Self-enforcing regional vaccination agreements

Abstract

In a highly interconnected world, immunizing infections are a transboundary problem, and their control and elimination require international cooperation and coordination. In the absence of a global or regional body that can impose a universal vaccination strategy, each individual country sets its own strategy. Mobility of populations across borders can promote free-riding, because a country can benefit from the vaccination efforts of its neighbours, which can result in vaccination coverage lower than the global optimum. Here we explore whether voluntary coalitions that reward countries that join by cooperatively increasing vaccination coverage can solve this problem. We use dynamic epidemiological models embedded in a game-theoretic framework in order to identify conditions in which coalitions are self-enforcing and therefore stable, and thus successful at promoting a cooperative vaccination strategy. We find that countries can achieve significantly greater vaccination coverage at a lower cost by forming coalitions than when acting independently, provided a coalition has the tools to deter free-riding. Furthermore, when economically or epidemiologically asymmetric countries form coalitions, realized coverage is regionally more consistent than in the absence of coalitions.

Full text

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Research
Cite this article: Klepac P, Megiddo I, Grenfell
BT, Laxminarayan R. 2016 Self-enforcing
regional vaccination agreements. J. R. Soc.
Interface 13: 20150907.
http://dx.doi.org/10.1098/rsif.2015.0907
Received: 22 October 2015
Accepted: 23 December 2015
Subject Areas:
computational biology, biomathematics
Keywords:
transboundary movement, regional
cooperation, epidemic dynamics,
SIR model, metapopulation
Author for correspondence:
Petra Klepac
e-mail: pklepac@alum.mit.edu
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2015.0907 or
via http://rsif.royalsocietypublishing.org.
Self-enforcing regional vaccination
agreements
Petra Klepac1, Itamar Megiddo2, Bryan T. Grenfell3,4,5
and Ramanan Laxminarayan2,3,6
1
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
2
Center for Disease Dynamics, Economics and Policy, Washington, DC 20036, USA
3
Ecology and Evolutionary Biology, and
4
Woodrow Wilson School of Public and International Affairs, Princeton
University, Princeton, NJ 08544, USA
5
Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
6
Public Health Foundation of India, New Delhi 110070, India
IM, 0000-0001-8391-6660
In a highly interconnected world, immunizing infections are a transbound-
ary problem, and their control and elimination require international
cooperation and coordination. In the absence of a global or regional body
that can impose a universal vaccination strategy, each individual country
sets its own strategy. Mobility of populations across borders can promote
free-riding, because a country can benefit from the vaccination efforts of
its neighbours, which can result in vaccination coverage lower than the
global optimum. Here we explore whether voluntary coalitions that
reward countries that join by cooperatively increasing vaccination coverage
can solve this problem. We use dynamic epidemiological models embedded
in a game-theoretic framework in order to identify conditions in which
coalitions are self-enforcing and therefore stable, and thus successful at
promoting a cooperative vaccination strategy. We find that countries can
achieve significantly greater vaccination coverage at a lower cost by forming
coalitions than when acting independently, provided a coalition has the tools
to deter free-riding. Furthermore, when economically or epidemiologically
asymmetric countries form coalitions, realized coverage is regionally more
consistent than in the absence of coalitions.
1. Background
Infectious diseases are a transnational problem that cannot be solved by
countries acting unilaterally. Because infections easily spread from one country
to another, controlling infectious diseases regionally requires international
cooperation and coordination of efforts. The need for cooperation in control
infectious diseases was recognized as early as the 1850s, when advances in
transportation and ease of travelling facilitated the spread of cholera epidemics
across Europe and to North America [1]. However, cooperation has not yet been
formally included in the modelling framework used to design immunization
strategies. The World Health Organization issues important guidelines and rec-
ommendations, but compliance with those guidelines is voluntary, and control
strategies are usually set and implemented by countries independently. By
focusing on strongly immunizing vaccine-preventable diseases, here we use a
coupled economic and epidemiological model to explore factors that can motiv-
ate coalition formation and promote cooperation among countries in designing
and implementing regional immunization strategies.
The performance of a vaccination campaign depends on transmission rates,
classically framed in terms of the basic reproduction ratio, R
0
, or the expected
number of new cases caused by a single infected case in an immunologically
naive population [2]. Paediatric mass vaccination at a level pagainst an
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immunizing infection reduces R
0
to an effective value, R
E
¼
R
0
(1 2p), which leads to a well-known threshold for herd
immunity, p
c
¼121/R
0
[2]. The underlying model involves
a homogeneous, well-mixed population, but the qualitative
prediction is robust: immunizing above the herd immunity
threshold, p
c
, leads to the local elimination of transmission
and the prevention of disease [2,3].
In order to decide on the best strategy, it is necessary to
take economic as well as epidemiological factors into consider-
ation. Vaccination programmes are costly, and when these
costs are explicitly balanced against the benefits of reduced
transmission and fewer cases, the best vaccination strategy
can lie anywhere from no intervention to local elimination,
depending on the relative costs of vaccination and infection
[4]; relatively non-pathogenic infections with expensive vac-
cines may generate an economic optimum vaccination rate
below the elimination level. While only four diseases are tar-
geted for global elimination (polio, guinea worm, malaria
and yaws), many more are controlled by routine vaccination
(e.g. rubella, mumps, rotavirus diarrhoea, Haemophilus influen-
zae type b, pertussis, diphtheria, tetanus, meningococcus C
and pneumococcus) that primarily provides early protection
against infections that are most dangerous for the very
young. Childhood immunizations prevent 2.5 million deaths
per year and have the potential to save another two million
deaths each year, mostly children under the age of five [5].
Here, we focus on strongly immunizing vaccine-preventable
diseases that are not necessarily aimed for global elimination,
and explore whether a cooperative regional vaccination
approach can improve national vaccination strategies.
To address the question of cooperation in a formal setting,
we model a region that aims to optimize its vaccination strat-
egy against a strongly immunizing infection. We assume that
vaccination needs to continue indefinitely even in the case of
local elimination to protect against imported infections or to
prevent a related pathogen to take advantage of the niche
vacated by elimination [6]. National vaccination strategies
reflect local interests, socioeconomic conditions and public
health priorities, and, as a result, can vary greatly within a
region. Disease dynamics in different countries of the region
are linked by cross-border movement of infected individuals,
and depend on the strength of population interchanges
between them. Countries with low vaccination coverage can
therefore act as a source of infection to their neighbours.
We allow countries to coordinate a regional vaccination
strategy by formation of coalitions through international
agreements and apply it to the control of infectious diseases
and the nonlinear dynamics that govern their spread. We
find that by forming coalitions and deciding on a joint vacci-
nation strategy, countries can achieve higher vaccination
coverage at a lower cost than when acting independently,
and that under certain conditions, a cooperative strategy of
this kind is stable. This result opens the way to the more
efficient use of existing public health resources.
2. Self-enforcing international agreements
Many environmental problems, such as depletion of the ozone
layer, pollution of air and the oceans, and climate change, have
a feature in common with infectious diseases, which is that
they are transboundary or global in nature and countries
cannot solve them by acting alone. The theory of international
environmental agreements (IEAs) offers useful insights for
studying transnational public goods, such as the protection
of the Earth’s ozone layer, greenhouse gas emissions
reduction, climate change mitigation and water management
[7,8]. To reach a common goal, countries form coalitions, but
there is no international body that can enforce these agree-
ments. The theory of IEAs tells us when such coalitions
succeed even though compliance is voluntary.
IEAs can be modelled in a game-theoretic framework
where countries first decide independently whether or not to
join the coalition, and then quantify their joint environmental
goals (e.g. pollution abatements) either simultaneously [7] or
with signatories taking the lead [8]. Because coalition mem-
bers’ abatement choice is an increasing function of the
number of member countries, the coalition implicitly employs
a carrot-and-stick mechanism: when a country joins, the
coalition rewards it by increasing abatement, and if it leaves,
then the coalition punishes it (and itself) by reducing it. At
the equilibrium, there is no incentive to leave (known as
internal stability) or join the coalition (known as external
stability)—the coalition is stable or self-enforcing [7–9].
Among identical countries, stable coalitions are rare and
agreements signed by all countries are unlikely owing to
free-riding [10]. If the difference between the global net
benefits under non-cooperative (countries acting indepen-
dently) and fully cooperative outcomes is large, so is the
incentive to free-ride, and the self-enforcing IEA cannot
support a large number of countries [8]. To increase partici-
pation and stability, coalition can employ a number of
measures, such as penalizing free-riding [8], offering transfers
[7,10,11], imposing trade sanctions [10,12] or linking environ-
mental protection to other international agreements, such as
those facilitating technology transfers [10,13].
3. Methods
We adapt the theory of IEAs to the particular problem of
transnational epidemiological dynamics. We incorporate
self-enforcing agreements in a metapopulation model for dynamics
of infectious diseases and consider their application for design and
implementation of regional control strategies for immunizing infec-
tions. We consider only strongly immunizing infections and
vaccines that mimic this immunity. Rapidly evolving pathogens
such as influenza require a more complex framework allowing
for different strain dynamics, host history of infection and
immunity [14,15] and are therefore not further considered here.
Coalitions are added to an explicit spatial SIR model
where npopulations are coupled through migration of infected
individuals (following [16,17])
_
Si¼
m
ið1piÞ
b
iX
n
j¼1
h
ijIjþ
m
i
2
43
5Si
_
Ii¼
b
iSiX
n
j¼1
h
ijIjð
m
iþniÞIi
_
Ri¼
m
ipiþniIi
m
iRi
_
Vi¼
m
ipi
m
iVi:
9
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
;
ð3:1Þ
Here, S
i
,I
i
and R
i
are respective proportions of susceptible,
infected and recovered individuals in population i, births are
balancing deaths at the rate
m
i
and the infection on average
lasts 1/n
i
. A proportion p
i
of the individuals are vaccinated at
birth (at the end of maternal immunity). Different populations
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are coupled through movement of infected individuals, captured
with the coupling matrix
h
,given by
h
ij ¼
ð1
h
0Þ, for i¼j
h
0
n1, for i=j:
(ð3:2Þ
The amount of transboundary movement across different
numbers of coupled countries (n) is symmetric to conserve
population sizes. This coupling reflects short trips made by
individuals, rather than permanent migration or relocation of
individuals.
For each population i, we distinguish between costs of
vaccination c(p
i
), that capture immunization programmes’
implementation and operation costs and increase exponentially
with the proportional increase in vaccination coverage (as sup-
ported by data, e.g. see figure 4 in [18]), and infection costs c
Ii
that capture direct and indirect costs of disease (e.g. morbidity,
mortality and loss of productivity) and so are proportional to
the equilibrium prevalence of infection [4],
cðpiÞ¼aiexp
i
cð
IiÞ¼cIi

Ii,)ð3:3Þ
with the total cost
p
i¼cðpiÞþcIi

Ii:ð3:4Þ
The cost of setting up a vaccination campaign in location iis
a
i
and the increase in vaccination cost for high coverage is cap-
tured by x(chosen to reflect that achieving 80% coverage costs
about five times as much to achieve 20% coverage).
We first consider countries that are identical in their parameters
for transmission rate, costs of vaccination and infection. In the
second part of the paper, we consider the interaction of asymmetric
countries to capture the heterogeneity in countries’ epidemiological
and economic conditions.
3.1. Self-enforcing vaccination agreements
Drawing on the theory of IEAs [8,11,19], we introduce self-
enforcing agreements to the management and control of immu-
nizing infections. Initially, we model coalition formation for
symmetric countries.
We set up a two-stage game. In the first stage, countries
decide whether or not to join a coalition. In the second stage,
the set of kcountries comprising the coalition chooses their
vaccination coverage p
sthat minimizes the combined costs of
vaccination and disease burden of the coalition (
p
C
¼k
p
s
because countries are identical)
p
s¼min
pk
p
s,ð3:5Þ
while incorporating the non-signatories’ reaction function in
their cost minimization. Countries outside of the coalition (non-
signatories) then minimize their local costs
p
ns
independently,
p
ns ¼min
p
p
ns,ð3:6Þ
3.2. Stability
A coalition of kcountries is self-enforcing or stable if no member
has the incentive to leave (internally stable), and no non-member
has the incentive to join (externally stable). A coalition is intern-
ally stable if the local cost of each member country (
p
s
)islower
in the status quo than its cost should it leave the coalition,
p
sðkÞ
p
nsðk1Þ. A coalition is externally stable if each non-
member’s local cost is lower than its cost should it choose to
accede,
p
nsðkÞ
p
sðkþ1Þ. By definition, k¼0 is internally and
k¼nis externally stable [7–9].
3.3. Travel restrictions
To increase cooperation and deter free-riding and defecting, the
coalition can impose a travel restriction by limiting movement of
infected individuals across its borders. This can be achieved,
for example, by reducing overall travel, by requiring proof of vac-
cination or by introducing border surveillance systems to detect
symptomatic individuals. Travel restriction acts here as a punish-
ment (like trade sanctions in [10,12]). Sanctions have a cost for
signatories as well as for non-signatories, and the coalition will
implement them only when their benefits outweigh their costs.
We then look at the effects of sanctions on the stability of
coalitions and on willingness to accede to a coalition. Following
[20], we assume that the coupling parameter
h
can be reduced by
imposing travel restrictions that limit cross boundary movement
of infected between signatories and non-signatories. We let Qbe
the total number of direct and indirect costs involved in travel
restrictions and assume that Qaffects the coupling parameter
between signatories (s) and non-signatories (ns) so that
h
q¼
h
0hðQÞ,
such that
Q[½0, Qmax,hð0Þ¼1, hðQmax Þ¼0, dh
dQ,0
and dh2
dQ2.0:ð3:7Þ
hðQÞ¼ QQmax
Qmax

2
:ð3:8Þ
When there are no restrictions imposed, the coupling par-
ameter
h
q
is equal to
h
0
. Full intensity of travel restrictions
results in complete isolation (there is no coupling), preventing
any cross-border movement. Both signatories and non-
signatories incur the direct and indirect cost of travel restrictions
(for example, direct costs by implementing the restrictions and
indirect ones through loss of trade),
p
q,s¼cðpsÞþcIsIsþðnkÞQ
q
and
p
q,ns ¼cðpnsÞþcIns

Ins þkQ
q,
9
>
>
>
=
>
>
>
;
ð3:9Þ
where qis a scaling parameter. For the coalition size kand the
cost of travel restrictions Qbetween any two countries, the
total cost incurred by the coalition is (n2k)Qand the sum of
the costs incurred by non-members is kQ.
In the second stage of the game, in addition to choosing the
vaccination level countries simultaneously choose the intensity of
travel restrictions. The coalition optimal strategy is the combina-
tion of vaccination coverage (p
s) and travel restriction intensity
(h(Q*)) that minimize the joint coalition costs
p
C¼k
p
q,s,given
the non-signatories choice.
p
s¼min
p,Qk
p
q,s:ð3:10Þ
3.4. Heterogeneity
As the cost of implementing travel restrictions can be prohibitive,
we next consider ways to promote coalition participation
in the absence of restrictions. While we first considered a meta-
population of identical countries, we now include regional
heterogeneity by allowing epidemiological and economic
parameters to vary between countries (equation (3.1)).
When countries are asymmetric, there are n
k
ways to
make a coalition of size kamong the total of ncountries, and
2
n
2(nþ1) possible coalitions in total (i.e. 1 ,kn). To mini-
mize its combined costs of vaccination and infection in the
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coalition C(
p
C
), the coalition now optimizes the vaccination
level for each of the countries in coalition ðp
i,i[CÞresulting
in a vector p¼ðp
iÞof optimal strategies
p¼min
p
p
C¼min
pX
i[C
p
i,ð3:11Þ
where
p
i
is given by equation (3.4).
In this case, coalitions of the same size can experience differ-
ent optimal levels of vaccination coverage and associated
prevalence and costs, depending on which countries are inside
(or outside) the coalition. Furthermore, in coalitions of size k,
country ican have different optimal vaccination strategies p
i
depending on cost and epidemic parameter values of other
members and non-members. The range of optimal outcomes p
i
is illustrated using summary statistics showing the mean
values and fifth and 95th quantiles.
3.5. Numerical simulations
All simulations were coded using the MATLAB programming
language version R2012a and its Optimization Toolbox and
performed on Princeton University’s Adroit computing cluster
(eight node Beowulf cluster). Equilibrium values are determined
using the fsolve function, which finds a root of the nonlinear
system of equations with the equilibrium value of the non-coupled
system as the initial condition for the solver. The minimization of
different cost functions is performed using the nonlinear program-
ming solvers fminbnd and fmincon, which respectively find a
minimum of constrained nonlinear single-variable (for a local
optimum of a single country) and multivariate functions (for a
global optimum of the system of ncountries). The minimization
procedure is constrained over the interval 0 p
i
1, where 1 ,
i,nand is subject to adjoint equations of the model described
by equations (3.1) and (3.2). For the model with travel restrictions,
additional constraints are given by equation (3.7). Simulations of
the model with heterogeneity find regional and local optima as
described above for all the countries over all 2
n
2npossibilities
(the non-cooperative outcome and all the possible coalitions).
4. Results
To study the effect of coalitions on vaccination coverage, we
refine a basic two-patch SIR model for immunizing infections
that includes economic constraints [4] in two significant ways.
First, we explore a system of ncountries coupled through
transboundary movement of infected individuals. Second,
we combine the game theory of international agreements
with the dynamic epidemiological model to allow relatively
complex patterns of coalitions in vaccine deployment
(Methods). Our analysis first focuses on a set of identical
countries (equal epidemiological and economic parameters).
If countries act independently, the result is the Nash equili-
brium [7,8]; each one chooses a vaccination strategy that
minimizes its local costs, and no country can profit by unilaterally
changing its strategy (local optimum—figure 1, red line). Full
cooperation is achieved if all countries try to minimize their com-
bined costs, or if a global planner can enforce a cost-minimizing
policy (global optimum—figure 1, green line). For highly coupled
regions, the independent, non-cooperative optimum results in
lower vaccination coverage (but at a higher cost) than the fully
cooperative outcome described by the global optimum [4].
Increasing the coupling or the number of interconnected
countries increases this mismatch between global (figure 1a,
green line and electronic supplementary material, figure S1)
and independent optima (figure 1 and electronic supplementary
material, figureS1 red line). The realized coverage in each country
is lower, but the sustained cost is higher (figure 1b,c and electronic
supplementary material, figure S1). Note that for the set of par-
ameters used in this example (and assuming we cannot stop
immunizing), the global optimum is not elimination—the cover-
age is below the herd immunity threshold, and the disease
prevalence is above zero.
4.1. Coalitions
The optimal outcome for the members of coalitions is to
increase their vaccination coverage compared with the case
when countries act independently. As a result of high cover-
age in the coalition (figure 1c, black line), non-signatories can
experience fewer incoming infections and may experience
lower costs compared with non-cooperative outcome (when
no countries form coalition). The non-signatories select a
level of vaccination coverage that minimizes their individual
costs (figure 1b,c grey line). Higher coverage in the coalition
reduces the prevalence and infection-related costs, making
further resources available for vaccination. The coverage in
the coalition depends on the number of signatories: when a
2 4 6 8 10 12 14
0.474
0.478
0.482
0.486
no. countries in coalition
costs
2 4 6 8 10 12 14
0.57
0.59
0.61
0.63
0.65
no. countries in coalition
realized coverage
51015
0.60
0.62
0.64
0.66
(a)(b)(c)
no. connected countries
realized coverage
independent opt.
global optimum
in coalition
outside coalition
independent optimum
global optimum
Figure 1. Multi-patch SIR model. (a) Global, fully cooperative optimum (green line) and the independent, non-cooperative optimum (Nash equilibrium), given in
red for increasing numbers of identical interconnected countries. Parameters: R
0
¼5, coupling strength ¼10
m
/(n21), a
i
¼0.1, c
Ii
¼5. Costs (b) and realized
coverage (c) for a system of 15 identical interconnected countries, for increasing coalition size (x-axis). Green and red lines show global and independent optima (as
in (a)), black lines show the optimum realized by the countries in the coalition, grey lines show optimum for countries that have not joined the coalition. Other
parameters as in (a).
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country joins, the coalition’s vaccination coverage increases
(figure 1c, black line).
4.2. Stability
Stability of a coalition depends on countries’ costs in the
coalition versus outside the coalition (see schematics in
figure 2a,b for unstable coalitions). Signatories to small
coalitions have a lower cost inside the coalition than what
their cost would be should they leave the coalition; these
coalitions are internally stable (figure 2c–e and figure 3a–c
blue shading, k3). For larger coalitions that achieve high
coverage, non-signatories benefit from the avoided incoming
infections, and therefore they have a lower cost by acting
independently than if they joined the coalition. In this case,
there is no incentive to join: the coalition is externally stable
(figure 2c–e and figure 3a–c orange shading,k3). The
larger the coalition, the higher immunization costs become rela-
tive to infection costs, increasing incentives to free-ride. At the
equilibrium, countries have no incentive either to leave or to
join the coalition: the coalition is stable (figure 2c–e and
figure 3a–c purple shading, k¼3 in this case). In stable
coalitions, countries voluntarily adhere to the regional strategy
and cooperatively increase their coverage: agreements are self-
enforcing. Similar to the solution in environmental agreements
[7,8], the self-enforcing vaccination agreement cannot support
a large number of identical countries.
4.3. Travel restrictions
To prevent disease spread, policymakers can implement
control at borders, require proof of vaccination or impose
travel restrictions. Travel restrictions have direct costs for
both non-signatories and for the coalition (electronic sup-
plementary material, figures S2gand S3ggrey line), but
they benefit the coalition in two ways. First, they directly
reduce the number of infections imported into signatory
countries. Second, travel restrictions isolate non-signatories,
stopping them from free-riding on elevated immunization
levels in the coalition; a cost for non-signatories (compare
grey lines in figure 3a,d,g). The combination of isolation
and lack of benefits from free-riding, leads to a shift in
costs that deters free-riding, and non-signatories are incenti-
vized to expand their immunization coverage (figures S2e
and S3ein electronic supplementary material, grey line;
note the increase in non-signatories’ cost of coverage in
the presence of travel restrictions), resulting in a higher
vaccination coverage in the entire region. With limited
free-riding on its efforts, the coalition signatories also
increase their own immunization levels (figures S2eand
S3ein electronic supplementary material). Both signatories
and non-signatories incur costs of travel restrictions accord-
ing to their relative sizes, so travel restrictions are costly for
small coalitions and for individual non-signatories, relative
to large coalitions.
In our simulations with n¼15 countries, when travel
restrictions are expensive to implement, other than the
grand coalition where k¼n, only coalitions of size k10
choose to use the strategy (figure 3fand electronic sup-
plementary material, figure S2d,g). When travel restrictions
are inexpensive, excluding the grand coalition, coalitions of
size k6 choose to implement them (figure 3iand electronic
supplementary material, figure S3d,g). For coalition members,
(a)
(c)(e)(d)
(b)
51015
2
4
6
8
10
12
14
internally stable
self-
enforcing
51015
2
4
6
8
10
12
14
no. countries in coalition
no. countries
51015
2
4
6
8
10
12
14
no. countries in coalition
no. countries
no travel restrictions expensive restrictions
no. countries in coalition
no. countries
inexpensive restrictions
externally
stable
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e
e
e
ble
ble
b
e
e
e
ab
ab
a
b
bl
b
sta
sta
s
a
a
a
y
y
y
s
s
ally
ally
a
y
y
y
na
na
n
a
al
a
ern
ern
e
n
n
n
nte
nte
n
e
er
e
in
in
n
nt
n
i
Figure 2. Coalition stability. (a) Externally unstable coalition—benefits of the coalition are greater than the free-riding pay-off, giving non-signatories the incentive to
join. (b) Internally unstable coalition—benefits of free-riding are greater than benefits from coalition giving signatories an incentive to defect. (c–e) Effects of travel
restrictions on coalition stability for different numbers of interconnected countries n(x-axis) and for increasing numbers of signatories k(y-axis). Blue shading shows
internally stable coalitions, externally stable coalitions are shaded orange, and their overlap shows self-enforcing coalitions. Coalitions in yellow are neither externally nor
internally stable, and area in white shows unfeasible coalitions. R
0
¼5, coupling strength ¼20
m
/(n21), a
i
¼0.1, c
Ii
¼5. (c) No travel restrictions, (d) expensive
travel restrictions, q¼1000, (e) inexpensive travel restrictions, q¼5000.
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the benefits of reduced disease burden from protection by
travel restrictions (electronic supplementary material figures
S2fand S3f, black line) outweigh the direct costs of implement-
ing restrictions (compare black lines in figures S2f,g and S3f,g
in electronic supplementary material). The restriction is
therefore a credible threat.
Because of the resulting change in costs, the payoffs are
larger inside the coalition, and fully cooperative coalition
(k¼n) becomes stable or self-enforcing (purple shading
figure 2d,e, figure 3d–i, see also electronic supplementary
material, figures S2 and S3). Note that the travel restrictions
are not implemented in any of the stable coalitions—it is
the credible threat of restrictions that encourages both joining
and remaining in the coalition.
4.4. Heterogeneity
Finally, we account for the spatial heterogeneity in costs,
disease burden or resources between countries in a
model without travel restrictions. We show the results for a
metapopulation of eight countries where cost of infection
varies linearly across countries from c
I1
¼1 for country 1,
and c
I8
¼15 for country 8 in figure 4 (see also electronic sup-
plementary material, figure S4). In this case, R
0
is the same for
all countries (R
0
¼5, p
c
¼0.8). The difference in cost
parameters leads to a range of local optimal vaccination
levels for different countries in a non-cooperative setting
(red lines in figure 4band electronic supplementary material,
figure S4). Heterogeneities in costs of vaccination or R
0
lead
to qualitatively similar results (see electronic supplementary
material, figures S5 and S6). With the heterogeneity in costs
of infection, local optima range from no vaccination for
country 1, to vaccinating above the elimination threshold p
c
for country 8 (red line in figure 4branges from 0 to greater
than 0.8). A country with low vaccination coverage can
now act as a source of infections for its well-vaccinated neigh-
bour. Even though vaccination coverage in country 8 is above
the elimination threshold p
c
, its prevalence is above zero
(figure 4c); it cannot reach elimination because of incoming
infections from other countries.
0
0.1
0.2
0.3
0.4
quarantine level
51015
0.478
0.480
0.482
0.484
0.486
no. countries in coalition
costs
51015
0.57
0.59
0.61
0.63
0.65
no. countries in coalition
realized coverage
51015
0.1
0.3
0.5
0.7
no. countries in coalition
quarantine level
0.475
0.480
0.485
(a)(b)(c)
(d)(e)(f)(d)(e)(f)
(g)(h)(i)
costs
0.58
0.60
0.62
0.64
0.66
realized coverage
0.478
0.480
0.482
0.484
0.486
costs
0.58
0.60
0.62
0.64
0.66
realized coverage
0
0.1
0.2
0.3
0.4
quarantine level
externally
stable
internally
stable
self-enforcing
global optimum
outside coalition
in coalition
independent opt.
Figure 3. Multi-patch SIR model with 15 coupled countries. Costs (a,d,g), realized coverage (b,e,h) and intensity of travel restrictions (c,f,i) for three control scen-
arios: no travel restrictions (a–c); expensive travel restrictions, q¼500 (d–f); inexpensive travel restrictions, q¼1000 (g–i). Green lines show vaccination
coverage at the global optimum (b,e,h) and the corresponding costs (a,d,g). Red lines show realized coverage and corresponding costs when countries are
acting independently (Nash equilibrium). Black and grey lines show optimal coverage and corresponding costs for signatories and non-signatories, respectively.
Internally stable coalitions are shaded blue, externally stable coalitions are shaded orange, and their overlap shows self-enforcing coalitions. Coalitions in white
are unstable. Parameters: R
0
¼5, coupling strength ¼20
m
/(n21), a
i
¼0.1, c
Ii
¼5.
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With eight countries, there are 247 possible coalitions of
k2 and one non-cooperative outcome, in which none of
the countries form a coalition (see electronic supplementary
material, figure S4 for an overview of all of the 248 optim-
izations). Resulting optima for a given country can vary
greatly in different coalitions of the same size, depending
on the parameter values of other countries inside and out-
side of the coalition (electronic supplementary material,
figure S4). For example, there are 35 coalitions of four
countries with country 1 as a member, and in those
coalitions, the optimal coverage for country 1 varies from
0% to 12% (electronic supplementary material, figure S4).
Figure 4 summarizes these results with circles showing the
mean optimal values for a given country and a coalition
of a given size, and whiskers showing the fifth and 95th
quantiles. For each country, we plot its summary statistics
for increasing coalition sizes—coalition size of 1 shows the
non-cooperative outcome (also given by the red line) and
coalition size 8 shows the fully cooperative outcome (also
given by the green line).
coalition
size
coalition
size
coalition
size
independent opt.
global optimum
in coalition
outside coalition
0.2
0.3
0.4
0.5
0.6
0.7
(a)
(b)
(c)
costs
0
0.2
0.4
0.6
0.8
1.0
coverage
12345678
0
0.02
0.04
0.06
0.08
0.10
0.12
prevalence
countr
y
independent opt.
global optimum
in coalition
outside coalition
1818181818181818
12345678
country
1818181818181818
12345678
country
1818181818181818
Figure 4. SIR model for a system of eight asymmetric interconnected countries showing summary statistics for costs (a), coverage (b) or prevalence (c). For each
country (shown on x-axis in black), the coalition sizes (indicated in grey on x-axis) are ordered from 1 (non-cooperative outcome) to 8 (fully cooperative outcome).
There are 8
k

possible coalitions of size k, and here we show the mean value and range of optimization outcomes for a given country in a coalition of a given
size. Circles show mean costs (a), coverage (b) or prevalence (c) for each country and each coalition size when that country is in coalition (black) and outside of
coalition (grey). Whiskers show fifth and 95th quantiles. Red and green lines show independent and global optimum for each country, respectively. Cost of infection
parameter varies linearly across countries from c
I1
¼1 for country 1, and c
I8
¼15 for country 8. R
0
¼5, coupling strength ¼10
m
/(n21), a
i
¼0.1. All 248
optimizations are shown in electronic supplementary material, figure S4.
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In a fully cooperative outcome (figure 4 green lines), opti-
mal vaccine coverage is considerably higher particularly for
the countries with previously low coverage (e.g. compare
red and green lines in figure 4 for country 1: vaccination
coverage increases from 0 to greater than 30%). The cost of
elevating coverage for these countries is high, but the savings
of their neighbours through avoided infections more than
compensate for these costs. As more countries join the
coalition, the vaccination coverage among signatories
increases (figure 4b, black circles) and approaches the global
optimum (figure 4b, green lines). Coverage among non-
signatories (figure 4bgrey dots) remains comparable to the
non-cooperative outcome (figure 4bred lines) although they
enjoy slightly lowered costs owing to free-riding (figure 4a,
grey dots). Compared with the non-cooperative outcome
(figure 4b, red lines), differences in coverage levels and preva-
lence decrease as the coalition approaches full cooperation
(figure 4b,c, note the decreased range of green compared
with than red lines; see also electronic supplementary
material, figure S6 where reductions in prevalence are
achieved at very little cost). Overall, the fully cooperative
coalition achieves higher vaccination coverage at a lower
cost than smaller coalitions, with some countries benefiting
more than others (figure 4aand see also electronic sup-
plementary material, figure S4). Countries with low
perceived cost of infection (countries 1 and 2 in figure 4a)
experience an increase in their costs compared with non-
cooperative outcome. If overall costs of the coalition decrease
when it becomes fully cooperative (all countries are mem-
bers), then its members can promote participation by
compensating the countries that would otherwise incur an
increase in costs. The coalition can become fully cooperative
in eight different ways—each respective country can be the
last one to join. Because of asymmetries in parameter
values, the costs incurred when increasing the size of the
coalition from seven to eight members will differ in each of
these scenarios. Regardless of which country joins the
coalition last the overall benefit of the coalition is positive
(figure 5), even though some countries can incur a cost
from joining a coalition (electronic supplementary material,
figure S7). With heterogeneity, the differences in incurred or
perceived costs between countries have the potential to be
used as compensation to increase coalition participation,
leading to elevated and more consistent vaccination coverage
in the region.
Heterogeneities in costs or epidemic parameters can be
exploited in terms of redistribution of benefits obtained by
increased vaccination coverage to stabilize the fully coopera-
tive coalition even without the threat of sanctions. A more
comprehensive analysis of asymmetries is needed to fully
identify where and how transfer systems can be used to
increase global welfare in infectious diseases prevention.
4.5. Caveats and future directions
The epidemic model is deliberately kept simple in this initial
study, but is subject to important caveats. Vaccines often
induce immunity that wanes over time (e.g. pertussis [21])
or can provide strong selection pressure that allows for emer-
gence and spread of viral immune escape variants [22–24].
Demographic parameters such as birth rate can greatly influ-
ence and drive the epidemic dynamics [25–27], whereas age
structure affects the spread of the disease and case fatality
patterns [28,29], suggesting that infection costs should also
vary with age. Stochastic effects, amplified by seasonality in
transmission, can lead to fade-outs at lower levels of coverage
than predicted by deterministic models [30– 32].
Here we model dynamics of infectious diseases and vac-
cination strategies on a population level. While individuals’
vaccine-seeking behaviour and response to interventions
contributes to the outcome and success of public health cam-
paigns, incorporating this behaviour in mechanistic models
of disease dynamics can be challenging [33] and is not further
considered here.
Finally, we look only at interactions between countries.
Donors, non-governmental organizations and foundations
play very important roles in the global health arena (e.g.
Rotary International and the Bill & Melinda Gates Foun-
dation in polio eradication and the Carter Center in the
case of neglected tropical diseases). Their incentives and
interventions, together with the political and economic set-
ting in which control strategies are framed, will be a
contributing determinant of the success of coordinated
control efforts.
5. Conclusion
The extensive literature on the spread of infectious diseases
[2,3] and the economics of their control by vaccination [34–
36] has not addressed the question of when regional
coalitions for regional disease control through vaccination
form and under what conditions they are stable. The theory
of IEAs [12] offers a useful baseline on coalition formation
and here we extend it and apply it to infectious diseases
and their nonlinear dynamics.
We study international coordination of immunization
efforts by linking self-enforcing coalitions with epidemiologi-
cal dynamics in a game-theoretic setting where countries are
coupled by transnational movement of infections. The effec-
tiveness of a coalition and the attractiveness of free-riding
are also a function of coupling, or the interconnectedness of
populations. As the transboundary coupling in this model
represents short trips made by the individuals, and not
1 2 3 4 5 6 7 8
0
0.005
0.010
0.015
0.020
0.025
last country to join a fully cooperative coalition
total coalition savings
Figure 5. Overall coalition savings when a coalition becomes a fully coop-
erative coalition (all countries participate, k¼n), with a country that
joins the coalition last indicated on the x-axis. Parameter values as in figure 4.
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permanent migration, the strength of coupling is expected to
be high and is set to 10 or 20 times the population turnover in
our analysis. International transportation data show that
coupling rates are very heterogeneous and region-specific,
and the values we consider are well within the range
observed in data. In 2005, there were more than 440 million
international tourist arrivals in Europe [37] compared with
nine million births [38], whereas Africa during the same
period had 37 million international arrivals [39] and around
30 million births [40].
When identical countries are coupled through the trans-
boundary movement of infections, global health benefits are
substantially higher in the fully cooperative outcome than
when countries act independently. That cooperative out-
comes are better than non-cooperative ones is considered
conventional wisdom, especially in economic literature
[8,10,41–43]), but this wisdom has not fully percolated to
other fields. By cooperating in the efforts to control infectious
diseases by means of immunization, we find that countries
can achieve much higher vaccination coverage at a lower
cost than when acting independently. This suggests that
coalitions may be helpful in improving the use of existing
resources and open up the funding for increasing vaccination
coverage or for other public health issues. However, because
of incentives to free-ride, large coalitions cannot be sustained
in a self-enforcing manner in the absence of sanctions (as it is
widely reported in economic literature [8,10,41– 43]).
Sanctions are commonly used in IEAs to increase coalition
participation and to ensure that signatories are meeting the
goals [10,12]. In the case of vaccination agreements, nonlinea-
rities in infectious disease dynamics provide a trade-off
between investing in the population immunity and the preva-
lence of infection in the population. With a threat of travel
restrictions, instead of free-riding on the high coverage in the
coalition, the non-signatories invest in higher local immunity
to avoid high costs of infection. As a result, non-signatories
realize higher vaccination coverage in the presence than in
the absence of imposed restrictions. The threat of sanctions
in this case leads to a substantial increase in vaccination
coverage both inside and outside the coalition.
Travel restrictions have been used to control the spread of
SARS [44] and Ebola virus [45,46], but their effectiveness is
limited [47,48] especially for diseases with presymptomatic
transmission like influenza [49– 51]. Furthermore, travel
restrictions are considered controversial owing to their
adverse economic impact [52,53] and prohibitive cost of
implementation. Epstein et al. [54] estimate that extensive
restrictions would cost the US 0.8% of its GDP, amounting
to over $130 billion based on 2013 data [55]. We therefore
also consider factors that could stabilize coalitions in the
absence of any restrictions.
Countries in a region can vary greatly in their epidemic,
demographic or socioeconomic characteristics. This existing
heterogeneity can be used to promote coalition participation
even in the absence of hard-to-implement restrictions.
Heterogeneity in cost and epidemiological parameters
results in diverse local vaccination optima and asymmetric
countries experience varying levels of savings (or costs) by
joining the coalition. This heterogeneity can increase
coalition participation when countries that benefit from the
coalition compensate others to join and increase their vacci-
nation coverage. As disparities in realized coverage among
countries decrease inside the coalition, the vaccination
coverage in the region becomes not only higher, but also
more consistent.
While many countries are adopting policies for univer-
sal coverage of childhood vaccines, vaccination coverage is
far from universal in many locations and increasing cover-
age in those places will require more investment. There are
particular disparities in local vaccination coverage in the
case of measles in sub-Saharan Africa [56], especially in
remote areas [57], making this system a good candidate
for policy interventions that foster coalitions. Another
example where a coalition approach would be useful is
in funding immunization campaigns with the new menin-
gococcal meningitis conjugate vaccine in the 25 countries of
the African meningitis belt [58 –60]. In those examples,
marginal benefit from vaccine uptake in a neighbouring
country can be significant enough to warrant regional
agreements.
Local infectious disease dynamics are one reason why
coalitions are so effective in increasing vaccination coverage.
Outbreaks, once sparked, depend predominantly on the local
effective transmission rate (R
E
.1)—i.e. the build-up of local
infectives [61]. Whereas importations of disease depend on
the herd immunity achieved by vaccination (supply of the
good by all countries) and the strength of connectivity with
other regions, the size of a local outbreak depends on the
number of non-immune individuals. If everybody in the
population is immunized, then the importations will not
lead to additional infections.
Most environmental issues are dynamically different. For
example, all countries contribute to the atmospheric accumu-
lation and mixing of ozone-depleting substances, such as
chlorofluorocarbons (CFCs). If only one country stops its
CFCs emissions, then the thickness of the ozone layer directly
above it will not improve; ozone layer protection requires
long-term commitment from nearly all countries in the
world. On the other hand, immunizing infections are domi-
nated by local nonlinearities arising from herd immunity.
Regional and global coordination is necessary to regulate dis-
ease importations and coordinate control efforts. In addition,
vaccines not only directly protect people who have been vac-
cinated, but also provide indirect protection to those
unvaccinated by reducing overall transmission. Local non-
linear dynamics of infectious diseases that unfold over
short periods and the indirect protection of vaccines make
the control of immunizing infections particularly fitting for
a regional approach.
Authors’ contributions. All authors contributed extensively to the work
presented in this paper. P.K. and I.M. conceived the study. P.K.,
I.M., B.T.G. and R.L. developed models. P.K. analysed models
numerically and produced all the figures. P.K., I.M., B.T.G. and
R.L. wrote the paper.
Competing interests. We declare we have no competing interests.
Funding. This research was supported by the Bill & Melinda Gates
Foundation, the RAPIDD programme of the Science & Technology
Directorate, Department of Homeland Security, and the Fogarty
International Center, National Institutes of Health (B.T.G.). P.K.
acknowledges the support from AXA Research Fund.
Acknowledgements. This work began as a part of the RAPIDD
Vaccine Refusal Workshop and was inspired by discussions with par-
ticipants of the workshop, especially Scott Barrett. L. Pomeroy and
A. Tavoni provided helpful comments on an earlier version of this
manuscript.
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