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Game Theory of Social Distancing in Response to an Epidemic

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Social distancing practices are changes in behavior that prevent disease transmission by reducing contact rates between susceptible individuals and infected individuals who may transmit the disease. Social distancing practices can reduce the severity of an epidemic, but the benefits of social distancing depend on the extent to which it is used by individuals. Individuals are sometimes reluctant to pay the costs inherent in social distancing, and this can limit its effectiveness as a control measure. This paper formulates a differential-game to identify how individuals would best use social distancing and related self-protective behaviors during an epidemic. The epidemic is described by a simple, well-mixed ordinary differential equation model. We use the differential game to study potential value of social distancing as a mitigation measure by calculating the equilibrium behaviors under a variety of cost-functions. Numerical methods are used to calculate the total costs of an epidemic under equilibrium behaviors as a function of the time to mass vaccination, following epidemic identification. The key parameters in the analysis are the basic reproduction number and the baseline efficiency of social distancing. The results show that social distancing is most beneficial to individuals for basic reproduction numbers around 2. In the absence of vaccination or other intervention measures, optimal social distancing never recovers more than 30% of the cost of infection. We also show how the window of opportunity for vaccine development lengthens as the efficiency of social distancing and detection improve.
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Game Theory of Social Distancing in Response to an
Epidemic
Timothy C. Reluga*
Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, United States of America
Abstract
Social distancing practices are changes in behavior that prevent disease transmission by reducing contact rates between
susceptible individuals and infected individuals who may transmit the disease. Social distancing practices can reduce the
severity of an epidemic, but the benefits of social distancing depend on the extent to which it is used by individuals.
Individuals are sometimes reluctant to pay the costs inherent in social distancing, and this can limit its effectiveness as a
control measure. This paper formulates a differential-game to identify how individuals would best use social distancing and
related self-protective behaviors during an epidemic. The epidemic is described by a simple, well-mixed ordinary differential
equation model. We use the differential game to study potential value of social distancing as a mitigation measure by
calculating the equilibrium behaviors under a variety of cost-functions. Numerical methods are used to calculate the total
costs of an epidemic under equilibrium behaviors as a function of the time to mass vaccination, following epidemic
identification. The key parameters in the analysis are the basic reproduction number and the baseline efficiency of social
distancing. The results show that social distancing is most beneficial to individuals for basic reproduction numbers around 2.
In the absence of vaccination or other intervention measures, optimal social distancing never recovers more than 30% of
the cost of infection. We also show how the window of opportunity for vaccine development lengthens as the efficiency of
social distancing and detection improve.
Citation: Reluga TC (2010) Game Theory of Social Distancing in Response to an Epidemic. PLoS Comput Biol 6(5): e1000793. doi:10.1371/journal.pcbi.1000793
Editor: Carl T. Bergstrom, University of Washington, United States of America
Received December 3, 2009; Accepted April 23, 2010; Published May 27, 2010
Copyright: ß 2010 Timothy C. Reluga. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was funded in part by the National Science Foundation (DMS-0920822) and the Bill and Melinda Gates Foundation (Grant Number 49276).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The author has declared that no competing interests exist.
* E-mail: timothy@reluga.org
Introduction
Epidemics of infectious diseases are a continuing threat to the
health of human communities, and one brought to prominence in
the public mind by the 2009 pandemic of H1N1 influenza [1].
One of the key questions of public health epidemiology is how
individual and community actions can help mitigate and manage
the costs of an epidemic. The basic problem I wish to address here
is how rational social-distancing practices used by individuals
during an epidemic will vary depending on the efficiency of the
responses, and how these responses change the epidemic as a
whole.
Social distancing is an aspect of human behavior particularly
important to epidemiology because of its universality; everybody
can reduce their contact rates with other people by changing their
behaviors, and reduced human contact reduces the transmission of
many diseases. Theoretical work on social distancing has been
stimulated by studies of agent-based influenza simulations
indicating that small changes in behavior can have large effects
on transmission patterns during an epidemic [2]. Further research
on agent-based models has argued that social distancing can arrest
epidemics if started quickly and maintained for a relatively long
period [3]. Compartmental epidemic models have also been used
to study social distancing by including states that represent
individuals employing specific behaviors. For instance, Hyman
and Li [4] formulate and begin the analysis of flu disease
transmission in SIR models where some individuals decrease their
activity levels following infection. Reluga and Medlock [5] uses
this approach to show that while social distancing can resemble
immunization, it can generate hysteresis phenomena much more
readily than immunization.
Rather than treating behaviors as states, some models treat
behaviors as parameters determined by simple functions of the
available information. Reluga et al. [6] studies dynamics where
contact rates can depend on the perceived disease incidence.
Buonomo et al. [7] investigates the impact of information
dynamics on the stability of stationary solutions in epidemic
models. Chen [8] considers a similar system but allows individuals
to learn from a random sample of neighbors. Funk et al. [9]
considers the information dynamics associated with social
distancing in a network setting by prescribing a reduction in
contacts based on proximity to infection. Related work by Epstein
et al.[10] explicitly considers the spatial and information dynamics
associated in response to an ongoing epidemic.
Building on the ground-breaking work of Fine and Clarkson
[11], there has been substantial recent interest in the application of
game theory to epidemiology [12–17]. The games studied so far
have primarily considered steady-state problems, and have not
allowed for dynamic strategies. One notable exception to this is the
work of Francis [18], which determines the time-dependent game-
theoretical solution of a vaccination problem over the course of an
epidemic. In another, van Boven et al. [19] studies the optimal use
of anti-viral treatment by individuals when they take into account
the direct and indirect costs of treatment.
PLoS Computational Biology | www.ploscompbiol.org 1 May 2010 | Volume 6 | Issue 5 | e1000793
To study the best usage of social distancing, we apply
differential-game theory at a population-scale. Differential games
are games where strategies have a continuous time-dependence; at
each point in time, a player can choose a different action. For
instance, a pursuit-game between a target and a pursuer is a two-
player differential game where each player’s strategies consist of
choosing how to move at each successive time until the target is
caught by the pursuer or escapes. Geometrically, one might think
of differential games as games where strategies are represented by
curves instead of points. Two-player differential-game theory was
systematically developed by Isaacs [20] as an extension of optimal
control theory [21–23]. Here, we employ an extension of
differential game theory to population games of the form described
by Reluga and Galvani [24]. The analysis in this paper will be
limited to the simplest case of the Kermack–McKendrick SIR
model with strong mixing [25].
In the Model section, we formulate an epidemiological-
economics model for an epidemic, accounting for the individual
and community costs of both social distancing practices and
infection. We then use differential game theory and numerical
methods to identify the equilibrium strategies over the course of an
epidemic. Numerical methods are used to investigate the finite-
time problem where vaccines become available after a fixed
interval from the start of the epidemic and the infinite-horizon
problem without vaccination. Fundamental results on the value
and timing of social distancing are obtained.
Model
In this article, social distancing refers to the adoption of
behaviors by individuals in a community that reduce those
individuals’ risk of becoming infected by limiting their contact with
other individuals or reducing the transmission risk during each
contact. Typically, social distancing incurs some costs in terms of
liberty, social capital, time, convenience, and money, so that
people are only likely to adopt these measures when there is a
specific incentive to do so. In addition to the personal
consequences, the aggregate effects of social distancing form an
economic externality, reducing the overall transmission of disease.
This externality needs to be accounted for in the determination
individuals optimal strategies, but, by definition, depends on the
choice of strategy.
To resolve this interdependence, we formulate our analysis as a
population game where the payoff to each individual is determined
by the individual’s behavioral strategy and the average behavioral
strategy used by the population as a whole. The model is related to
that previously studied by Chen [26]. We will use c
s
to represent
one specific individual’s strategy of daily investment in social
distancing. The population strategy
c
s
is the aggregate daily
investment in social distancing by the population. The overbar
notation is used to indicate that the aggregate investment
c
s
should
be thought of as an average investment aggregated over all
individuals in the population. In the limit of infinitely large
populations, c
s
and c
s
can be thought of as independent because
changes in one person’s behavior will have little affect on the
average behavior. Similarly, the epidemic’s dynamics depend on
the population strategy
c
s
but are independent of any one
individual’s behavior c
s
.
The effectiveness of social distancing is represented by a
function s(c
s
), which is the relative risk of infection given a daily
investment c
s
in social distancing practices. If there is no
investment, the relative risk s(0)~1. As the daily investment c
s
increases, the relative risk s(c
s
) decreases, but is bounded below by
0. We expect diminishing returns with increasing investment, so
we will also make the convenient assumption that s(c
s
) is convex.
Consider a Susceptible-Infected-Recovered (SIR) epidemic
model with susceptible (X), infected and infectious (Y), and
removed (Z) states. Suppose an epidemic starts with Y(0) cases in
a community of N~X(0)zY(0) total individuals (taking
Z(0)~0) and proceeds until time t
f
, at which point all the
individuals in the susceptible state are vaccinated. This epidemic is
fast relative to demographic processes and we do not distinguish
among the possible states of individuals leaving the infectious state,
so the population size N can be treated as constant. Between time
0 and time t
f
, the dynamics are described by
dX
dt
~{s(
c
s
)bYX, ð1aÞ
dY
dt
~s(
c
s
)bYX{cY , ð1bÞ
dZ
dt
~cY, ð1cÞ
where b is the transmission rate and c is the removal rate. This
SIR model assumes the population is homogeneous, strongly
mixed, and that the duration of infections is exponentially
distributed. At the start of the epidemic when there are few cases
of infection (Y (0)&0), the basic reproduction number R
0
~bN=c.
The total cost of the epidemic to the community, J, is the sum
of the direct costs plus the indirect costs of any economic
repercussions from the epidemic. To keep our analysis tightly
focused, we will only consider direct costs of the epidemic,
including the daily costs from infection, daily investments in social
distancing, and the costs of vaccination. Mathematically,
J~{
ð
t
f
0
(c
s
Xzc
I
Y )e
{ht
dt{c
V
X(t
f
)e
{ht
f
{c
I
Y (t
f
)e
{ht
f
1zh
ð2Þ
where c
I
is the daily cost of each infection, c
V
is the cost of
vaccination per person, and h is the discount rate. Note that while
the cost of infection c
I
is a constant, the investment in social
distancing
c
s
is a function of time. The last term in Eq. (2) is called
a salvage term and represents the cumulative costs associated with
individuals who are sick at the time the vaccine is made available
Author Summary
One of the easiest ways for people to lower their risk of
infection during an epidemic is for them to reduce their
rate of contact with infectious individuals. However, the
value of such actions depends on how the epidemic
progresses. Few analyses of behavior change to date have
accounted for how changes in behavior change the
epidemic wave. In this paper, I calculate the tradeoff
between daily social distancing behavior and reductions in
infection risk now and in the future. The subsequent
analysis shows that, for the parameters and functional
forms studied, social distancing is most useful for
moderately transmissible diseases. Social distancing is
particularly useful when it is inexpensive and can delay
the epidemic until a vaccine becomes widely available.
However, the benefits of social distancing are small for
highly transmissible diseases when no vaccine is available.
Game Theory of Social Distancing
PLoS Computational Biology | www.ploscompbiol.org 2 May 2010 | Volume 6 | Issue 5 | e1000793
(t
f
). The assumption that the entire remaining susceptible
population is vaccinated at time t
f
and that vaccination takes
effect instantly is, of course, unrealistic, but does provide an
approximation to the delayed release of a vaccine.
To simplify our studies, we will work with the dimensionless
version of the equations by taking:
S ~
b
c
X , I~
b
c
Y, R~
b
c
Z ,
^
tt~ct,
c~
c
s
c
I
, k~
c
V
c
I
,
^
hh~
h
c
,
^
ss(
c)~s(c
I
c):
ð3Þ
Under this choice of units, time will be measured in terms of
disease generations, social distancing costs will be measured
relative to the daily cost of infection, and population sizes will be
measured relative to the critical population size necessary to
sustain an epidemic.
Epidemics usually start with one or a few index cases, so we
focus on scenarios where I(0)&0. The dynamics can be described
in terms the shape of s(
^
cc), the discount rate
^
hh, and a single initial-
condition parameter
S (0)&
b
c
N: ð4Þ
From this, it follows that R
0
&S(0). Since epidemics are often
much faster than human demographic processes governing the
discount rate [27], we will also take
^
hh~0 in all calculations.
Henceforth, we will drop the hat-notation and work with the
dimensionless parameters. The dimensionless equations are
dS
dt
~{s(
c)IS, ð5aÞ
dI
dt
~s(
c)IS{I, ð5bÞ
J~{
ð
t
f
0
(cSzI)e
{ht
dt{
I(t
f
)e
{ht
f
1zh
{kS(t
f
)e
{ht
f
ð5cÞ
with the constraint that
c§0. Note that we drop the function
notation when necessary to simplify the presentation.
For our further analysis, we will assume
s(c)~
1
1zmc
, ð6Þ
with the maximum efficiency of social distancing s(0)~{m. Eq.
(6) is nicely behaved for numerical solutions because of its
relatively fat tail.
The Social Distancing Game
We now formulate a differential game for individuals choosing
their best social distancing practices relative to the aggregate
behavior of the population as a whole. The following game-
theoretic analysis combines the ideas of Isaacs [20] and Reluga
and Galvani [24]. The premise of the game is that at each point in
the epidemic, people can choose to pay a cost associated with
social distancing in exchange for a reduction in their risk of
infection. The costs of an epidemic to the individual depend on the
course of the epidemic and the individual’s strategy of social
distancing. The probabilities p(t) that an individual is in the
susceptible, infected, or removed state at time t evolve according
to the Markov process
_
pp~Q(t; c)p ð7Þ
where c is the individual’s daily investment as a function of the
epidemic’s state-variables and the transition-rate matrix
Q(t; c)~
{s(c)I 00
s(c)I {10
010
2
6
4
3
7
5
: ð8Þ
Note that both c and I change over time. Along the lines discussed
above, c and
c represent different quantities in our analysis; c
represents one individual’s investment strategy and the population
strategy
c represents an aggregated average of all individual
investments. We also note that there are several different ways c
and
c can be parameterized. They may be parameterized in terms
of time, as c(t) and
c(t), or in implicit feedback form c(V
S
, I) and
c(V
S
, I), or in explicit feedback form c(S, I) and c(S, I). The form
used will be clear from the context.
Since the events in the individual’s life are stochastic, we can not
predict the exact time spent in any one state or the precise payoff
received at the end of the game. Instead, we calculate expected
present values of each state at each time, conditional on the
investment in social distancing. The expected present value is
average value one expects after accounting for the probabilities of
all future events, and discounting future costs relative to immediate
costs. The expected present values V of each state evolve
according to the adjoint equations
{
_
VV~ Q
T
{hI

Vzv ð9Þ
where v
T
(t; c)~{c(t), {1, 0½. The components V
S
(t; c,c),
V
I
(t; c,c), and V
R
(t; c,c) represent the expected present values of
being in the susceptible, infected, or removed state at time t when
using strategy c in a population using strategy
c. The expected
present values depend on the population strategy
c through the
infection prevalence I.
The adjoint equations governing the values of each state are
derived from Markov decision process theory. They are
{
dV
S
dt
~{hV
S
z(V
I
{V
S
)s(c)I{c, ð10aÞ
{
dV
I
dt
~{hV
I
zV
R
{V
I
{1, ð10bÞ
{
dV
R
dt
~{hV
R
, ð10cÞ
with the constraints that c(t)§0 for all time t. Solution of (10)b
and (10)c gives
V
I
(t
f
)~{1=(1zh), V
R
(t
f
)~0: ð11Þ
If it is impossible to make a vaccine, the equations must be solved
over an infinite horizon. Over an infinite horizon, V
S
(?)~{c=h,
Game Theory of Social Distancing
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assuming c becomes constant. In the case of no discounting (h~0),
we still have V
S
(?)~0 provided c(S(t),0)~0 for sufficiently large
t. In the case where a perfect vaccine is universally available at
terminal time t
f
, the value of the susceptible and removed states
differs by the cost of vaccine k for t§t
f
. To avoid complications
with the choice of whether-or-not to vaccinate, we take k~0 so
V
S
(t
f
)~0. This is reasonable in scenarios where the cost of the
vaccine is covered by the government.
The dynamics are independent of R, so we need not consider
removed individuals further. Taking h~0 and V
I
~{1, we need
only study the reduced system
dS
dt
~{s(
c)IS, ð12aÞ
dI
dt
~s(
c)IS{I, ð12bÞ
{
dV
S
dt
~{ 1zV
S
ðÞs(c)I{c, ð12cÞ
with boundary conditions
S (0)~R
0
{I
0
, I(0)~I
0
, V
S
(t
f
)~0: ð12dÞ
The other conditions must be calculated from the solution of the
boundary-value problem and provide useful information. {V
S
(0)
will be the expected total cost of the epidemic to the individual.
The final size of the epidemic is given by S(0){S(t
f
).
Game Analysis
Solving a game refers to the problem of finding the best strategy
to play, given that all the other players are also trying to find a best
strategy for themselves. In some games, there is a single strategy
that minimizes a player’s costs no matter what their opponents do,
so that strategy can very reasonably be referred to as a solution. In
many games, no such strategy exists. Rather, the best strategy
depends on the actions of the other players. Any strategy played by
one player is potentially vulnerable to a lack of knowledge of the
strategies of the other players. In such games, it is most useful to
look for strategies that are equilibria, in the sense that every
player’s strategy is better than the alternatives, given knowledge of
their opponent’s strategies. A Nash equilibrium solution to a
population game like that described by System (12) is a strategy
that is a best response, even when everybody else is using the same
strategy. i.e. given V
S
(t; c,c), c
is a Nash equilibrium if for every
alternative strategy c, V
S
(0; c,c
)ƒV
S
(0; c
,c
). A Nash equilib-
rium strategy is a subgame perfect equilibrium if it is also a Nash
equilibrium at every state the system may pass through. I will not
address the problem of ruling out finite-time blowup of the
Hamilton–Jacobi equation and establishing existence and unique-
ness of subgame perfect equilibria. But numerical and analytical
analyses strongly support the conjecture that the stategies
calculated here are the unique global subgame perfect equilibria
to the social distancing game.
The equilibria of System (12) can be calculated using the
general methods of Isaacs [20]. The core idea is to implement a
greedy-algorithm; at every step in the game, find the investment
that maximizes the rate of increase in the individual’s expect value
V
S
. We represent strategies as functions in implicit feedback form.
c(V
S
,I) is the amount an individual invests per transmission
generation when the system is at state V
S
,I.Ifc
(V
S
,I) is a
subgame perfect equilibrium, then it satisfies the maximum
principle
c
(V
S
,I)~ argmax
c§0
{(1zV
S
)s(c)I{c
ð13Þ
when
c~c
everywhere. So long as s(c
) behaves well, in the sense
that it is differentiable, decreasing, and strictly convex, then c
is
uniquely defined by the relations
c
~0if{s(0)I(1zV
S
)ƒ1,
{s(c
)I(1zV
S
)~1 otherwise,
where s~
ds
dc
:ð14Þ
Figure 1 shows the interface in V
S
|I phase space separating the
region where the equilibrium strategy will include no investment in
social distancing (c
~0,s(c
)~1) from the region where the
equilibrium strategy requires investment in social distancing
(c
w0,s(c
)v1).
Two cases are immediately interesting. The first is the infinite-
horizon problem what is the equilibrium behavior when there is
never a vaccine and the epidemic continues on until its natural
end? The second is the finite-horizon problem if a vaccine is
introduced at time t
f
generations after the start of the epidemic,
what is the optimal behavior while waiting for the vaccine? In both
of these cases, it is assumed that all players know if and when the
vaccine will be available.
The infinite-horizon and finite-horizon problems are distin-
guished by their boundary conditions. In the finite-horizon case,
we assume all susceptible individuals are vaccinated at final time
t
f
,soV
S
(t
f
)~0, V
I
~{1, S(0)~S
0
, I(0)~I
0
&
>
0 while V
S
(0)
and S(t
f
) are unknown. In the limit of the infinite-horizon case
(t
f
??), we solve the two-point boundary value problem with
terminal conditions V
S
(?)~0, I(?)~0, and initial conditions
S(0)~S
0
, I(0)~I
0
&
>
0 while V
S
(0) and S(?) are unknown. But
these conditions are insufficient to specify the infinite-horizon
problem. The plane I~0 is a set of stationary solutions to Eq. (12),
so we need a second order term to uniquely specify the terminal
condition when we are perturbed slightly away from this plane.
Figure 1. Contour plots of relative risk surface for equilibrium
strategies. The relative risk is presented in feedback form with implicit
coordinates s(c
(V
S
,I)) (left) and transformed to explicit coordinates
s(c
(S,I )) (right) for the infinite-horizon problem with maximum
efficiency m~10. The greater the value of the susceptible state (V
S
),
the greater the instantaneous social distancing. We find that increasing
the number of susceptible individuals always decreases the investment
in social distancing, and the greatest investments in social distancing
occur when the smallest part of the population is susceptible. Note that
in the dimensionless model, the value of the infection state V
I
~{1.
doi:10.1371/journal.pcbi.1000793.g001
Game Theory of Social Distancing
PLoS Computational Biology | www.ploscompbiol.org 4 May 2010 | Volume 6 | Issue 5 | e1000793
Using Eq. (12), we can show solutions solve the second-order
terminal boundary condition
LV
S
LI
S ,I~0ðÞ~
{1
1{S
ð15Þ
for 0ƒSv1 as t
f
??.
Most of the equilibria we calculate are obtained numerically.
Some exceptions are the special cases where S(0)~0, I(0)w0.
Under these conditions, solutions can be obtained in closed-form.
First, I (t)~I(0)e
{t
. While mI(1zV
S
)v1, c
~0 and
V
S
(t)~½V
S
(0)z1e
I(0)(1{e
{t
)
{1: ð16Þ
When matched to the terminal boundary condition, we find that if
we write V
S
in feedback form as a function of I rather than t,
V
S
(I)~e
{I
{1 ð17Þ
is a solution so long as mIe
{I
v1 for all I. Inspecting the
inequality condition, we find that this holds as long as mve.
Results
A problem with solving Eq. (12) under Eq. (14) is that it requires
I(t) to be known from past time and V
S
(t) to be known from
future time. This is a common feature of boundary-value
problems, and is resolved by considering all terminal conditions
S(t
f
). Using standard numerical techniques, identifying an
equilibrium in the described boundary-value problem reduces to
scalar root finding for S (t
f
) to match the given S(0). The special
form of the population game allows the solution manifold to be
calculated directly by integrating backwards in time, rather than
requiring iterative approaches like those used for optimal-control
problems [23]. Code for these calculations is available from the
author on request.
Before presenting the results, it is helpful to develop some
intuition for the importance of the maximum efficiency m of
investments in social distancing. Given m for an arbitrary relative
risk function s, then in the best-case scenarios, where diminish-
ments on returns are weakest, one would have to invest atleast
1=m of the cost of infection per disease generation to totally isolate
themselves. The units here are derived from dimensional analysis.
This could be invested for no more than m generations, before
one’s expenses would exceed the cost of becoming infected. When
returns are diminishing, fewer than m generations of total isolation
are practical. Thus, the dimensionless efficiency m can be thought
of as an upper bound on the number of transmission generations
individuals can afford to isolate themselves before the costs of
social distancing outweigh the costs of infection.
For the infinite-horizon problem, an example equilibrium
strategy and the corresponding dynamics in the absence of social
distancing are shown in Figure 2. We can show that if social
distancing is highly inefficient (the maximum efficiency
mve&2:718), then social distancing is a waste of effort, no
matter how large R
0
. If social distancing is efficient, then there is a
threshold value of R
0
below which social distancing is still
impractical because the expected costs per day to individuals is too
small compared to the cost of social distancing, but above which
some degree of social distancing is always part of the equilibrium
strategic response to the epidemic (Figure 3).
The exact window over which social distancing is used depends
on the basic reproduction number, the initial and terminal
conditions, and the efficiency of distancing measures. The
feedback form of equilibrium strategies, transformed from (V
S
,I)
coordinates to the (S,I) coordinates of the phase-space is
represented with contour plots in Figure 1. Among equilibrium
strategies, social distancing is never used until part-way into the
epidemic, and ceases before the epidemic fully dies out.
The consequences of social distancing are shown in Figure 4.
The per-capita cost of an epidemic is larger for larger basic
reproduction numbers. The more efficient social distancing, the
more of the epidemic cost can be saved per person. However, the
net savings from social distancing reaches a maximum around
R
0
~2, and never saves more than 30% of the cost of the epidemic
per person. For larger R
0
’s, social distancing is less beneficial.
We can also calculate solutions of the finite-time horizon
problem where a vaccine becomes universally available at a fixed
time after the detection of disease (Figure 5). If mass vaccination
occurs soon enough, active social distancing occurs right up to the
date of vaccination. Using numerical calculations of equilibria over
finite-time horizons, we find that there is a limited window of
opportunity during which mass vaccine can significantly reduce
Figure 2. Epidemic solutions with equilibrium social distancing and without social distancing. Social distancing reduces the epidemic
peak and prolongs the epidemic, as we can see by comparing a time series with subgame-perfect social distancing (top left) and a time series with
the same initial condition but no social distancing (bottom left) (parameters R
0
~S(0)~4:46, m~20). In the phase plane (right), we see that both
epidemics track each other perfectly until S&3, when individuals begin to use social distancing to reduce transmission. Eventually, social distancing
leads to a smaller epidemic. The convexity change appearing at the bottom the phaseplane orbit with social distancing corresponds to the cessation
of social distancing.
doi:10.1371/journal.pcbi.1000793.g002
Game Theory of Social Distancing
PLoS Computational Biology | www.ploscompbiol.org 5 May 2010 | Volume 6 | Issue 5 | e1000793
the cost of the epidemic, and that social distancing lengthens this
window (Figure 6). The calculations show that increases in either
the amount of time before vaccine availability or the basic
reproduction number increase the costs of the epidemic. Smaller
initial numbers of infections allow longer windows of opportunity.
This is as expected because the larger the initial portion of the
population infected, the shorter the time it takes the epidemic to
run its full course.
Discussion
Here, I have described the calculations necessary to identify the
equilibrium solution of the differential game for social distancing
behaviors during an epidemic. The benefits associated with the
equilibrium solution can be interpreted as the best outcome of a
simple social-distancing policy. We find that the benefits of social
distancing are constrained by fundamental properties of epidemic
dynamics and the efficiency with which distancing can be
accomplished. The efficiency results are most easily summarized
in terms of the maximum efficiency m, which is the percent
reduction in contact rate per percent of infection cost invested per
disease generation. As a rule-of-thumb, m is an upper bound on
the number of transmission generations individuals can isolate
before the costs of social distancing outweigh the costs of infection.
Social distancing is not practical if this efficiency is small compared
to the number of generations in the fastest epidemics (m v2:718).
While social distancing can yield large reductions in transmission
rate over short periods of time, optimal social-distancing strategies
yield only moderate reductions in the cost of the epidemic.
Our calculations have determined the equilibrium strategies
from the perspective of individuals. Alternatively, we could ask
what the optimal social distancing practices are from the
perspective of minimizing the total cost of the epidemic to the
community. Determination of the optimal community strategy
Figure 3. Social distancing threshold. This is the threshold that dictates whether or not equilibrium behavior involves some social distancing. It
depends on both the basic reproduction number R
0
and the maximum efficiency m, and is independent of the exact form of s. As rough rules of
thumb, if R
0
v1 or mv2:718, then equilibrium behavior involves no social distancing.
doi:10.1371/journal.pcbi.1000793.g003
Figure 4. Total costs and savings. Plots of the total per-capita cost of an epidemic {V
S
(0) (left) under equilibrium social distancing for the
infinite-horizon problem with several efficiencies m under Eq. (6), and the corresponding per-capita savings (right). Savings in expected cost
compared to universal abstention from social distancing are largest for moderate basic reproduction numbers, but are relatively small, even in the
limit of infinitely efficient social distancing. The m~? case corresponds to infection of the minimum number of people necessary to reduce the
reproduction ratio below 1.
doi:10.1371/journal.pcbi.1000793.g004
Game Theory of Social Distancing
PLoS Computational Biology | www.ploscompbiol.org 6 May 2010 | Volume 6 | Issue 5 | e1000793
leads to a nonlinear optimal control problem that can be studied
using standard procedures [23]. Yet, practical bounds on the
performance of the optimal community strategy can be obtained
without further calculation. The optimal community strategy will
cost less than the game-theoretic solution per capita, but must cost
more than 1{1=R
0
, as that is the minimum number of people
who must become sick to reduce the effective reproduction
number below the epidemic threshold. Preliminary calculations
indicate that optimal community strategies and game equilibrium
strategies converge as R
0
grows, and significant differences are
only observable for a narrow window of basic reproduction
numbers near 1.
The results presented require a number of caveats. I have, for
instance, only considered one particular form for the relative risk
function. Most of the analysis has been undertaken in the absence
of discounting (h~0), under the assumption that the epidemic will
be fast compared to planning horizons. Discounting would
diminish importance of long term risks compared to the instant
costs of social distancing, and thus should diminish the benefits of
social distancing. The benefits of social distancing will also be
diminished by incorporation of positive terminal costs of
vaccination (kw0). Realistically, mass vaccination cannot be
accomplished all-at-once, as we assume. It’s much more likely that
vaccination will be rolled out continuously as it becomes available.
This could be incorporated into our analysis, for instance, by
including a time-dependent forcing. Other approaches include
extending the model to incorporate vaccination results of Morton
and Wickwire [28], or to allow an open market for vaccine
purchase [18].
The simple epidemic model is particularly weak in its prediction
of the growths of epidemics because it assumes the population is
randomly mixed at all times. We know, however, that the contact
Figure 5. Solutions when vaccine becomes available after a fixed time. These are time series of an equilibrium solution for social distancing
when mass vaccination occurs 8:6 generations (left) and 6:5 generations (right) after the start of the epidemic. Investments in social distancing begin
well after the start of the epidemic but continue right up to the time of vaccination. Social distancing begins sooner when vaccine development is
faster. For these parameter values (m~20, S(0)~R
0
~3, I(0)~3|10
{6
), individuals save 50% of the cost of infection per capita (left) and 80% of the
cost of infection (right).
doi:10.1371/journal.pcbi.1000793.g005
Figure 6. Windows of Opportunity for Vaccination. Plots of how the net expected losses per individual ({V
S
(0)) depend on the delay
between the start of social-distancing practices and the date when mass-vaccination becomes universally available if individuals use a Nash
equilibrium strategy. The more efficient social distancing, the less individuals invest prior to vaccine introduction. The blue lines (m~0) do not use
social distancing, as the efficiency is below the threshold. The dotted lines represent the minimal asymptotic epidemic costs necessary to stop an
epidemic.
doi:10.1371/journal.pcbi.1000793.g006
Game Theory of Social Distancing
PLoS Computational Biology | www.ploscompbiol.org 7 May 2010 | Volume 6 | Issue 5 | e1000793
patterns among individuals are highly structured, with regular
temporal, spatial, and social correlations. One consequence of
heterogeneous contact structure is that epidemics proceed more
slowly than the simple epidemic model naively predicts. Thus, the
simple epidemic model is often considered as a worst-case-
scenario, when compared with more complex network models
[29,30] and agent-based models [31–33]. In the context of social
distancing, it is not immediately clear how weaker mixing
hypotheses will affect our results. Weakened mixing will prolong
an epidemic, increasing the window over which social distancing is
needed. But under weakened mixing, individuals may be able to
use local information to refine their strategies in ways analogous to
the ideas of Funk et al. [9] and Perisic and Bauch [34]. In general,
the analysis of aggregate games with stochastic population
dynamics require a significant technical leaps, and are the subjects
of active research.
One of the fundamental assumptions in our analysis is that there
are no cost-neutral behavior changes that can reduce contact rates.
In fact, life-experience provides good evidence that many
conventional aspects of human behavior are conditional on
cultural norms, and that different cultures may adopt alternative
conventions. The introduction of a new infectious disease may
alter the motivational pressures so that behavioral norms that were
previously equivalent are no longer, and that one norm is now
preferred to the others. In such cases, there are likely to be
switching costs that retard the rapid adoption of the better
behaviors that conflict with cultural norms. The rate of behavior
change, then, would be limited by the rate of adoption of
compensatory changes in cultural norms that reduce the cost of
social distancing.
Another deep issue is that behavior changes have externalities
beyond influencing disease incidence, but we have not accounted
for these externalities. People’s daily activities contribute not just to
their own well-being but also to the maintenance of our economy
and infrastructure. Social distancing behaviors may have serious
negative consequences for economic productivity, which might
feed back into slowing the distribution of vaccines and increasing
daily cost-of-living expenses.
We can extend our analysis to include economic feedbacks by
incorporating capital dynamics explicitly. Individuals may accu-
mulate capital resources like food, water, fuel, and prophylactic
medicine prior to an epidemic, but these resources will gradually
be depleted and might be difficult to replace if social distancing
interferes with the economy flow of goods and services. Further
capital costs at the community and state scales may augment
epidemic valuations. These factors appear to have been instru-
mental in the recent US debate of school-closure policies. One
feature of a model with explicit capital dynamics is the possibility
of large economic shocks. This and related topics will be explored
in future work.
These calculations raise two important mathematical conjec-
tures which I have not attempted to address. The first is that the
social distancing game possesses a unique subgame-perfect Nash
equilibrium. There is reasonable numerical evidence of this in
cases where the relative risk function s is strictly convex, and
stronger unpublished arguments of this in cases of piecewise linear
s. I believe this will also be the case for non-convex but monotone
relative risks under some allowances of mixed-strategies. A second
conjecture, not yet addressed formally, is that increases in the
efficiency of social distancing always lead to greater use of social
distancing, all other factors being equal. This seems like common
sense, but the precise dependence of Figure 1 on the efficiency has
yet to be determined mathematically.
As with all game-theoretic models, human behavior is unlikely
to completely agree with our equilibria for many reasons,
including incomplete information about the epidemic and vaccine
and strong prior beliefs that impede rational responses. On the
other hand, our approach is applicable to a large set of related
models. We can analyze many more realistic representations of
pathogen life-cycles. For instance, arbitrary infection-period
distributions and infection rates can be approximated using a
linear chain of states or delay-equations [24]. Structured
populations with metapopulation-style mixing patterns may also
be analyzed. I hope to apply the methods to a wider variety of
community-environment interactions in the future.
Acknowledgments
The author thanks A. Bressan, A. Galvani, and E. Shim for helpful
discussion, and two anonymous referees for their valuable criticisms.
Author Contributions
Conceived and designed the experiments: TR. Performed the experiments:
TR. Analyzed the data: TR. Wrote the paper: TR.
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Game Theory of Social Distancing
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A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the "bang-bang" type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.
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Optimal control by immunization of a general deterministic model of an epidemic is examined when cost is measured by the maximum number of infectives plus a measure of the control effort expended. Results guaranteeing that the optimal control has one switch are presented, as are conditions under which the optimal control is to allow the epidemic to run its course unchecked. Optimal control by more efficient removal of infectives is also considered.
Book
The author presents an introduction to the theory of biologial conservation, including a wealth of applications to the fishery and forestry industries. The mathematical modelling of the productive aspects of renewable-resource management is explained, including both economic and biological factors, with much attention paid to the optimal use of resource stocks over time. This book includes chapters on the theory of resource regulation and on stochastic resource models, sections on irreversible investment, game-theoretic models, and dynamic programming.