Decameron by Boccaccio, Romeo and Juliet by Shakespeare,
and The Plague by Camus. A case of smallpox is vividly depicted in Dickens’ Bleak House. Besides
the disease, these works provide insights into how individuals behave to avoid the disease and what
preventive steps authorities undertake. For example, in Romeo and Juliet friar John is detained whentrying to deliver Juliet’s letter from Verona to Rome in Mantua. The movements between the two cities
are limited to stop the spread of the plague.
3
Ours is not the first model of behavior during an epidemic. An account of the related literature
follows below.

1dividuals imply a higher likelihood that one gets infected and, therefore, more distancing.
The difficulty in fully characterizing the equilibrium is that the prevalence of the diseasedepends on previous distancing behavior by all individuals. Nevertheless, we show thatan equilibrium exists and that it is unique. Depending on parameters, two types of dynamics can arise. The disease either never takes off or it spreads through the population.
In the latter case, the epidemic has a single peak: it propagates through the population
until it reaches the peak prevalence, then it recedes and eventually dies out. Importantly,
susceptible individuals distance throughout the epidemic, though the intensity of theirdistancing varies with the number of actively infected individuals. While distancing affects the entire path of the epidemic, it has important consequences for three crucial andcommonly discussed measures: the conditions for an epidemic to start, its peak, and its
final size.
First, we define a basic reproduction number taking distancing into account—the
behavioral basic reproduction number. The behavioral basic reproduction number consistsof the classical, epidemiological basic reproduction number, R0 , multiplied by a behavioral
term.4 We show that the disease propagates itself if and only if the behavioral basic
reproduction number is larger than one. Interestingly, the behavioral basic reproduction
number is concave in the transmission rate. Therefore, the disease spreads only forintermediate values of the transmission rate. This novel finding stands in stark contrast
with the predictions offered by the SIR model without distancing (see, for example, Brauer
and Castillo-Chavez, 2012) where a disease spreads if its transmission rate is sufficiently
high. With distancing, a higher transmission rate is required for the disease to take off
as the caution of individuals lowers the behavioral basic reproduction number. However,if the transmission rate is too high, individuals distance with such fervor that the disease
never takes off.5
Second, we derive results pertaining to peak prevalence of the disease. Peak prevalence
is of crucial importance to understand whether a disease might cause the health system
to reach its capacity. For example, the 1918 influenza pandemic hit an unprepared healthsystem which soon became overwhelmed (see Jester et al., 2018; Schoch-Spana, 2001). In
March 2020—less than a month after the coronavirus erupted in Italy—, the healthcare
system in Northern Italy was under such severe pressure that some pneumonia patients4
We also derive an analogous behavioral effective reproduction number. That the basic reproduction
number itself may be misleading to understand epidemic dynamics has been recognized before. For
example, Caley et al. (2008) find that the observed attack rate of the 1918-1919 influenza pandemic wassubstantially lower than the one expected based on the basic reproduction number and attribute thisdiscrepancy to social distancing.
5
The importance of behavior at the outset of an epidemic for the entire trajectory of the epidemic is
discussed, for example, in Brauer (2019).

2

could not be treated.6 Hence, the goal of many officials and societies became to flatten
the curve to avoid the active number of infected individuals exceeding the health caresystem’s capacity.7 To improve the understanding of the effects that behavior has onpeak prevalence, we study analytically how changes in the disease’s transmission rateand changes in the cost of distancing affect the peak prevalence of a disease.
An increase in the cost of distancing unequivocally leads to a reduction in distancing
and therefore to a higher peak prevalence of the disease. However, peak prevalence is
non-monotonic in the transmission rate. If the transmission rate is high enough for the
disease to spread but not too high, an increase in the transmission rate leads to anincrease in the peak prevalence. In contrast, when the transmission rate is sufficiently
high, an increase in the transmission rate decreases the peak prevalence and causes a
flattening of the curve.8 A decrease in the transmission rate—which in the SIR model
without behavior always reduces peak prevalence—may have undesirable consequencesin the short run by raising peak prevalence. Intuitively, there are two forces at play when
the transmission rate is lowered: (i) conditional on behavior, the spread of the disease isslower, (ii) conditional on a particular prevalence, individuals face a lower infection risk
and engage in less protective distancing. We show that the latter affect may outweighthe former leading to a higher prevalence of the disease.9
It is important to emphasize that the same cannot occur with a decrease in thecost of distancing. To the best of our knowledge, the body of work that studies nonpharmaceutical interventions models these either as reductions in the transmission rate(see, for example, Kruse and Strack, 2020; Rachel, 2020a) or as directly choosing thesocial activity level of individuals (see, for example, Acemoglu et al., 2020; Alvarez et al.,forthcoming; Farboodi et al., 2020)—which are equivalent approaches in the SIR dynamics without behavior. Our results suggest that modeling individual distancing choices
explicitly requires a careful choice of modeling interventions as qualitative implications
differ through the behavioral channel. On the one hand, those interventions affecting therate at which the disease propagates conditional on meetings, e.g., mandatory mask mandates, should be modeled as a decrease in the transmission rate.10 On the other hand,6

See https://www.nytimes.com/2020/03/12/world/europe/12italy-coronavirus-health-care.html.
McCabe et al. (2020) provide a calibration assessment of the risk of health system capacities being
exceeded in the winter of 2020/21 due to COVID-19 patients in different European countries undervarious levels of lockdown effectiveness. They highlight that capacity constraints may become a severe
concern again.
8This finding is in accord with the result that for too high transmission rates individuals distance somuch that the disease does not spread in the first place.
9
This intuition is reminiscent of risk compensation introduced by Peltzman (1975): an intervention
that improves an individual’s security can lead to more risky behavior. For a survey, see Hedlund (2000).
10Note that this result does not necessarily imply that mandating mask wearing in public spaces will
worsen the epidemic; it may flatten the curve as well. However, we want to highlight the possibility ofthis perverse effect arising. Indeed, Chernozhukov et al. (2021) show that mask mandates have reduced
7

3

those interventions that directly affect the incentives to distance, e.g., restaurant, bar or
museum closures, should be modeled as a decrease in the cost of distancing. Awarenessof these potential detrimental effects of lowering the transmission rate is particularly important given that analytical results are rare and the majority of work on optimal policy
is computational. Getting parameters in the calibration wrong can have adverse effects
on the consequences of policy advice.
Third, we find that the possible detrimental short-run effects of a decrease in thetransmission rate disappear in the long run. The total number of infected individuals
throughout the epidemic is monotonically increasing in both the cost of distancing and thetransmission rate. Conversely, the limiting number of susceptible individuals is decreasingin these parameters, but smaller than the ratio of recovery and transmission rate. In the
SIR model without distancing, the number of infected individuals starts decreasing once
the number of susceptible individuals is sufficiently low, in particular, once it falls below
γ/β. When the number of susceptible individuals is too small, the pool of infectedindividuals is being depleted due to the rate of recovery being greater than the inflow of
newly infected individuals. The number of susceptibles converges to a number strictly
larger than 0 and smaller than γ/β; for a derivation, see Brauer and Castillo-Chavez(2012). Our model predicts a larger final size (more susceptibles, i.e., less total infections)than the standard SIR model due to distancing. Indeed, our model converges to the SIR
model without distancing when the cost of distancing grows and so does the final size ofthe epidemic. Notably, as long as the cost of distancing is large enough for the disease to
spread, the final size is below γ/β—even with distancing.
With these findings, we highlight an important trade-off between short-run mitigation,i.e., flattening the curve to avoid an overburdened health system, and long-run size of
epidemics when considering the transmission rate. This trade-off arises due to the varying
degree to which behavior matters during an epidemic. At the peak, the infection risks
are high and individuals’ distancing decisions have a strong impact on the dynamics
of the epidemic. When an epidemic fades out, however, behavior is of less importanceas individual risks are low and the standard SIR mechanics dominate the behavioral
effects. However, the trade-off disappears once policies are considered that directly affectdistancing incentives of individuals and both short-run mitigation and long-run size ofthe epidemic are obtained with similar policies, i.e., lowering the cost of distancing.
Next we provide bounds for the relevant values of cost of getting infected by studying amodel in which the cost of infection is endegenously derived at each point in time. Finally,
we provide a connection between models with explicit distancing decisions of individuals
the number of COVID-19 cases and deaths in the US.

4and models that generalize the contact and transmission rates of the standard SIR model,
as in Capasso and Serio (1978).
Related literature. We provide a brief account of the related literature. We apologize for any omissions that might arise due to the speed with which new work is produced.
Sethi (1978) introduces a behavioral component in an SIS model, but analyzes only theplanner’s problem. For subsequent references on behavior in SIS models, see Toxvaerd(2019).
Reluga (2010) proposes an SIR model with behavior and provides mostly numericalresults. Chen (2012) introduces an SIR model with a constant cost of infection, similar
to what we analyze, but a more general contact function. His focus is on conditions on
the contact functions that deliver uniqueness of the Nash equilibrium in each period for agiven prevalence of the disease. Fenichel et al. (2011) and Fenichel (2013) study a modelin which the cost of getting infected is endogenous. They derive necessary conditions forequilibria and perform numerical analysis. Rachel (2020a) and Toxvaerd (2020) analyze
a model of behavior with a linear cost of distancing and an endogenous time-varying costof getting infected.11 They derive the necessary conditions for an equilibrium and derive
two different paths that satisfy the necessary conditions. Farboodi et al. (2020) study asimilar model numerically. Dasaratha (2020) analyzes a model similar to ours where theinfected individuals do not necessarily know whether they are infected. The complexityof his model requires that he mostly focuses on local results. Among other things, heshows that a marginal decrease in transmission rate around the peak of epidemic can leadto an increase in prevalence. McAdams (2020) proposes a model in which an individual’s
benefit of social activities depends on the actions of other individuals and shows that
there is a unique equilibrium of social activity choices in each period. McAdams (2021)
provides an excellent account of the rapidly growing literature.
The majority of the literature on behavior and policy over an epidemic focuses on
numerical results and simulations. Makris and Toxvaerd (2020) numerically study how
the expectation of the arrival of a pharmaceutical innovation affects individuals’ optimal distancing. Toxvaerd and Rowthorn (2020) compare the individual and planner’s
decisions to apply treatments and vaccinations as pharmaceutical interventions during
an epidemic. Giannitsarou et al. (forthcoming) provide numerical projections for the
COVID-19 pandemic, based on a model with endogenous distancing. Acemoglu et al.
(2020) analyze optimal lockdowns as a direct reduction of individuals’ activity levels andcalibrate the model to the COVID-19 pandemic in the US.
11

Rachel (2020b) builds on this work to study lockdown effectiveness and the possibility of a second
wave occurring.

5

2

The Model

We study behavior in an otherwise standard SIR model. A continuum of individuals,
indexed by i and normalized to unity, are infinitely lived with time indexed by t ∈ [0, ∞).
Each individual can be in one of the three states: susceptible, infected (and infectious),or recovered. Susceptible individuals might get infected in which case they transition
into the infected state. Infected individuals can recover, but cannot become susceptible
again. Recovered individuals acquire permanent immunity.12 We denote the share of the
population that is susceptible at time t by S(t), infected by I(t) and recovered by R(t).
At each moment in time, susceptible individual i chooses how much activity to engage
in, denoted by εi (t) ∈ [0, 1]. The individuals enjoy the activity, but it exposes them tothe danger of infection; hence, termed exposure. The converse, di (t) := 1 − εi (t), is themeasure of distancing an individual engages in. While susceptible, the individual incurs
a flow playoff, πS . Distancing is uncomfortable and comes at the cost 2c (di (t))2 . Getting
infected comes at a cost −η, η < 0. An individual discounts the future at rate ρ > 0.
Individuals meet through a pairwise-matching technology where each individual has anequal chance of meeting any other individual—regardless of which state they are in. The
only matches of consequence—that is, the only matches with an infection risk—are theones between a susceptible and an infected individual. The probability that a susceptible
individual who chooses exposure level εi (t) meets an infected individual at time t isεi (t)I(t). The rate at which the infection is transmitted, upon such a match, is β > 0.
The rate at which a susceptible individual i who chooses exposure εi (t) gets infected is,therefore, βεi (t)I(t).13 Finally, infected individuals recover at rate γ > 0.14
At each point in time t, a susceptible individual i solves the problemcmax πS − (1 − εi (t))2 + βI(t)εi (t)η.
εi (t)∈[0,1]21Let ε(t) := S(t)R

ε (t)di be the average exposure at time t.
i∈S(t) i

12(1)

Analogously, define

The standard SIR model is suitable for viral diseases which are transmitted directly from human tohuman in a given period of time. Such viral diseases include measles, chickenpox (varicella), mumps,
rubella, smallpox, influenza, poliomyelitis, Ebola virus disease, SARS, MERS, and COVID-19.
13
We implicitly assume that infected individuals choose full exposure. Though strong, the assumption
is not as stark as it might at first seem. It is straightforward to accommodate exposure of infected withsome parameter e, as long as it is fixed over time. Then, the same model as ours can be obtained bydefining β̃ = eβ.
14
Following Keeling and Rohani (2008), it is straightforward to incorporate fatalities from the disease
into the SIR model by introducing a probability of death before recovery, σ, and rescaling the recovery
rate γ to γ/(1 − σ).

6d(t) := 1 − ε(t) as the average distancing at time t. Then, the model is governed by thefollowing dynamics
Ṡ(t) = −βε(t)I(t)S(t),˙ = βε(t)S(t)I(t) − γI(t),
I(t)

(2)

Ṙ(t) = γI(t),

(4)

(3)

with the assumption that there is a seed of infected, I(0) = I0 ∈ (0, 1), and susceptible
individuals, S(0) = S0 = 1 − I0 . The number of susceptible individuals is ever shrinking.
The number of infected individuals, on the other hand, is growing as long as the inflow
of infections, βε(t)S(t)I(t), is larger than the outflow of recoveries, γI(t), and shrinking
˙ = −γI(t). The net inflow of individuals
otherwise. It should be noted that Ṡ(t) + I(t)into the pool of susceptibles and infected is negative due to recoveries. Needless to say,since S, I and R are the only three states
S(t) + I(t) + R(t) = 1
at each instance of time.
Definition 1. A symmetric equilibrium (an equilibrium, for short) is a tuple of functions(S, I, R, (εi )i∈[0,1] ) with the following three properties: (i) (S, I, R) follow (2), (3) and (4)with the initial condition (S(0), I(0), R(0)) = (S0 , I0 , 0), where ε is the average exposure;(ii) each εi solves (1), that is, εi is a best-response to (S, I, R), where the average exposure
ε is induced by (εj )j6=i ; and (iii) ε = εi for all i ∈ [0, 1].

The first-order condition to the individual’s problem yields the individual’s optimal
distancing choiceηβdi (t) := 1 − εi (t) = min − I(t), 1 .
c(5)When − ηβI(t) exceeds unity, individuals fully distance. Distancing at time t dependsconly on the infected population at time t—up to constants β, c and η. In a symmetricequilibrium, εi = ε for all i. By equation (5), exposure in a symmetric equilibrium isηβε(t) = max 1 +I(t), 0 .
c

7Plugging this equation into the SIR dynamics yields
ηβI(t),0 ,Ṡ(t) = −βS(t)I(t) max 1 +cηβI(t)
˙ = βS(t)I(t) max 1 +I(t), 0 − γI(t),cṘ(t) = γI(t),

(6)
(7)
(8)

with the initial condition (S(0), I(0), R(0)) = (S0 , I0 , 0) and I0 = 1 − S0 ∈ (0, 1). Since
this is an initial value problem, the following result obtains. All the proofs are collectedin the Appendix.
Proposition 1. A symmetric equilibrium exists and is unique.

3

Analysis of the SIR Model with Behavior

While the above system of differential equations does not have a tractable closed-form
solution, we nevertheless establish qualitative properties of the equilibrium and the resulting epidemic dynamics. In this section, we proceed in four steps. First, we deriveconditions for an outbreak to occur and show that, in that case, any epidemic is singlepeaked. Second, we derive an implicit equation for the solution path in the (S, I)-phasespace. Third, we show how peak prevalence is affected by changes in the transmissionrate, β, and the cost of distancing, c. Fourth, we derive how the final size of the epidemic
is affected by the transmission rate and the cost of distancing.

3.1

Single-peaked Epidemic and Its Onset

We start by showing that the number of active cases peaks at most once.
˙ t̂) = 0, then I(¨ t̂) < 0 and at t̂ distancing is maximized.
Proposition 2. If t̂ is such that I(

Proposition 2 implies that if I has a critical point, this critical point has to be a local
maximum. Together with the continuous differentiability of I, this implies that I canhave at most one peak. The infection either immediately dies out, or becomes an epidemic
with a single peak. Moreover, distancing is maximized at the peak of the epidemic.
In the standard SIR model, the infection propagates itself only if the basic repro-8duction number, R0 := βγ S0 , is larger than 1; see Heesterbeek and Dietz (1996).15 The
basic reproduction number is the number of infections an infected person would cause
during her infection if introduced into a population of susceptible individuals of size S0 .
Namely, each infected causes βS0 transmissions per unit of time while her infection lasts
1/γ periods on average. From a practical point of view, the observed and measurable
variable is how many secondary infections have been caused given an individual’s behavior. Therefore, incorporating the behavioral component in the basic reproduction number
is of paramount importance. For a discussion of this issue—diseases that after an outbreak grows more slowly than expected—, see, for example, Brauer (2019). We define
the behavioral basic reproduction number as:
R0b :=βS0 ε(0).
γ(9)Notice that R0b = ε(0)R0 . That is, the behavioral basic reproduction number consistsof the virus-inherent basic reproduction number R0 = βγ S0 and a behavioral factor ε(0).
˙Equation (3) at t = 0 can now be rewritten as I(0)= Iγ0 (R0b − 1). Therefore, the˙infection spreads, I(0)> 0, if and only if R0b > 1, paralleling a similar resort in the model
without distancing. However, while in the standard SIR model R0 is increasing in β,
the behavioral basic reproduction number R0b is non-monotonic here and, in particular,
concave. This follows from ε(0) = 1 + ηβ
I and (9); recall that η < 0. This finding has
c 0important implications on which types of an infection will spread.
˙
Proposition 3. Fix I0 ∈ (0, 1). Then, I(0)> 0 if and only if R0b > 1. Moreover:γ
˙
1. If I0 < 1−14ηγ , there exist β and β, with 1−I< β < β < − ηIc0 , such that I(0)> 0 if0c

and only if β ∈ (β, β). In that case ε admits an interior solution for all t.
˙ ≤ 0 for all t.
2. If I0 ≥ 1−14ηγ , then I(t)c

In the standard SIR model, for the infection to spread, β must be high enough. In
particular, β > Sγ0 . In the model with distancing, instead, the transmission rate has tobe large enough to also overcome the initial distancing:
β>

γ
γ
> .16
(1 − d(0))S0S0Thus, our model predicts that a higher transmission rate is needed for the epidemic tostart than in the standard SIR model. However, what differentiates the model with
15

Depending on the source R0 is defined either as β/γ or βS0 /γ. We use the latter definition as itallows for an easier presentation of results.
16
It should be noted that d(0) depends on β as well.

9Figure 1: The Onset of an Epidemic. Left: Constant cost of infection model; Right:
Standard SIR model. The solid line in each panel depicts the combination of (β, I0 ) with
˙
I(0)
= 0.
distancing from the standard SIR model even more starkly is that the infection peters
out if the transmission rate is too high. If the disease is highly contagious, individuals
are much more cautious, up to the point that their resolute distancing alone is sufficient
to stop the disease in its tracks from the onset.17 In the face of preventive behavior the
infection, therefore, spreads only if its transmission rate is large enough, but not toolarge, as illustrated in the left panel of Figure 1.18
Proposition 3 derives conditions on the transmission rate such that an epidemic takes
off. The same question can be analyzed along other dimensions. For example, the
CDC has adopted a categorization for influenza viruses along the severity-transmissibility
dimensions (see Reed et al., 2013). In our model, this can be interpreted as categorizing
the combination of the cost of infection, −η, and the transmission rate, β. The range ofparameters (β, −η) such that the disease takes off is depicted in Figure 2.
Recall that in the SIR model without behavior—which is nested in our model as
the case η = 0—the epidemic takes off whenever βS0 > γ. This corresponds to thered dashed line in the figure. As the cost of the infection, −η, increases, individuals’
distancing incentives start to matter for the onset of an epidemic. In particular, for a
fixed β, the higher the cost of infection, the more individuals engage in distancing. If thecost of infection becomes very large, it prevents the disease from spreading altogether:˙I(0)< 0. There is a cutoff cost of infection such that the disease will never spread when−η > 4γc SI00 as getting infected is so costly for individuals that their distancing behavior
will compensate any transmission rate β.
17
An informal discussion of the role of disease-intrinsic parameters and its effect on the outbreak ofan epidemic can be found in Christakis (2020).
18
We use parameters for COVID-19 in our simulations. A summary and justification of the parameterschosen can be found in Appendix B. We also describe our numerical algorithm there.

10Figure 2: The Onset of an Epidemic Along the Severity and Transmissibility Dimensions.
˙The solid line depicts the combination of (β, −η) with I(0)= 0. The red dashed line
depicts the set of transmission rates for the standard SIR model without behavior (which˙
has −η = 0) such that I(0)> 0.
It follows that a disease can only spread if its (β, −η)-combination is intermediate. For
a given β, the infection cost must not be too high; while for a given −η, the transmission
rate must neither be too high nor too low. The existence of an upper and a lower boundfor β follows the same intuition as the one applying for Proposition 3.

3.2

The Dynamics

It is often useful to think of the model in terms of a phase diagram in the (S, I)-space,a graph showing how the number of infected individuals changes with the number ofsusceptible individuals. To find the solution path (S, I) := (S(t), I(t))t≥0 in the phasespace, one derives the quotient differential equation
γ1
1
dI

= −1 +dS
β S max 1 + βηI,0c
by dividing equation (3) with equation (2) and using (5) for ε.

11

(10)

Proposition 4. Suppose d(0) < 1.19 The solution path (S, I) is implicitly determined by

S=
2

exp

exp
!


β η
c
1+
2γc
βη

2

β2η
2γc



c
S + I + βη

1
+ 2β
S0

s

(−η)
2γc

2 
.

Z β q −η (1+ c )
2γc

β

q

βη

e
−η
2γc

−v 2

(11)

dv

(S+I+ βηc )Figure 3: Solutions Paths for Different Transmission Rates.
Figure 3 depicts the solution paths in the phase space for different transmission rates.
It is apparent that these are not monotonically ordered.
To compare the dynamics of our model with the standard SIR model, denote byˆ R̂) the proportion of susceptible, infected, and recovered individuals in the standard(Ŝ, I,ˆ
SIR model. We take the same initial condition (Ŝ(0), I(0),R̂(0)) = (S0 , I0 , 0) and thesame parameters (β, γ). The dynamics of the standard SIR model is obtained by replacingˆ R̂, 1) in equations (2), (3) and (4). The solution path (Ŝ, I)ˆ of the(S, I, R, ε) with (Ŝ, I,
standard SIR model is captured bydIˆ
dŜ= −1 +

19

γ1
.
β Ŝ

(12)

The assumption is made for ease of exposition directly on d(0); Formula (5) provides the corresponding assumptions on primitives. Also, if d(0) = 1, then individuals engage in full distancing up to somepoint, after which an equation analogous to (11) determines the dynamics of the epidemic.

12

Figure 4: Solution Paths for the SIR Models With and Without Behavior. The solid curve
depicts the solution path for our SIR model with behavior. The dashed curve depicts thesolution path of the SIR model without behavior. The dotted vertical line depicts theherd-immunity threshold in the SIR model without behavior.
ˆ liesComparing equations (10) and (12), we can show that the solution path (Ŝ, I)
above the path (S, I) in the phase space. More generally, we show that the solution path
(S, I) moves upwards as the cost of infection (−η) decreases. The SIR model without
distancing can be recovered as the special case of our model with no cost of infection(η = 0). Hence, for any level of the susceptible population, the corresponding numberof active infections is lower in the SIR model with distancing. These comparative statics
are depicted in Figure 4.
˙
Proposition 5. Assume that I(0)> 0. The following hold:

1. As η increases, the solution path (S, I) moves upwards (lies above the original soluˆ of the standard SIR model, whichtion path). In particular, the solution path (Ŝ, I)
is associated with η = 0, lies above the original solution path.
2. Moreover, for every t, S(t) ≥ Ŝ(t) and R(t) ≤ R̂(t).

Increasing the cost of infection, −η, pushes the solution path down, that is, it decreases
the infected population at any level of susceptibles in the phase space. Since the SIR
13model without distancing can be seen as the special case of η = 0, its solution path isabove the path for any η < 0. In particular, the peak prevalence in our model is belowthat of the standard model.
While the first part of the above result compares the infected population I for eachlevel of the susceptibles S, the second part provides the comparison of the susceptibles
S and the recovered R at each point in time. Namely, the susceptible population S(t) in
our model is always at least as large as in the standard SIR model while the recovered
population R(t) is at most as large as in the standard SIR model. The latter suggests thatRt
in the model with behavior is smaller
the cumulative infected population 0 I(s)ds = R(t)γthan in the standard SIR model. This result is intuitive as a higher cost of infection
incentivizes individuals to take greater precaution in order to reduce their infection risk.

3.3

Peak Prevalence

The peak prevalence of an epidemic has profound consequences on the overall provision ofhealth care services. A large number of infected individuals may lead to an overwhelming
demand of personal protective equipment such as face masks and that of medical devices
such as ICU beds and ventilators. The shortage of medical resources, in turn, may cause
a suboptimal treatment and health care coverage.20 The high demands of the epidemic
on the health system also divert medical resources from other important activities suchas vaccination against other infectious diseases and deliveries. What is more, health-care
workers themselves are at high risk of infection.21 All this implies that the peak prevalence and how it is affected by parameters—in particular, by the disease’s transmission
rate but also by the individual’s distancing cost—are of paramount interest for epidemic
preparedness and optimal policy responses.
˙˙When the epidemic takes off (I(0)> 0), the prevalence is maximized when I(t)=0—Proposition 2 implies that the local maximum is also global—that is, whenε(t)S(t) = γ/β.

(13)

Let I ∗ := maxt I(t). Note that I ∗ together with the corresponding S ∗ is a solution to (11)20

As references regarding capacity constraints of the health care system see, for example, Schoch-Spana(2001) for the 1918 influenza pandemic, Ferguson et al. (2020) for the COVID-19 pandemic, and Reedet al. (2013) for influenza epidemics.
21
Elston et al. (2017) survey the health impact of the 2014-15 Ebola outbreak in West Africa. For
Siera Leone, they report a 20 % decrease in measles coverage, an overall 20-23 % decrease in deliveries
and Caesarian sections. They also report that 10.7 % of the health-care workforce were infected and 6.9% died from Ebola virus disease.

14

and (13). Despite this system of equations being intractable, the phase diagram analysis
allows for several important comparative statics.
ˆ∗ˆIn the standard SIR model with R0 > 1, the peak prevalence I := maxt I(t) is
given by Iˆ∗ = 1 − βγ + βγ log βSγ 0 ; see, for example, Brauer and Castillo-Chavez (2012) or
Hethcote (2008). The peak prevalence is attained when the population Ŝ(t) of susceptibles
reaches the threshold of herd immunity βγ . When the peak prevalence I ∗ of our modelis attained, the population S(t) of susceptibles is larger than βγ . Since the solution path
ˆ our model has a smaller peak prevalence than the SIR(S, I) is below the path (Ŝ, I),
model without behavior, I ∗ < Iˆ∗ .
We study how the peak prevalence changes with changes in the parameters β and c.
To focus on the case in which the infection can take place, we assume I0 < 1−14ηγ so thatc˙I(0)> 0 if and only if β ∈ (β, β); see Proposition 3. When this assumption fails, theinfection dies out so that I ∗ = I0 .
Proposition 6. The following hold:
1. Fix γ, c and η and let I0 < 1−14ηγ . Then, I ∗ is non-monotonic in β ∈ (β, β). In
cparticular, there exist β1 < β2 such that I ∗ is increasing in β for β ∈ (β, β1 ) anddecreasing in β for β ∈ (β2 , β).
2. The peak prevalence I ∗ is non-decreasing in c. It is strictly increasing in c whenever
˙
I(0)
> 0.
In the SIR model without behavior, the peak prevalence Iˆ∗ is monotonically increasing in the transmission rate β, as illustrated in the right panel of Figure 5: the moretransmissive a virus is, the higher the peak prevalence, provided that the infection takes
off. In contrast, in our model, a higher rate of transmission leads to, ceteris paribus,
more distancing. This effect can be so strong that a higher transmission rate reduces
the peak prevalence and flattens the infection curve. More precisely, the above result
establishes a non-monotonicity of peak prevalence in β. When the rate of transmission islow, the peak of the infection is increasing in β. However, when the rate of transmission ishigh, the peak prevalence decreases with β. The left panel of Figure 5 illustrates this nonmonotonicity of the peak in the transmission rate. Proposition 6 leads to a rather striking
conclusion: a measure imposed to fight the epidemic through a decreased β could havea daunting short-run effect; for example, if the potential resulting increase in prevalence
leads to stress of the health care system. We want to emphasize that this effect arises
only for a subset of potential parameters. In particular, Chernozhukov et al. (2021) show
15

Figure 5: Peak Prevalence as Function of Transmission Rate for the SIR Models withand without Behavior. The left panel depicts peak prevalence of the SIR model withbehavior. Initially, a higher transmission rate increases the peak. However, beyond acertain threshold, the peak decreases in the transmission rate as the distancing behavior
outweighs the direct effect of a higher transmission rate. The right panel depicts peakprevalence of the SIR model without behavior.
that the introduction of mask mandates, which we consider to be a β-reducing policy,
reduced the number of active cases in 2020 during the COVID-19 pandemic in the US.
At the same time, Knotek II et al. (2020) report survey evidence that some individuals
view mask wearing as a substitute for physical distancing. This issue has been discussed
in Howard et al. (2021).22
On the other hand, an increase in the cost of distancing always increases the peakprevalence. Not surprisingly, a higher cost of distancing leads to less distancing, all else
equal. The disparity in effects of changes in c and β on peak prevalence can be most
readily seen by studying how the slope of the solution path at a fixed point in the phasespace varies with changes in the two parameters. Differentiating the slope with respect
to the cost of distancing parameter yields
∂
∂c



dI
dS


=−

γ ∂ε
< 0,
βε2 S ∂cwhere the inequality follows from the observation that for a fixed I the exposure increases
if the cost of distancing increases. Importantly, the only effect an increase in the cost hason the solution path is through the change in distancing. By an implication, the slopeof a solution path with a higher cost is smaller than the slope of a solution path with asmaller cost of distancing at any point of intersection. The fact that they start from thesame point, (S0 , I0 ), then implies that everywhere else the solution path corresponding
22

This indirect effect of a measure reducing individual risk on taking less precautions is reminiscent ofrisk compensation. See, for example, Peltzman (1975) and Hedlund (2000).

16

to a higher cost must be above the one with the lower cost.
The change in the transmission rate, on the other hand, has a more nuanced effect.
Differentiating the slope of the solution path at a fixed point yields
∂
∂β



dI
dS


=−

γ
β 2 εS−γ ∂ε,βε2 S ∂βrevealing that an increase in β has two effects. A higher β makes the infection moreperilous. The increase in the transmission rate, holding everything else fixed, results inmore secondary infections from each infected individual, thereby increasing the velocityof the spread of the disease. Such a direct effect is absent from changes in the costof distancing. The second, indirect, effect is due to the response of distancing to thechange in the transmission rate. A more infectious disease results in more distancing andthus dampens the evolution of the epidemic. The two effects run in opposite directions.
Depending on which of the two dominates, an increase in β can lead to either a smaller
or a larger slope of the solution path.
The above finding has an important implication on how various preventive policies
should be studied in models with an epidemiological component. Such models commonly
adopt two apparatus: behavior is either modeled implicitly by changes in β in the standard SIR model or by directly imposing behavioral changes in models with behavior.23
Our results point to the importance of differentiating between changes in the transmission rate and changes in the cost of distancing. For example, if a government imposes
temporary restaurant closures to slow the spread of the disease, this gives individuals
fewer reasons to go out and should be modeled as a decrease in the cost of distancing,and not as a decrease in the transmission rate.24

3.4

Final Size of the Epidemic

In the long run, the epidemic dies out, I∞ := lim I(t) = 0. After S falls below γ/β, sot→∞does εS, necessitating a reduction in the incidence rate. An important long-run property of the disease is S∞ := lim S(t), the number of remaining susceptible individualst→∞once the epidemic is over. Note that S∞ is well-defined because S is weakly decreasing.
Conversely, 1 − S∞ is the size of the epidemic. In the SIR model without distancing,
23

Examples of models of the former include Brauer (2019); Capasso and Serio (1978); Kruse and Strack(2020) while examples of the latter include Acemoglu et al. (2020); Alvarez et al. (forthcoming); Farboodi
et al. (2020); Rachel (2020a).
24
Note that one can view holidays or vacation periods as an increase in the cost of distancing which
affects individual behavior as well.

17



γ 25
Ŝ∞ := lim Ŝ(t) ∈ 0,
. At the end of the epidemic, a strictly positive fraction oft→∞β
the population remains susceptible, Ŝ∞ > 0. The upper bound on this number is givenby γ/β. This follows immediately in the standard model without distancing as the number of infected individuals is increasing whenever Ŝ(t) > βγ . Only after this thresholdhas been reached, the share of susceptibles falls and the epidemic starts to falter. The
following result establishes how the model with distancing compares with respect to thesize of the epidemic.
Proposition 7. The following chain of inequalities holds:
β

0 < S0 e− γ ≤ Ŝ∞ ≤ S∞ <γ.
βFigure 6: Final Size of Susceptibles as Function of Transmission Rate. The solid blue
curve depicts the final size of susceptibles in our SIR model with behavior. The red
dotted line shows as a comparison the final size in the SIR model without behavior whichlies always below the solid blue curve. The dashed black line depicts the herd-immunity
threshold that is the upper bound for the final size of susceptibles in both models.
Not all too surprisingly, the model with distancing predicts a smaller final size ofthe epidemic, 1 − S∞ , than the model without distancing. More surprising is the fact
that—even with distancing—the final number of susceptibles, S∞ , cannot exceed γ/β.
25

In particular, see e.g., Brauer and Castillo-Chavez (2012) or Hethcote (2008), 1 − Ŝ∞ = βγ log

18



S0
Ŝ∞



.

The reason is that in the limit I tends to 0. If S∞ was strictly above γ/β, then, as the
epidemic would be winding down, so would the distancing. Since the exposure would beclose to one, εS ≈ S > γ/β. But then (3) implies that the epidemic should reignite,
contradicting the supposition that it was winding down.
The following result studies the effect of the transmission rate and the cost of distancing on the final size of the epidemic and is graphically represented in Figure 6.
Proposition 8. S∞ is decreasing in β, for β ∈ [0, − ηIc0 ], and c.
˙
The size of the epidemic, 1 − S∞ , is monotone in β and c, as long as I(0)> 0.26Higher β leads to a larger size of infection, as does an increase in the cost of distancing.
The monotonicity of the final size of the epidemic in the transmission rate is in contrast
with the result that the peak prevalence is non-monotonic in the same parameter. While
policies that affect β might have perverse effects in the short run—e.g., a decrease in
the transmission rate, β, may lead to an increase in peak prevalence—the effects in thelong run are desirable. If the hospital capacities are not at their limits, such policies willachieve the desired result, a reduction in cumulative infections, in the long run. In the
short run, however, one needs to be circumspect if the medical capabilities are at or close
to the capacity and a trade-off between short-run prevalence and long-run epidemic size
might occur.
Intuitively, the effect of changes in the transmission rate on the final epidemic size
is monotone and resembles the comparative statics of the standard SIR model because
when the epidemic vanishes, i.e., when I(t) approaches 0, individuals stop distancing.
Hence, the only effect that changes in β have in this final phase of the epidemic is the
direct effect on infections.

4

Endogenous Cost of Infection

In this section, we present a model with an endogenous cost of infection and establish aconnection to the model with a constant cost of infection. As before, the individuals at
each point in time decide how much to distance, which determines how likely they areto get infected. An individual’s flow payoff from being in state θ ∈ {S, I, R} is πθ . We
assume πS ≥ πR ≥ πI .27 The endogeneity of costs of infection results from differences
To see this, note that β < − ηIc0 .
Models with endogenous cost of infection have been presented in Reluga (2010); Fenichel et al. (2011);
Fenichel (2013); McAdams (2020); Rachel (2020a); Toxvaerd (2020), among others. Yet, analytical
26

27

19

in the flow payoff across the states and the individual taking future infection risks into
account.
A susceptible individual i with exposure εi (t) enjoys the instantaneous payoff πS −c(1 − εi (t))2 . Let pi (t) be the probability of being infected at time t. Then,2ṗi (t) = εi (t)βI(t)(1 − pi (t)),(14)

with pi (0) = 0.28 Once an individual gets infected, her progression to recovery is independent of her behavior. Her continuation payoff from the moment she became infected,
VI , is:29
1
VI =
ρ+γ



γ
πI + π R .
ρ(15)

A susceptible individual who faces average exposure ε from her peers solves the problem
Z ∞
n
o
c
max
e−ρt (1 − pi (t))[πS − (1 − εi (t))2 ] + pi (t)ρVI dt(16)εi (·)∈[0,1] 0
2s.t.
ṗi (t) = βεi (t)I(t)(1 − pi (t)),
pi (0) = 0,
and the underlying dynamics given by (2), (3) and (4) with the initial condition (S(0), I(0), R(0)) =(1 − I0 , I0 , 0) and I0 ∈ (0, 1).30 The individual’s payoff can be thought of as the expectedvalue of being susceptible or infected at each point in time where the flow payoff of ancharacterizations of equilibria are rather elusive.
28
We model the behavior of susceptible individuals. Hence, the probability that they are infected at
the outset is zero.
29
Suppose that an individual gets infected at time τ . The (conditional) probability that the individual
will have been recovered after time τ + t is 1 − e−γt . Therefore,Z ∞
VI (τ ) =e−ρt e−γt πI + (1 − e−γt )πR dt,
0which is independent of τ . See also Toxvaerd (2020).
30
The payoff in (16) can be obtained from


Z ∞
c
ṗi (t)e−ρt (1 − pi (t)) πS − (1 − εi (t))2 +VI dt.
2
1 − pi (t)0With probability 1 − pi (t) individual i has not been infected by time t and receives the flow payoff(πS − 2c (1 − εi (t))2 )dt. In addition, with probability ṗi (t)dt she becomes infected and receives the lumpsum payoff VI . The above payoff is obtained by integration by parts. This approach was used in Toxvaerd(2020). For the approach dealing with all three states (S, I and R) see Rachel (2020a).

20infected individual is ρVI . An individual’s behavior affects her probability of infection
directly, but none of the population dynamics as she is small.
We study symmetric equilibria.
Definition 2. A symmetric equilibrium (an equilibrium, for short) is a tuple of functions(S, I, R, (εi , pi )i∈[0,1] ) with the following three properties:(i) (S, I, R) follow (2), (3) and (4) with the initial condition (S(0), I(0), R(0)) = (S0 , I0 , 0),
where ε is the average exposure;(ii) each εi solves (16), that is, εi is a best-response to (S, I, R), where the average
exposure ε is induced by (εj )j6=i ; and(iii) ε = εi for all i ∈ [0, 1].

In equilibrium, each pi is determined by the average exposure ε and I, and thus p = pi
for each i ∈ [0, 1]. For ease of exposition, we denote an equilibrium by (S, I, R, ε, p).

Assumption 1. πS − 2c > ρVI .
Even if a susceptible individual is fully distancing, her flow payoff of being suceptible
is greater than the flow payoff of being infected. Differently, the infection is so severe thatan individual faced with the choice between fully distancing forever and being infectedwith certainty chooses the former.
The current-value Hamiltonian of problem (16) isc
Hi = (1 − pi (t))[πS − (1 − εi (t))2 ] + pi (t)ρVI + ηi (t)βεi (t)I(t)(1 − pi (t)),2(17)

where ηi (t) is the current-value co-state variable. It represents the marginal value of anincrease in the probability of being infected at time t. The optimality condition withrespect to exposure εi (t) at time t is∂Hi
= (1 − pi (t))[c(t)(1 − εi (t)) + βηi (t)I(t)] = 0.
∂εi (t)Assuming that pi (t) < 1, which will be verified in Remark 1, the optimality conditiondelivers optimal distancingβdi (t) = − ηi (t)I(t),
(18)
c
21

provided that the entire distancing path admits an interior solution, i.e., that di (t) ∈ [0, 1]
for all t. One should keep in mind that the marginal value of an increased probabilityof infection, ηi (t), is negative. The extent to which an individual distances is, ceterisparibus, increasing in the infection rate, β, and the size of the infected population, I(t),and decreasing in the cost parameter, c, and the co-state, ηi (t). Importantly, the decisions
today influence the probability of getting infected both today and in the future, which inturn affects the distancing decisions today—a fact that is captured by the co-state ηi (t).
The current-value co-state variable ηi follows the adjoint equation∂Hi∂pi (t)η̇i (t) = ρηi (t) −c= ηi (t) (ρ + εi (t)βI(t)) + πS − (1 − εi (t))2 − ρVI .
2(19)

The transversality condition is lim e−ρt ηi (t) = 0. In equilibrium, η = ηi for all i. Usingt→∞the adjoint equation and the transversality condition, we solve for η.
Lemma 1. Suppose that the rest of the population is following the strategy ε, and εi isthe individual i’s best response. ThenZ ∞ηi (t) = −e−ρ(s−t) 1 − pi (s)c2πS − (1 − εi (s)) − ρVI ds.
1 − pi (t)
2

t



(20)

Let (S, I, R, ε, p) be an equilibrium. ThenZ ∞η(t) = −−ρ(s−t) S(s)

e
t


c
2
πS − (1 − ε(s)) − ρVI ds.
S(t)
2


(21)

We term πS − 2c (1−ε(t))2 −ρVI the susceptibility premium at time t. It is the differencein flow payoffs between being susceptible and being infected. The cost of getting infected,−η(t), is the discounted value of the susceptibility premium over time weighted by theconditional probability of being susceptible at each time in the future, s ≥ t, S(s).31 TheS(t)assumption πS − 2c > ρVI implies that η is negative. In other words, getting infected withcertainty is worse than being susceptible and fully distancing: −η(t) > 0. Distancing
over a period of time reduces the quality of life and, thus, the susceptibility premium.
However, it also decreases the probability that the individual will get infected and rewards
her with the premium for a longer period of time. The functional form of ηi demonstrates
the difficulty of the dynamic problem. Optimal exposure at time t depends on exposureof the remaining individuals through the effect it has on the spread of the infection, as
well as on the exposure of individual i at each instance in the future and the effect that
31

Note that formula (21) extends to more general specifications of the cost of distancing.

22this future exposure has on the benefit of getting infected today.
Alternatively, one can decompose η in two parts
η(t) = − (VS (t) − VI (t))
where
Z ∞
VS (t) =−ρ(s−t)

e
t





 
S(s) cS(s)2πS − (1 − ε(s)) + 1 −ρVI dsS(t)2
S(t)

is the continuation payoff of being susceptible andVI (t) = VI ,is the continuation payoff of being infected.
The above discussion implies that characterizing the set of equilibria analytically isuntenable. To verify, whether a distancing function ε can be part of an equilibrium, one
needs to posit that the individuals use it, derive S, I, R and η, and then verify that ε
is indeed a best reply given the dynamics. This task is made more challenging by thefact that even the SIR model without distancing does not have a tractable closed-form
solution and that η is pinned down only in the limit rather than at any point.
However, we can make use of the model with an endogenous cost of infection to
inform our parameter choices in the constant cost of infection model. The following
lemma provides bounds for η, which enable us to connect the two models.
Lemma 2. Let (S, I, R, ε, p) be an equilibrium. ThenπS − ρVI − 2cπS − ρVI≤ η(t) ≤ −
,
−
ρ
ρ+β
and

(22)

πS − ρVI.
ρ(23)

πS − ρVI − 2c
η(t) ≤ −.
ρ(24)

lim η(t) = −t→∞

If η̇(0) > 0, thenAs time passes, η eventually converges to the lower bound—that is, the cost of getting
infected approaches its upper bound. The bound is attained when individuals choose full
exposure in perpetuity without facing any risk of becoming infected. This is the scenario
23

in which getting infected would be most costly as there is no need to distance and no risk
of future infection. The convergence to this bound is intuitive: as times goes to infinity
the infection dies out and obviates the need for distancing.
The above lemma also provides an upper bound on η. This bound applies even if
η is locally decreasing at time 0. When η is increasing at the onset, which occurs if
π −ρV − c
I0 is sufficiently small, the upper bound − S ρ I 2 is approximately tight. This boundcorresponds to the cost of infection when individuals are fully distancing from now until
eternity.

Figure 7: Peak Prevalence and Solution Path of the Endogenous Cost of Infection Model.
In the left panel, the solid blue curve shows the peak prevalence in the endogenous
cost of infection model as a function of the transmission rate. The dashed and dottedcurves reproduce the constant cost of infection model’s peak prevalence using the derived
bounds on η. In the right panel, the solid blue curve represents the solution path of theendogenous cost of infection model. The dashed and dotted curves represent the constantcost of infection model’s solution paths using the derived bounds on η.
Lemma 2 connects the solution paths of the model analyzed here and the model witha fixed cost of infection. Towards that, let (S, I, R, ε, p) be an equilibrium of the endogenous infection cost model with η being the corresponding co-state given by (21). Let
ηL and ηH be the lower and the upper bound on η as given by Lemma 2. Finally, let(Sj , Ij , Rj , εj ), for j ∈ {L, H}, be the equilibria of the model with the constant cost of
infection corresponding to the lower and upper bounds of η.

Proposition 9. In the phase space, the graph of (SH , IH ) is above that of (S, I), which,in turn, is above that of (SL , IL ).

We numerically solve the endogenous cost of infection model using commonly usedparameters for COVID-19 following Farboodi et al. (2020) with our objective function.
24

The results for the solution path in the phase diagram are depicted in Figure 7. First, thecurves corresponding to the endogenous cost of infection model are between the curves ofthe model with the constant cost of infection, with the cost of infection being evaluated at
the lower and the upper bound. Importantly, the non-monotonicity of the peak prevalencein β is not an artifact of the constant cost of infection model which can be seen in theleft panel of the figure.

5

Behavioral Foundation of Non-linear Contact Rates

The standard SIR model assumes a constant rate of disease transmission which does notincorporate behavioral responses by individuals. As a remedy, SIR models have been
proposed in which the transmission rate depends on the current infected population; thepioneering paper is Capasso and Serio (1978), for an overview of literature see Funk et al.
(2010) and Verelst et al. (2016).
Our model provides a behavioral foundation of such SIR models with non-linear contact rates. While the subsequent literature examines a particular functional form of atime-varying contact rate that yields a certain solution path, our modeling can recover
the underlying cost of distancing that yields the given functional form. In principle, we
can relate the effective disease transmission rate and the cost of distancing.
In the SIR model of Capasso and Serio (1978) the force of infection βε(t)I(t) at time
t is given by g(I(t)), where g : [0, 1] → [0, ∞) is a non-negative bounded continuouslydifferentiable function which satisfies the following three assumptions: (i) g(0) = 0; (ii)g 0 (0) > 0; and (iii) g(x) ≤ g 0 (0)x for all x ∈ [0, 1]. Given this specification, they study the
dynamics of the solution path (S, I) qualitatively in the phase space. They also provide
a particular solution path by considering a particular function g. We will recover theunderlying cost of distancing for a general g and the particular g in Capasso and Serio(1978).
To that end, with some abuse of notation, denote by c : [0, 1] → [0, ∞) the costfunction of distancing. The cost of distancing at time t, therefore, is c(d(t)). Assume
that c is twice-continuously-differentiable, increasing, convex, and c0 (0) = 0. In thisset-up, the first-order condition of problem (1) with respect to distancing is
c0 (d(t)) = ηβI(t).

25Since c is strictly convex, g can be recovered asg(I(t)) = βI(t)(1 − (c0 )−1 (ηβI(t))).
It is easy to verify that g satisfies the three required assumptions.
Capasso and Serio (1978) consider the particular function g given byg(I(t)) =βI(t)1 + I(t)α.

It can be seen that the underlying cost function satisfiesc(d(t)) = −ηβα((1 − d(t)) − log(1 − d(t))).

(25)

In their specification, thus, the cost of distancing is unbounded when an individual fully
distances. This observation allows us to reinterpret the model of Capasso and Serio(1978) as a model of endogenous behavioral responses to a constant infection risk withthe cost of distancing function given in equation (25); in such a model, each individual
weights the cost and benefit of distancing at each point in time. Interestingly, Farboodi
et al. (2020) use precisely this cost function in their numerical analysis of the COVID-19
pandemic.32

32

Note that they have an additional but inconsequential +1 in their cost function so that c(0) = 0.

26

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30

A

Appendix

Proof of Proposition 1. Any individual’s problem (1) is concave; therefore, the firstorder condition given by (5) is also sufficient. This pins down the individual’s optimal
distancing in the SIR dynamics. Thus, in any equilibrium, if it exists, (S, I, R) is characterized by the system of differential equations dtd (S, I, R) = F (t, S, I, R), where F isdefined by (6), (7), and (8). The initial condition is (S(0), I(0), R(0)) = (S0 , I0 , 0). Then,the initial value problem admits a unique solution (S, I, R) on [0, ∞), as the system
satisfies the standard conditions. Namely, the function F is continuous on the domain
D = [0, ∞) × [0, 1]3 , and F is uniformly Lipschitz continuous in (S, I, R): there exists a˜ R̃)k ≤ Lk(S, I, R) − (S̃, I,
˜ R̃)k
Lipschitz constant L satisfying kF (t, S, I, R) − F (t, S̃, I,
for each t ∈ [0, ∞). See, for example, Walter (1998). Now, ε = εi is uniquely determined,
and hence the model admits a unique (symmetric) equilibrium.

Proof of Proposition 2. Let t̂ be as in the supposition of the proposition. We first show
˙ = I(t)(βε(t)S(t)−γ) ≥ε(t̂) ∈ (0, 1). Observe that I(·) is always positive as I follows I(t)˙−γI(t) and I(0) > 0. Thus, evaluating I(t)= 0 at t̂ yields βS(t̂)ε(t̂) = γ. Hence,ε(t̂) ∈ (0, 1).
¨ t̂) exists. We show:
Next, since I˙ is differentiable at t̂, it follows that I(¨ t̂) = β Ṡ(t̂)I(t̂)ε(t̂) + S(t̂)I(˙ t̂)ε(t̂) + S(t̂)I(t̂)ε̇(t̂) − γ I(
˙ t̂)
I(


= βI(t̂) Ṡ(t̂)ε(t̂) + S(t̂)ε̇(t̂)
= βS(t̂)I(t̂) −βI(t̂)ε2 (t̂) + ε̇(t̂)= −βS(t̂)I 2 (t̂)ε2 (t̂) < 0.
˙ t̂) = 0, the third from (2), and the forth fromThe second equality follows from I(ε̇(t̂) =ηβ ˙I(t̂) = 0,c(26)which, in turn, follows from optimality condition (5) and ε(t̂) ∈ (0, 1). Equation (26) also
yields the second assertion.
˙
Proof of Proposition 3. The first statement follows from the equality I(0)= Iγ0 R0b − 1 .

31Part (1): From (7) it follows that


 
βη
˙
I0 (1 − I0 ) − γ > 0.
I(0)> 0 if and only if I0 β 1 +c
˙Therefore, I(0)> 0 if and only if β ∈ (β, β) where β and β are solutions to the quadratic
equation


ηI0
β 1+
β (1 − I0 ) − γ = 0.
c

(27)

It can be seen that the discriminant of the quadratic equation is positive if and only if
I0 < 1−14ηγ . Note that the solid curve in the left panel of Figure 1 corresponds to equationc(27). Since ε(0) = 1 + ηIc0 β < 1 and I0 > 0, the left-hand side of the above equation isγγ
˙
negative at β = 1−I. Thus, β > 1−I. If β = − ηIc0 , then ε(0) = 0 and I(0)< 0. Thus,00
β < − ηIc0 .
For the second assertion, let β ∈ (β, β). Then, it follows from the previous arguments
˙that ε(0) > 0 and I(0)> 0. Suppose to the contrary that 1 + ηβI(t) = 0 at some t.
cηβWithout loss, assume that 1 + c I(s) > 0 for all s ∈ [0, t). At t, ε(t) = 0, Ṡ(t) = 0, and˙ = −γI(t) < 0. Then, there exists ŝ ∈ (0, t) such that I(ŝ)˙I(t)= 0, i.e., S(ŝ)ε(ŝ) = βγ .
c
˙Thus, before I hits β(−η)
at t, (S, I) has to satisfy I(ŝ)= 0 and Ṡ(ŝ) < 0. That is, at ŝ,ηβηβ
I peaks. Then, 0 = 1 + c I(t) > 1 + c I(ŝ) > 0, a contradiction.

Part (2): Let I0 ≥ 1−14ηγ . Then, the quadratic equation (27) has at most one solution.
c˙˙ ≤ 0 for some t (take t = 0) thenThus, I(0)≤ 0. Proposition 2 then implies that if I(t)˙I(s)< 0 for all s > t.

Proof of Proposition 4. By Assumption d(0) < 1, it can be seen that ε(t) ∈ (0, 1) for
all t. Then, we havedSββ
= Sε =d(S + I)
γ
γ



βη 2
βη
− S +S+(S + I)S ,cc

where the first equality follows from dividing (2) by the sum of (2) and (3), and thesecond uses (5) and simple manipulations. The above expression can be rewritten asdd(S + I)  1
β β 2η
1
β 2η
++(S + I)
=
,
S
γ
γc
S
γc

which is a linear first-order differential equation with respect to S1 and (S + I). For ease32

of notation, let y = S1 and x = S + I. Then,dy+dx
Let µ(x) := exp

R 

2
β
+ βγcη xγβ β 2η
β 2η
+
x y=
.
γ
γc
γc

(28)


dx be the integrating factor. We have
β 2η
2γc

µ(x) = k · exp


2 !
c
x+
,
βη

(29)

where k is the constant of integration. Then, equation (28) reduces to


 
d
β β 2η
β 2ηd[µ(x)y] = µ(x)y++x y = µ(x)
.
dx
dx
γ
γc
γc

(30)

Integrating the outer most sides of Expression (30) and using (29) yield
"
exp

β 2η
2γc



c
x+
βη

2 ! #1
y
S+I

β 2η
=
γc

Z 1

β 2η
2γc

exp
S+I


2 !
c
x+
dx.
βη

(31)

The left-hand side of (31) reduces to
exp

β 2η
2γc



c
1+
βη

2 !

For the right-hand side of (31), let v = β
hand side of (31) reduces to
s
−β

β 2η
2γc

1
− exp
S0

2(−η)
γc

q

−η
2γc





2γc

β

−η
2γc

2 !

1
.
S


q
−η
c
dv
x + βη
. Since dx= β 2γc, the right-

Z β q −η (1+ c )
q

c
S+I +
βη

βη

2

e−v dv.

(S+I+ βηc )Hence, we can rewrite equation (31) as

2 !
β η
c
1
S+I +
= exp
2γc
βη
S
2

exp

s

2 !
Z β q −η (1+ c )
2γc
βη
β η
c
1
2(−η)
2
1+
+β
e−v dv,
q
−η
c
2γc
βη
S0
γc
β 2γc (S+I+ βη )2and finally we obtain (11), as desired.
Proof of Proposition 5. Part (1): We prove the assertion with respect to ηc . Denote bya point (S, I(S)) on the solution path. Differentiating the quotient differential equation

33

dI
with respect to ηc at a fixed point (S, I(S)) yields
dS

∂
∂ ηc



dI  η 
dS c


=−

γI
1
< 0.
S (1 + βI ηc )2

˜ the solution path associated with η̃ andNow, take ηc and η̃c̃ with ηc < η̃c̃ . Denote by (S̃, I)dI
˜ 0 −δ) > I(S0 −δ) for some small δ > 0. Now,c̃. Since dS< 0 at (S0 , I0 ), it follows that I(Sit is sufficient to show that two curves I and I˜ do not intersect. Suppose to the contrary
˜
that I and I˜ did intersect. Then, for S := sup{S ∈ (0, S0 ) | I(S)
= I(S)}, it would have

dI
dI˜
dI
(S) < 0 at
to be the case that dS̃ (S) > dS (S). However, this cannot happen as ddη dScany point (S, I(S)).
Part (2): We first show S(t) ≥ Ŝ(t) for all t ≥ 0. Suppose to the contrary that there
˙
exists some t̃ such that S(t̃) < Ŝ(t̃). At time 0, S(0) = Ŝ(0) and Ṡ(0) > Ŝ(0). Thus,
there exists an interval in which S(·) > Ŝ(·). Then there would have to exist t0 such that˙
S(t0 ) = Ŝ(t0 ) and Ṡ(t0 ) < Ŝ(t0 ). However, it follows from S(t0 ) = Ŝ(t0 ) and the previous
ˆ 0 ), and thus
argument that I(t0 ) ≤ I(t˙ˆ 0 ) = Ŝ(tṠ(t0 ) = −βε(t0 )S(t0 )I(t0 ) > −βS(t0 )I(t0 ) > −β Ŝ(t0 )I(t0 ),which is impossible.
Next, we show R(·) ≤ R̂(·). Suppose to the contrary that R(t0 ) > R̂(t0 ) for somet0 ∈ (0, ∞). Define t1 := sup{t ∈ [0, t0 ] | R(t) = R̂(t) and R(s) > R̂(s) for all s ∈ˆ 1 ). There(t, t0 ]}.33 Since R(t1 ) = R̂(t1 ), it follows that S(t1 ) + I(t1 ) = Ŝ(t1 ) + I(tˆexists a small δ ∈ (0, t0 − t1 ) such that I(t)< I(t) for all t ∈ (t1 , t1 + δ) becauseRtRtˆR(t) − R(t1 ) = γ t1 I(s)ds, R̂(t) − R̂(t1 ) = γ t1 I(s)ds,
and R(t1 ) = R̂(t1 ). Also, it
follows from the previous argument that Ŝ(t) ≤ S(t) for all t ∈ (t1 , t1 + δ). Thus,R̂(t) > R(t) for all t ∈ (t1 , t1 + δ), a contradiction.
Proof of Proposition 6. We prove the result with respect to c first, then with respect
to β.
Part (2): The proof of Proposition 3 has established that
 


βη
˙
I(0)
> 0 if and only if I0 β 1 +I0 (1 − I0 ) − γ > 0.
c
˙Therefore, I(0)> 0 if and only if c > c, where c can be recovered from the above
inequality; remember that η < 0. The peak prevalence when c ≤ c̄ is I0 . If c > c, thenthe peak prevalence is strictly greater than I0 . We show that the peak prevalence is33

If R(t) > R̂(t) for all t ∈ (0, t0 ] then t1 = 0. If not, t1 = sup{t ∈ [0, t0 ] | R(t) = R̂(t)}.

34strictly increasing in c > c. Differentiating (10) with respect to c, while holding S and I
fixed, yields34
∂
∂c



dI
dS


=

γηI
c2 S 1 + βI ηc2 < 0,where the inequality is due to η < 0. If two solution paths corresponding to c and c0 > cintersect at some point, the solution path corresponding to c0 has a smaller slope. A
certain point of intersection is the beginning of the infection (S0 , I0 ). At this point in thegraph with S on the horizontal and I on the vertical axis the solution path corresponding
to c0 is steeper; the solution paths are decreasing at (S0 , I0 ). Just below S0 , then, thesolution path corresponding to c0 is above the one corresponding to c. If they wereto intersect at some other S < S0 , the solution path corresponding to c0 would haveto intersect the solution path corresponding to c from above and stay below it. This
would contradict the finding that the solution path corresponding to c0 is above the one
corresponding to c for S slightly below S0 . Finally, given that the solution path underc0 is above the solution path under c, the peak of infection under c0 must be higher thanthe peak of infection under c.
Part (1): We break up the proof for β into two steps.
Step 1: I ∗ is decreasing in β for β ∈ [− 2Ic0 η , β]; notice that η is negative, thus, thelower bound is positive. The derivative of the quotient differential equation (10) at agiven point (S, I(S)) with respect to β is
∂
∂β



dI
(β)
dS



I(S)
γ 1 + 2 βη
c
.
=− 2
βη
β S (1 + c I(S))2(32)

The above derivative evaluated at (S0 , I0 ) is greater or equal to 0, for β ≥ −2Ic 0 η . Thismeans that at (S0 , I0 ), a higher β leads to a slower spread of the infection when thestarting β is high enough. At (S0 , I0 ) solution paths are decreasing, thus the positive
derivative with respect to β means that the solution path becomes flatter as β increases.
That is, around (S0 , I0 ) the solution path corresponding to a higher β is, therefore, below
the one with the lower beta.
∂dI
Moreover, ∂β(β)≥ 0 at (S0 , I0 ), for β ≥ −2Ic 0 η , implies that the same is true fordS
all (S, I) with I > I0 . This means that if two solution paths corresponding to some β
and β 0 > β in [− 2Ic0 η , β] intersect, then the solution path corresponding to β 0 must have alarger slope. One such point of intersection is (S0 , I0 ). Therefore, a solution path for β 0 is
below the one of β just below S0 and it cannot intersect it anymore as long as I ≥ I0 . In
34

˙
Recall that when I(0)> 0, exposure is interior for all t.

35

other words, the solution path of β 0 is strictly below the solution path of β for all I > I0 .
The maximum of I for β 0 is, therefore, strictly below the maximum of I for β.
Step 2: There exists a β1 such that I ∗ is increasing in β on (β, β1 ). We divide thisstep into three substeps. First, we show that the peak I ∗ is continuous in β. Then, wedI∂(β) < 0 along the entire solution path. Finally, we combine these twoshow that ∂β
dS

dI
∂
(β)
< 0 implying that theinsights to show that for β > β but sufficiently close ∂βdS
peak is increasing in β for β ∈ (β, β + δ) for some δ > 0.
Step 2.1: We argue that I ∗ is continuous in β ∈ (0, β). For β ∈ (0, β), I ∗ = I0 .
dI
For β ∈ (β, β), in the (S, I)-phase space, (S, I) = (S ∗ , I ∗ ) satisfies (11) and dS= 0,γ1i.e., S ∗ = β 1+ ηβ I ∗ . Substituting the latter equation into the former and rearranging, wecobtains2 !
Z q −η (β+ c )
2γc
η
η
c
1
(−2η)
−v 2

e
exp
+β
β+
dv
q
−η
γ
2γc
η
S0
γc
+βI ∗ + ηc
2γc 1+ βη I ∗
c

!2 


η
1
β 2η ∗γc β+I exp + βI ∗ +
.
=
βη
∗
γ
c
2γc 1 + c I
ηThis implies that I ∗ is differentiable and thus continuous in β for β ∈ (β, β).
Finally, if β = β then (S ∗ , I ∗ ) = (S0 , I0 ) satisfies the above implicit equation. Thus,
I ∗ is also continuous at β = β.
∂Step 2.2: ∂β

dI
(β)
dS



< 0, along the entire solution path whenever it holds along theβηpath that 1 + 2 c I(S) > 0. This is satisfied for β = β.

dI∂Recall that the sign of ∂β(β)
at each (S, I(S)) is determined by the negative ofdS
the sign of 1 + 2 βηI(S). Thus, it is sufficient for the derivative to be negative along thecentire path that 1 + 2 βηI ∗ > 0.
c
Observe that the solution β of equation (27) is given by
c
β=−
2ηI0


1 !
ηγ I0 21− 1+4.
c S0

36

Therefore,βη
1 + 2 I0 =c1
ηγ I0 2
1+4
> 0,
c S0

where the inequality follows due to the assumption on I0 in the statement of the result.
Consequently,
∂
∂β




dI
(β) < 0.
dS

Step 2.3: I ∗ is increasing in β for β ∈ (β, β + δ) for some δ > 0.
βηSince 1+2 c I0 > 0, there exists a δ1 > 0 such that 1+2 βη
I > 0 for all β ∈ [β, β +δ1 ).
c 0
By continuity of the peak, for every δ2 > 0, there exists a δ3 > 0, such that β ∈[β, β + δ3 ) implies I ∗ (β) < I0 + δ2 . Choose δ2 to correspond to the δ1 argued above Step
2.2, and let δ3 corresponds to such δ2 . This guarantees that we consider β to lie in arange such that the peak is sufficiently low to ensure that the slopes of the solution paths
can be ordered by comparing β.
∂dI
I(S) > 0 and therefore ∂β(β) < 0.
By Steps 2.1 and 2.2, for any such β, 1 + 2 βηcdSThis implies that whenever two solution paths corresponding to different β in (β, β + δ3 )intersect at a point, the one with the higher β has the smaller slope. Indeed, one pointof intersection is (S0 , I0 ). The solution path with a higher β must be steeper than theother path; the two are decreasing at (S0 , I0 ). Suppose the two solution paths were to

dI
∂
(β)
< 0,
intersect at some S < S0 and let S̃ be the largest such S. Then due to ∂βdS
the solution path with the higher β would need to intersect the solution path with thelower beta from above, and fall below it for S > S̃. But this contradicts the fact that at
S0 the solution path corresponding to a higher β is above the one with the lower beta.
β

Proof of Proposition 7. First, we show S0 e− γ ≤ Ŝ∞ . Let R̂∞ := lim R̂(t). For thet→∞purpose of illustration, let ε̂(·) = 1. It follows from the SIR dynamics that






Z ∞
Z ∞
β
β
ˆ
ˆ
Ŝ∞ = S0 exp −βε̂(s)I(s)ds ≥ S0 exp −β
I(s)ds = S0 exp − R̂∞ ≥ S0 e− γ .
γ00The first equality follows in the same way as (38) is derived, by letting t0 = t and
t1 = ∞. The first inequality follows because ε̂(·) ≤ 1. Indeed, it follows with equality
in the standard SIR model ε̂(·) = 1. The second equality follows from integrating (4)ˆ respectively). The second inequality follows(precisely, with R and I replaced by R̂ and I,37because R̂∞ ≤ 1.
Second, we show Ŝ∞ ≤ S∞ . Since Ŝ(t) ≤ S(t) holds by Proposition 5 (2), letting
t → ∞ yields the desired result.
Third, we show S∞ < βγ in two steps. The first step establishes S∞ ≤ βγ . Suppose
not. As S(t) is weakly decreasing throughout, there exists a δ > 0 such that S(t) ≥ δ + βγ
for all t ≥ 0. Since lim ε(t) = 1 and δ > 0, for a given κ ∈ (0, δ), there exists t1 ∈ [t0 , ∞)t→∞such that δε(t) − βγ (1 − ε(t)) > κ for all t ≥ t1 . Then, for all t ≥ t1 , we have˙ = βI(t)(S(t)ε(t) − γ ) ≥ βI(t)((δ + γ )ε(t) − γ ) > βI(t)κ,
I(t)
β
β
β
˙
˙
> βκ (note that, since I(t)≥ −γI(t), I(t) is always positive: I(t) ≥that is, I(t)I(t)I(0)e−γt > 0). Thus, I(t) ≥ I(t1 )eβκt , which yields I∞ = +∞. This is a contradiction to I∞ = 0.

The second step establishes S∞ 6= βγ . Suppose to the contrary S∞ = βγ . Then,dI(S∞ ) = −1 + βγ S1∞ = 0 as lim ε(t) = 1. However, note that
dS
t→∞



d dI
γ
1
1
1
dε(I(S)) dI(S∞ ) = −+dS dS
β ε(I(S))S S ε(I(S)) dI(S) dS
γ
1
=−
<0
β ε(I(S))S 2
dI(S∞ ) = 0, where ε(I(S)) = 1 + β ηc I(S). Thus, there is a δ > 0 such that for
as dS
dI
S ∈ (S∞ , S∞ + δ), dS(S∞ + δ) < 0 and, hence, that I(S∞ + δ) < 0, a contradiction. Thus,γ
S∞ < β .

Proof of Proposition 8. It follows from Proposition 5 that S∞ is increasing in c. Thus,
we show that S∞ is decreasing in β for the following three cases: (1) β ∈ [0, β]; (2)cβ ∈ [β, β]; and (3) β ∈ [β, (−η)I].
0˙Case 1. Let β ∈ [0, β]. In this case, I ∗ = I0 , and I(t)< 0 for all t ∈ (0, ∞). Also,ηε(·) ∈ (0, 1) as ε(t) = 1 + β c I(t) is decreasing in I(t), I(t) > 0 is decreasing, andc
β < − ηI(0).

38The derivative of the quotient differential equation with respect to β at (S, I(S)) is
∂ dI
γ
1
=−
∂β dS
βS β(1 + βη
I(S))
c




βη
1 + 2 I(S) < 0.
cThis implies that, for any β, β 0 ∈ [0, β] with β < β 0 , the solution path associated with β 0
has a flatter slope than the one associated with β at any point S ∈ (S0 , S∞ (β)), whereS∞ (β) is S∞ associated with β. Thus, I(S∞ (β)) > 0 for the solution path associatedwith β 0 , and hence S∞ (β 0 ) < S∞ (β).

˙
Case 2. Let β ∈ [β, β]. In this case, I(0)≥ 0 and ε(·) ∈ (0, 1). Substituting (S, I) =(S∞ , 0) into (11) yields

S∞ =
exp

η
2γc


c
β+
η

exp
!
2

η
2γc



βS∞ + ηc

1
+ 2β
S0

s

2 

(−η)
2γc

Z q −η (β+ c )
2γc

q

η

−v 2

e
−η
2γc

.

(33)



2 !
c
βS∞ +
.
η
(34)

dv

(βS∞ + ηc )Rewriting Expression (33),

exp

η
2γc



c
β+
η

2 !

S∞
+2βS∞
S0

s

(−η)
2γc

Z q −η (β+ c )
2γc

q

η

e
−η
2γc

(βS∞ + ηc )−v 2

dv = exp

η2γc

For the right-hand side,∂(RHS) = exp
∂β
|

η
2γc


2 !



η
c
c
∂S∞
βS∞ +
βS∞ +
S∞ + β
.
η
γc
η
∂β
{z
}

=(RHS)For the left-hand side, we obtain
!

2 !
∂S∞
∂
η
c
S∞ η
1
∂β
(LHS) = exp
β+β(1 − S0 ) + +
∂β
2γc
η
S0 γc
γ
S∞
q
s
!
Z −η (β+ c )
∂S∞
2γc
η
(−η)
1
∂β
−v 2
+ 2βS∞
e dv
+
q
−η
2γc
β
S∞
βS∞ + ηc )
2γc (



2 !
η
∂S∞
η
c
S∞ + β
exp
βS∞ +
+ βS∞
.
γc
∂β
2γc
η
Equating the derivatives of the left-hand and right-hand sides and using Expression (34)39and rearranging yield
exp
= exp


2 ! 



c
β
1
∂S∞ 1
γ
βS∞ +
−
+
S∞ −
η
γ S∞
∂β
γ
β
!

2
 

η
c
S∞ 1
ηβ(1 − S0 ) γ
β+
1+
−
.
2γc
η
S0 γ
c
β
η
2γc

Thus,



2 
c






exp
β+η
β
1
∂S∞
S∞ 1
γ
1
γ
 
−
=
2  S γ ε(0) − β − γ S∞ − β .
γ S∞
∂β
0
η
c
exp 2γc βS∞ + η
η
2γc

(35)

Since S∞ < βγ follows from Proposition 7,35 it follows that


2 
c
exp
β+η
γ

∂S∞
S∞ 
S∞ ε(0) − β < 0.


=
−
1


γ

2
∂β
β 
S
S
−
0
∞
η
β
exp 2γc βS∞ + ηc




η
2γc

c
Case 3. The case with β ∈ [β, (−η)I] is analogous to Case 1, and thus the proof is0
omitted.

Remark 1. Recall that we have assumed pi (t) < 1 in deriving equation (18). We showβthat this is indeed the case in three steps. First, the proof of the inequality S0 e− γ ≤ S∞
in Proposition 7 holds for any SIR dynamics (2), (3) and (4) with ε(·) ∈ [0, 1]. Especially,
it holds for the model with the endogenous cost of infection in which η evolves according
1−p∞to (19). Second, 1−p(0)= SS∞0 > 0 holds, where the equality follows from observations in
the proof of Lemma 1 and the inequality from the first step. Third, pi , which follows (14),is weakly increasing and satisfies p = pi in equilibrium. Then, p(t) ≤ p∞ := lim p(t) < 1,t→∞
as desired.

Proof of Lemma 1. We prove (20) in two steps. First, it follows from (14) thatdṗi (t)log(1 − pi (t)) = −= −εi (t)βI(t).
dt
1 − pi (t)γ˙
In fact, Equation (35) itself yields S∞ 6= βγ . Since I(0)≥ 0, we have ε(0) ≥ βS
> βγ . Since the0γ
first-term of the right-hand side of (35) is not zero, it cannot be the case that S∞ = β .
3540Integrating both sides from some t0 to t1 > t0 and taking the exponential yield Z t1
1 − pi (t1 )βεi (t)I(t)dt .
= exp −
1 − pi (t0 )
t0

(36)

Second, since (19) is a linear first-order differential equation, let−ρtµ(t) := eZ texp −βεi (τ )I(τ )dτ0

be the integrating factor. Since dtd [µ(t)ηi (t)] = µ(t) (η̇i (t) − (ρ + βεi (t)I(t))ηi (t)), it follows that


d
c
2
[µ(t)ηi (t)] = µ(t) (πS − ρVI ) − (1 − εi (t)) .
dt
2Integrating both sides on [t, ∞) and using the transversality condition giveZ te exp −βεi (τ )I(τ )dτ ηi (t)
0


Z s
Z ∞

c
−ρs
εi (τ )I(τ )dτ(πS − ρVI ) − (1 − εi (s))2 ds.
e exp −β
=
2
0
t
−ρt

Thus,

Rs

−β 0 εi (τ )I(τ )dτ 
c
2


(π
−
ρV
)
−
(1
−
ε
(s))
ds
ηi (t) = −
e
S
I
i
Rt
2
t
exp −β 0 εi (τ )I(τ )dτ
Z ∞


c
−ρ(s−t) 1 − pi (s)2e=−(πS − ρVI ) − (1 − εi (s)) ds,
1 − pi (t)2tZ ∞−ρ(s−t) exp(37)

where the last equality used (36).
Next, we show (21) in two steps. First, observe that (2) can be rewritten asdlog(S(t)) = −βε(t)I(t).
dt
Integrating both sides from some t0 to t1 > t0 and taking the exponential yield Z t1S(t1 )= exp −
β(t)ε(t)I(t)dt .
S(t0 )
t0

41

(38)

Second, in an equilibrium, (37) reduces to

Rs

exp
−β
ε(τ
)I(τ
)dτ
c
2
0


(π
−
ρV
)
−
(1
−
ε(s))
ds
η(t) = −e−ρ(s−t)SI
Rt2t
exp −β 0 ε(τ )I(τ )dτZ ∞
S(s) ce−ρ(s−t)=−(πS − ρVI ) − (1 − ε(s))2 ds,
S(t)
2
t
Z ∞

where the last equality used (38).

Proof of Lemma 2. We first show (23). We rearrange (21) asZ ∞e

η(t) =−ρ(s−t) S(s)

t

S(t)

Z ∞
(ρVI − πS )ds +e−ρ(s−t)tS(s) c(1 − ε(s))2 ds.
S(t) 2(39)

For the first term of (39), since ρVI − πS < 0,
ρVI − πS
=
ρ

Z ∞
e

−ρ(s−t)Z ∞(ρVI − πS )ds ≤te−ρ(s−t)t

S(s)S(∞) ρVI − πS
(ρVI − πS )ds ≤
.
S(t)
S(t)
ρ

I
As t → ∞, the first term of (39) converges to − πS −ρV. For the second term of (39),ρ
observe I∞ := lim I(t) = 0. This is because, if I∞ > 0, then R is unbounded, whicht→∞
is impossible. By optimality condition (18), lim εi (t) = 1. Then, for any small numbert→∞κ > 0, there exists t0 ∈ [0, ∞) such that if t ≥ t0 then

Z ∞
0≤

e
t

−ρ(s−t) S(s) cS(t) 2Thus,(1 − ε(s)) ds ≤t

Z ∞0 ≤ lim

t→∞

Z ∞

2

e

cκ2
(1 − ε(s)) ds ≤.
S(t) 22ρ−ρ(s−t) S(s) c

t

cκ2
c
e−ρ(s−t) (1 − ε(s))2 ds ≤
.
2
2ρ

2

Since κ is arbitrary, the second term of (39) converges to zero. Hence, we obtain (23), as
desired.
As for the bounds, the lower bound is obtained by replacing ε(t) = 1 and S(s)= 1 forS(t)all s ≥ t in (21). For the upper bound, it follows from (19) that η̇i (t) > 0 if and only if
η(t) > −πS − ρVI − 2c (1 − ε(t))2.
ρ + ε(t)βI(t)π −ρV − cIf η(t) satisfies η(t) > − S ρ+βI 2 , then from time t on η is always increasing, which
contradicts the statement that η converges to its lower bound as time goes to infinity.
Next, assume η̇(0) > 0. Observe that η is bounded because it is continuous and42

converges to the finite lower bound (23). Letting tη be a time at which η attains amaximum, it follows from the assumption η̇(0) > 0 that η̇(tη ) = 0. Thus,
η(tη ) = −πS − ρVI − 2c (1 − ε(tη ))2.
ρ + ε(tη )βI(tη )Substituting for βI(t) from equation (18) for optimal distancing and rearranging yield
πS − ρVIc+ (1 − ε2 (tη ))ρ2ρcπS − ρVI − 2.
≤−ρη(tη ) = −(40)

π −ρV + c

Finally, we show that the upper bound − S ρ I 2 is approximately tight when η̇(0) >0. Substituting (40) into optimality condition (18) yields the following quadratic equation
with respect to ε(tη ):πS − ρVI − 2cβI(tη ) 2βε (tη ) + ε(tη ) − 1 + I(tη )
= 0.
2ρ
c
ρ
This quadratic equation admits a unique solution ε(tη ) ∈ [0, 1]:ρε(tη ) = −βI(tη )
Since 1 −√s
1−

1 + 2x ≈ −x and 1 −ρε(tη ) ≈βI(tη )

βI(tη )
ρβI(tη )
1+2
ρ

√


!
πS − ρVI − 2cβ
1 − I(tη ).
cρ

1 + 2x ≥ −x,πS − ρVI − 2cπS − ρVI − 2cββ.
1 − I(tη )= 1 − I(tη )cρcρπ −ρVI − 2c.
ρComparing the last equation with optimality condition (18), we obtain η(tη ) ≈ − S

Proof of Proposition 9. Recall thatγdI,
= −1 +dS
βS max 0, 1 + βηIc
and denote the solution path of the model with endogenous cost of infection by (Se , Ie )and its co-state by ηe . By construction, ηe (·) ∈ [ηL , ηH ]. Therefore, for any fixed values
dIL
dIH
dIe
of S and I, the following chain of inequalities obtains: dS
≤ dS
≤ dS
. Finally, recalleHLthat all three solution paths go through (S0 , I0 ).

43Consider first the solution path of the model with an endogenous cost of infectiondIL
dIe, at any point ofand the model with the fixed cost of infection ηL . Since dS≥ dSeLintersection the solution path of the model with the fixed cost ηL intersects the model withthe endogenous cost from below. Hence, for δ > 0 small enough IL (S0 − δ) ≤ I(S0 − δ).
But then there can be no intersection for any S < S0 as otherwise at such an intersectiondIL
dIe. Thus, IL (S) ≤ Ie (S).
< dS
dSL
e
The proof for the case with the fixed cost of infection ηH is analogous and, therefore,
IH (S) ≥ Ie (S).

B

Parameters and Computational Algorithm

We simulate the model at a daily frequency. We follow Farboodi et al. (2020) for mostmodel parameters as summarized in Table 1. We set γ = 1/7, assuming that the average
length of disease is 7 days. For the transmission rate β for the baseline simulation of the
˙without
endogenous cost of infection model, we assume that the initial growth rate I(0)I(0)
behavior is 0.3. Since it is given as β − γ for the dynamics of the standard SIR modelwith S0 = 1, we set β = 0.3 + γ = 0.443. This gives R0 = 3.1 without behavior. We
vary β for various numerical simulations. For I0 , we match 194 people who died fromCOVID-19 in the US on or before March 18, a week after the pandemic declaration of
the WHO on March 11, 2020. Given a population of 328 million and an IFR of 0.0062,
we set I0 = 0.95 × 10−4 . We take ρ = ρ̃ + λ = (0.05 + 0.67)/365, where ρ captures a 5
percent annual discount rate, and λ implies an expected time until the arrival of a cureof 1.5 years as in Alvarez et al. (forthcoming) and Farboodi et al. (2020).
For the flow payoff, we normalize it to be −(1 − ε(t))2 . Thus, we set c = 2 and πS = 0.
To compute the parameter η of the constant cost of infection model, we follow the same
steps as in Farboodi et al. (2020). We assume the value of a statistical year of life to be
US $ 270, 000 and an average remaining life expectancy of COVID-19 victims to be 14.5
years, which gives US$ 3, 915, 000 where the numerical values are taken from Hall et al.
(2020). Hence, to avoid a 0.1 percent probability of death an individual would be willingto pay US$ 0.001 × 3, 915, 000. Using the discount rate to translate this into flow units
we obtain US$ ρ · 3, 915 as the willingness to pay to avoid the 0.1 percent probabilityof death. To translate this into utils, we also use the US per capita consumption fromHall et al. (2020) of US$ 45,000 per year implying that an individual is willing to give up
3,915ρ·365= 31.755ρ in terms of annual consumption units, i.e., ε = 1 − 31.755ρ, to avoid
45,000a 0.1 percent risk of death. Applying the assumed utility function, an individual, who is44

Table 1: Table of Baseline Parameters for Numerical Analysis.
ParameterγβDescription
Recovery Rate
Transmission Rate

Value
1/7
0.3 + γI0

Initial Seed of Infections

0.95 × 10−4

ρ̃
λ
c
πS
η

Discount Rate
Arrival Rate of Cure
Cost of Distancing
Flow Payoff of Susceptibles
Cost of Infection

0.05/365
0.67/36520{−2761.63, −2254.68}

SourceFarboodi et al. (2020)
Farboodi et al. (2020)
Based on death toll in the
US before March 19, 2020
Farboodi et al. (2020)
Farboodi et al. (2020)
Normalization
Normalization
Hall et al. (2020)

willing to give up 31.755 ρ units of consumption per period to avoid a 0.1 percent risk ofdeath, is indifferent between this and full exposure with a 0.001 risk of death which has
a utility cost of v:−(1 − 1)2(1 − 31.755ρ)2− 0.001v = −.
ρρMultiplying this value of life in utils by the death rate of 0.0062 (also from Hall et al.,2020) yields a cost of infection η = −2761.63.
For the endogenous cost of infection model, we set πR = 0 and πI = −399.96 so thatπI
VI = ρ+γ= η works as the lower bound of η(t) in the endogenous cost of infection model.
= −2254.68, which we also use in the constant
The upper bound of η is −2761.63 + c/2
ρcost of infection model.
We have solved the constant cost of infection model using the fourth-order RungeKutta method. For the endogenous cost of infection model, recall that the equilibriumof the model is characterized as follows. First, (S, I, R) follow (2), (3), and (4) with theinitial condition (S(0), I(0), R(0)) = (S0 , I0 , 0), where ε(t) = 1 + βη(t)I(t) is the averagecπS − ρVI
exposure. Second, η follows equation (19) with lim η(t) = −as in (23).
t→∞ρI
To numerically solve (S, I, R, η), we set η(T ) = − πS −ρVat T = 400 × 365 (days).
ρThen, given η, we solve for (S, I, R) with the initial condition. In turn, given (S, I, R), weI
solve for η with the terminal condition η(T ) = − πS −ρV. We iterate the procedure untilρthe sum of the distances of (S, I, R, η) in two successive iterations is below a thresholdvalue. To facilitate the computation, at each iteration, when S(t) − S(t + 1) and I(t + 1)
are below threshold values, we have terminated the simulation of (S, I, R) at t + 1, andI
we start the computation of η with η(t + 1) = − πS −ρV
and (S, I, R). Once the iterations
ρ
end, we have checked whether ε(τ ) ∈ [0, 1] for every time τ . The right panel of Figure 7
depicts the peak prevalence when ε(τ ) ∈ [0, 1] for every time τ .

45