ArticlePDF Available

Self-enforcing regional vaccination agreements

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
rsif.royalsocietypublishing.org
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
&2016 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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 [79].
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
ið1piÞ
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
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
2
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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
n1, 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
nsðk1Þ. 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 [79].
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 QQmax
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,s¼cðpsÞþcIsIsþðnkÞQ
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 ,kn). To mini-
mize its combined costs of vaccination and infection in the
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
3
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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 p¼ðp
iÞof optimal strategies
p¼min
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).
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
4
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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, k3). 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,k3). 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 k10
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 k6 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
y
y
y
ly
y
ally
a
ally
y
a
a
na
n
na
a
rn
n
ern
e
ern
n
te
e
e
xte
x
xte
e
ex
x
x
ex
e
ex
x
e
e
e
e
e
e
y
y
y
ally
ally
a
y
y
y
na
na
n
a
all
a
ern
ern
e
n
na
n
xte
xte
x
e
er
e
ex
ex
e
x
xt
x
e
e
e
e
e
e
e
le
e
ble
b
bl
ab
b
b
ab
a
b
b
sta
a
sta
s
t
bl
bl
b
b
b
b
b
b
t
t
le
e
e
e
e
ab
b
ble
b
ble
e
sta
a
ab
a
ab
b
y
sta
s
sta
a
lly
y
y
y
y
na
a
ally
a
ally
y
rn
n
na
n
na
a
te
e
ern
e
ern
n
in
n
nte
n
nte
e
in
in
n
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.
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
5
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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.
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
6
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
With eight countries, there are 247 possible coalitions of
k2 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.
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
7
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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 [2224].
Demographic parameters such as birth rate can greatly influ-
ence and drive the epidemic dynamics [2527], 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.
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
8
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
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
[8,10,4143]), 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.
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
9
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
References
1. Huber V. 2006 The unification of the globe by
disease? The International Sanitary Conferences on
cholera, 1851– 1894. Hist. J.49, 453. (doi:10.1017/
S0018246X06005280)
2. Anderson RM, May RM. 1991 Infectious diseases of
humans. Oxford, UK: Oxford University Press.
3. Keeling MJ, Rohani P. 2008 Modeling infectious
diseases in humans and animals. Princeton, NJ:
Princeton University Press.
4. Klepac P, Laxminarayan R, Grenfell BT. 2011
Synthesizing epidemiological and economic optima
for control of immunizing infections. Proc. Natl
Acad. Sci. USA 108, 14 366 –14 370. (doi:10.1073/
pnas.1101694108)
5. Levine OS, Bloom DE, Cherian T, de Quadros C, Sow
S, Wecker J, Duclos P, Greenwood B. 2011 The
future of immunisation policy, implementation, and
financing. Lancet 378, 439448. (doi:10.1016/
S0140-6736(11)60406-6)
6. Lloyd-Smith JO. 2013 Vacated niches, competitive
release and the community ecology of pathogen
eradication. Phil. Trans. R. Soc. B 368, 20120150.
(doi:10.1098/rstb.2012.0150)
7. Carraro C, Siniscalco D. 1993 Strategies for the
international protection of the environment.
J. Public Econ. 52, 309 328. (doi:10.1016/0047-
2727(93)90037-T)
8. Barrett S. 1994 Self-enforcing international
environmental agreements. Oxford Econ. Pap. 46,
878894.
9. d’Aspremont C, Jacquemin A, Gabszewicz JJ,
Weymark JA. 1983 On the stability of collusive price
leadership. Can. J. Econ. 17 25. (doi:10.2307/
134972)
10. Carraro C, Siniscalco D. 1998 International
environmental agreements: incentives and political
economy. Eur. Econ. Rev. 42, 561– 572. (doi:10.
1016/S0014-2921(97)00118-9)
11. Barrett S. 2001 International cooperation for sale.
Eur. Econ. Rev. 45, 1835– 1850. (doi:10.1016/
S0014-2921(01)00082-4)
12. Barrett S. 1997 The strategy of trade sanctions in
international environmental agreements. Resour.
Energy Econ. 19, 345 361. (doi:10.1016/S0928-
7655(97)00016-X)
13. Wagner UJ. 2001 The design of stable international
environmental agreements: economic theory and
political economy. J. Econ. Surv. 15, 377 411.
(doi:10.1111/1467-6419.00143)
14. Francis T. 1960 On the doctrine of original antigenic
sin. Proc. Am. Philos. Soc. 104, 572578.
15. Kucharski A, Gog JR. 2012 Influenza emergence in
the face of evolutionary constraints. Proc. R. Soc. B
279, 645652. (doi:10.1098/rspb.2011.1168)
16. Grenfell BT, Bolker BM, Kleczkowski A. 1995
Seasonality and extinction in chaotic
metapopulations. Proc. Royal Soc. Med. 259,
97103. (doi:10.1098/rspb.1995.0015)
17. Keeling M, Rohani P. 2002 Estimating spatial
coupling in epidemiological systems: a mechanistic
approach. Ecol. Lett. 5, 20– 29. (doi:10.1046/j.1461-
0248.2002.00268.x)
18. Freuling CM, Hampson K, Selhorst T, Schro
¨der R,
Meslin FX, Mettenleiter TC, Mu¨ ller T. 2013 The
elimination of fox rabies from Europe: determinants of
success and lessons for the future. Phil. Trans. R. Soc.
B368, 20120142. (doi:10.1098/rstb.2012.0142)
19. Telser LG. 1980 A theory of self-enforcing
agreements. J. Bus. 53, 2744. (doi:10.1086/
296069)
20. Rowthorn R, Laxminarayan R, Gilligan C. 2009
Optimal control of epidemics in metapopulations.
J. R. Soc Interface 6, 11351144. (doi:10.1098/rsif.
2008.0402)
21. Lavine JS, King AA, Bjornstad ON. 2011 Natural
immune boosting in pertussis dynamics and the
potential for long-term vaccine failure. Proc. Natl
Acad. Sci. USA 108, 7259 7264. (doi:10.1073/pnas.
1014394108)
22. Restif O, Grenfell BT. 2007 Vaccination and the
dynamics of immune evasion. J. R. Soc. Interface 4,
143 153. (doi:10.1098/rsif.2006.0167)
23. Park AW, Daly JM, Lewis NS, Smith DJ, Wood JLN,
Grenfell BT. 2009 Quantifying the impact of immune
escape on transmission dynamics of influenza. Science
326, 726– 728. (doi:10.1126/science.1175980)
24. Pepin KM, Volkov I, Banavar JR, Wilke CO, Grenfell
BT. 2010 Phenotypic differences in viral immune
escape explained by linking within-host dynamics
to host-population immunity. J. Theor. Biol. 265,
501 510. (doi:10.1016/j.jtbi.2010.05.036)
25. Grenfell B, Bjornstad O, Finkenstadt B. 2002
Dynamics of measles epidemics: scaling noise,
determinism, and predictability with the TSIR
model. Ecol. Monogr. 72, 185 202. (doi:10.1890/
0012-9615(2002)072[0185:DOMESN]2.0.CO;2)
26. Pitzer VE et al. 2009 Demographic variability,
vaccination, and the spatiotemporal dynamics of
rotavirus epidemics. Science 325, 290– 294. (doi:10.
1126/science.1172330)
27. Bonds MH, Rohani P. 2010 Herd immunity acquired
indirectly from interactions between the ecology of
infectious diseases, demography and economics.
J. R. Soc. Interface 7, 541547. (doi:10.1098/rsif.
2009.0281)
28. Andreasen V. 1989 Disease regulation of age-
structured host populations. Theor. Popul. Biol. 36,
214 239. (doi:10.1016/0040-5809(89)90031-2)
29. Metcalf CJE, Lessler J, Klepac P, Cutts F, Grenfell BT.
2012 Impact of birth rate, seasonality and
transmission rate on minimum levels of coverage
needed for rubella vaccination. Epidemiol. Infect.
140, 2290 2301. (doi:10.1017/S09502688120
00131)
30. Bolker B, Grenfell B. 1996 Impact of vaccination on
the spatial correlation and persistence of measles
dynamics. Proc. Natl Acad. Sci. USA 93, 12 648
12 653. (doi:10.1073/pnas.93.22.12648)
31. Ferrari MJ, Grais RF, Bharti N, Conlan AJK, Bjornstad
ON, Wolfson LJ, Guerin PJ, Djibo A, Grenfell BT.
2008 The dynamics of measles in sub-Saharan
Africa. Nature 451,679– 684. (doi:10.1038/
nature06509)
32. Ferrari MJ, Grenfell BT, Strebel PM. 2013 Think
globally, act locally: the role of local
demographics and vaccination coverage in the
dynamic response of measles infection to control.
Phil.Trans.R.Soc.B368, 20120141. (doi:10.
1098/rstb.2012.0141)
33. Funk S, Bansal S, Bauch CT, Eames KTD, Edmunds
WJ, Galvani AP, Klepac P. 2015 Nine challenges in
incorporating the dynamics of behaviour in
infectious diseases models. Epidemics 10, 21 25.
(doi:10.1016/j.epidem.2014.09.005)
34. Geoffard P-Y, Philipson T. 1997 Disease eradication:
private versus public vaccination. Am. Econ. Rev. 87,
222230.
35. Bauch CT, Earn D. 2004 Vaccination and the theory
of games. Proc. Natl Acad. Sci. USA 101, 13 391
13 394. (doi:10.1073/pnas.0403823101)
36. Bhattacharyya S, Bauch CT. 2010 A game dynamic
model for delayer strategies in vaccinating behaviour
for pediatric infectious diseases. J. Theor. Biol. 267,
276282. (doi:10.1016/j.jtbi.2010.09.005)
37. UNWTO. 2006 International tourist arrivals by
country of destination—Europe. Tourism Market
Trends, Annex 7.
38. Lanzieri G. 2006 Long-term population projections
at national level. Eurostat: Stat. Focus, Popul. Soc.
Conditions 3,17.
39. UNWTO. 2006 International tourist arrivals by
country of destination—Africa. Tourism Market
Trends, Annex 4.
40. UN. 2010 World population prospects, the 2010
revision. See http://esa.un.org/unpd/wpp/Excel-
Data/population.htm (accessed 1 March 2013).
41. Barrett S. 2005 The theory of international
environmental agreements. In Handbook of
environmental economics, vol. 3 (eds KG Maler,
JR Vincent), pp. 1457 1516. Amsterdam, The
Netherlands: Elsevier. See http://www.sciencedirect.
com/science/handbooks/15740099.
42. Barrett S. 2003 Increasing participation and
compliance in international climate change
agreements. Int. Environ. Agreements: Politics,
Law Econ.3, 349– 376. (doi:10.1023/B:INEA.
0000005767.67689.28)
43. Carraro C, Siniscalco D. 1995 R&D cooperation
and the stability of international environmental
agreements. CEPR Discussion Papers, No.1154.
London, UK: CEPR.
44. Anderson RM, Fraser C, Ghani AC, Donnelly CA, Riley
S, Ferguson NM, Leung GM, Lam TH, Hedley AJ.
2004 Epidemiology, transmission dynamics and
control of SARS: the 20022003 epidemic. Phil.
Trans. R. Soc. Lond. B 359, 10911105. (doi:10.
1098/rstb.2004.1490)
45. Bogoch II et al. 2015 Assessment of the potential
for international dissemination of Ebola virus via
commercial air travel during the 2014 west African
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
10
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
outbreak. Lancet 385, 29– 35. (doi:10.1016/S0140-
6736(14)61828-6)
46. Gostin LO, Lucey D, Phelan A. 2014 The Ebola
epidemic: a global health emergency. JAMA 312,
10951096. (doi:10.1001/jama.2014.11176)
47. Hollingsworth TD, Ferguson NM, Anderson RM.
2006 Will travel restrictions control the international
spread of pandemic influenza? Nat. Med. 12,
497499. (doi:10.1038/nm0506-497)
48. Poletto C et al. 2014 Assessing the impact of travel
restrictions on international spread of the 2014 West
African Ebola epidemic. Euro Surveill.19, 20936.
(doi:10.2807/1560-7917.ES2014.19.42.20936)
49. Ferguson NM, Cummings DAT, Fraser C, Cajka JC,
Cooley PC, Burke DS. 2006 Strategies for mitigating
an influenza pandemic. Nature 442, 448– 452.
(doi:10.1038/nature04795)
50. Colizza V, Barrat A, Barthelemy M, Valleron A-J,
Vespignani A. 2007 Modeling the worldwide spread
of pandemic influenza: baseline case and
containment interventions. PLoS Med. 4, e13.
(doi:10.1371/journal.pmed.0040013)
51. Bajardi P, Poletto C, Ramasco JJ, Tizzoni M, Colizza
V, Vespignani A. 2011 Human mobility networks,
travel restrictions, and the global spread of 2009
H1N1 pandemic. PLoS ONE 6, e16591. (doi:10.1371/
journal.pone.0016591)
52. Knobler S, Mahmoud A, Lemon S, Mack A, Sivitz L,
Oberholtzer K, Institute of Medicine (US) Forum on
Microbial Threats. 2004 Learning from SARS: preparing
for the next disease outbreak: workshop summary.
Washington, DC: National Academies Press (US).
53. Beutels P, Jia N, Zhou Q-Y, Smith R, Cao W-C,
de Vlas SJ. 2009 The economic impact of SARS in
Beijing, China. Trop. Med. Int. Health 14, 85– 91.
(doi:10.1111/j.1365-3156.2008.02210.x)
54. Epstein JM, Goedecke DM, Yu F, Morris RJ, Wagener
DK, Bobashev GV. 2007 Controlling pandemic flu:
the value of international air travel restrictions. PLoS
ONE 2, e401. (doi:10.1371/journal.pone.0000401)
55. The World Bank 2015 United States data 2013. See
data.worldbank.org. 1– 4.
56. Grout L et al. 2014 Local discrepancies in measles
vaccination opportunities: results of population-
based surveys in Sub-Saharan Africa. BMC Public
Health 14, 193. (doi:10.1186/1471-2458-14-193)
57. Metcalf CJE, Tatem A, Bjornstad ON, Lessler J,
O’Reilly K, Takahashi S, Cutts F, Grenfell BT. 2015
Transport networks and inequities in vaccination:
remoteness shapes measles vaccine coverage and
prospects for elimination across Africa. Epidemiol.
Infect. 143, 1457 1466. (doi:10.1017/S0950268814
001988)
58. LaForce FM, Okwo-Bele J-M. 2011 Eliminating
epidemic group A meningococcal meningitis in
Africa through a new vaccine. Health Aff.
(Millwood)30, 1049 1057. (doi:10.1377/hlthaff.
2011.0328)
59. Trotter CL, Cibrelus L, Fernandez K, Lingani C,
Ronveaux O, Stuart JM. 2015 Response thresholds for
epidemic meningitis in sub-Saharan Africa following
the introduction of MenAfriVac
w
.Vaccine 33,6212
6217. (doi:10.1016/j.vaccine.2015.09.107)
60. MenAfriCar consortium. 2015 The diversity of
meningococcal carriage across the African
meningitis belt and the impact of vaccination with
a group a meningococcal conjugate vaccine. J Infect
Dis 212, 1298 1307. (doi:10.1093/infdis/jiv211)
61. Bjornstad O, Finkenstadt B, Grenfell B. 2002
Dynamics of measles epidemics: estimating scaling
of transmission rates using a time series SIR model.
Ecol. Monogr. 72, 169184. (doi:10.2307/3100023)
rsif.royalsocietypublishing.org J. R. Soc. Interface 13: 20150907
11
on January 20, 2016http://rsif.royalsocietypublishing.org/Downloaded from
... To explore, scrutinize, and provide insights into questions surrounding EEOT, the epidemiology and health economics fields have a rich toolbox that captures the nonlinear transmission dynamics, temporal features, and economic implications of disease "control"-the disease reduction that occurs without necessarily achieving EEOT. Separately, a few studies have tried to grapple with questions around the economic implications of EEOT by employing game-theoretic approaches (3,10,11). These studies, usually focused on vaccine-preventable diseases, have designated different levels of coverage (e.g., vaccine coverage) necessary for control and EEOT, and the difference in costs between control and EEOT strategies constitutes the price for elimination (3,10,11). ...
... Separately, a few studies have tried to grapple with questions around the economic implications of EEOT by employing game-theoretic approaches (3,10,11). These studies, usually focused on vaccine-preventable diseases, have designated different levels of coverage (e.g., vaccine coverage) necessary for control and EEOT, and the difference in costs between control and EEOT strategies constitutes the price for elimination (3,10,11). While making the problem theoretically tractable, applications of such methods are quite narrow: in practice, addressing Significance While the health economic implications of disease elimination have been discussed before, the combination of uncertainty, cost effectiveness, and elimination has not been tackled before. ...
Article
Full-text available
Significance While the health economic implications of disease elimination have been discussed before, the combination of uncertainty, cost effectiveness, and elimination has not been tackled before. We propose a modification to the net-benefit framework to explicitly consider the implications of switching from an optimal strategy, in terms of cost-per-burden averted, to a strategy with a higher likelihood of meeting the global target of elimination. The modification proposed yields a methodology to quantify the efficiency of elimination and to aid discussions among stakeholders with different objectives. We apply our method to strategies against human African trypanosomiasis in three settings, but this method is flexible enough that it can be applied directly to any simulation-based studies of disease elimination efforts.
... For instance, with coauthors, I investigated the effect of lowering travel rates between locations [20] and of interconnection between a large urban centre and smaller satellite cities [22]. Other interesting issues studied include the effect of vaccination targeted at high risk areas [29], cooperation between governments on vaccination policy [123]. Geographically targeted vaccination has also been considered at smaller spatial scales; see, e.g., [26,29,101,118,121,135]. ...
Chapter
Daily number of hospitalizations and deaths are key outcomes in quantifying the outbreak of infectious diseases. For the purposes of understanding the disease spread trend and the effect of observations from previous days, the time series approach for modeling the outcomes can be used to conduct a cointegration analysis to identify the long-run relationship between those multiple processes that are key to understanding trends such as hospitalization and death. As an alternative perspective, relationships between outcomes can be model through a shared latent stochastic error term; here, we propose a novel framework to study the underlying correlation between two time series processes through this method, called joint modeling. This framework is used for our Ontario Covid-19 study, where a cointegration analysis utilizes statistical tests to identify the long-run relationship between the daily number of new hospitalizations 6 days prior and the daily number of new deaths. Additionally, we show that a joint autoregressive model can provide a framework to model the underlying correlation between the processes.
... Whether elimination can be achieved at the country level, regionally or globally, depends upon coordination of interventions across political boundaries. Given vast inequities in resource availability, the willingness of rich countries to support control efforts in poorer countries is likely to be key, and there are many configurations where this will be to mutual advantage, given the ever present risk of pathogen introduction (Klepac et al. 2016). In a globalized world, metapopulation dynamics might be leveraged to rapidly and economically achieve elimination goals (Ruktanonchai et al. 2020). ...
Article
Full-text available
When a novel pathogen emerges there may be opportunities to eliminate transmission - locally or globally - whilst case numbers are low. However, the effort required to push a disease to elimination may come at a vast cost at a time when uncertainty is high. Models currently inform policy discussions on this question, but there are a number of open challenges, particularly given unknown aspects of the pathogen biology, the effectiveness and feasibility of interventions, and the intersecting political, economic, sociological and behavioural complexities for a novel pathogen. In this overview, we detail how models might identify directions for better leveraging or expanding the scope of data available on the pathogen trajectory, for bounding the theoretical context of emergence relative to prospects for elimination, and for framing the larger economic, behavioural and social context that will influence policy decisions and the pathogen’s outcome.
... Modeling studies have also shown that coordinated influenza vaccine sharing would reduce the financial and infection burden of influenza outbreaks globally (17). Similar problems related to optimizing vaccine allocation have also been explored in networks with community structure (18), as well as in the face of economic constraints (19) and vaccination coalition formation (20), and for SARS-CoV-2 with age-(and contact-) heterogeneity (21,22). ...
Article
Full-text available
Stockpiling and control A triumph that has emerged from the catastrophe of the severe acute respiratory syndrome coronavirus 2 pandemic has been the rapid development of several potent vaccines. However, 18 months into the pandemic and more than 6 months after vaccine approval, wealthy countries remain the major beneficiaries. Wagner et al . model the consequences of vaccine stockpiling in affluent countries on disease rates in lower- and middle-income countries and the consequences for the eruption of new variants that could jeopardize the early success of vaccines. For countries that can readily access vaccines, it would be better to share vaccines equitably to lower disease burdens in countries with less access, reduce the cost of having to be constantly vigilant for case imports, and minimize virus evolution. —CA
... For instance, differences in budget reductions in public health spending were found to be associated with regionally varying measles, mumps and rubella coverage in Italy [23]. Moreover, variation in vaccination strategies due to local interests, socioeconomic conditions and public health priorities were also linked to regional differences in vaccination coverage [24]. ...
Article
Full-text available
Background The World Health Organization recommends vaccination rates of 75% against seasonal influenza for patients over 65 years old. In the 2013/2014 season, the German vaccination rates ranged between 14 and 65%. This study aimed to compare the attitudes, personal characteristics and vaccination behaviours of general practitioners (GPs) in regions with high and low vaccination rates in Germany. Methods In May 2016, a questionnaire was sent to 1594 GPs practising in 16 districts with the highest and the lowest vaccination rates in Western and Eastern Germany as described by the Central Research Institute of Ambulatory Health Care in Germany for the 2013/2014 season. Descriptive statistics and multiple regression analyses were computed to identify potential factors associated with high vaccination rates. Results A total response rate of 32% (515/1594 participants) was observed in the study. GPs reported their attitudes towards vaccination in general and vaccination against influenza as mostly ‘very positive’ (80%, n = 352 and 65%, n = 288, respectively). GPs practising in regions with low vaccination rates reported their attitudes towards vaccinations in general (p = 0.004) and towards influenza vaccination (p = 0.001) more negatively than their colleagues from regions with high vaccination rates. Multiple logistic regression identified an increasing influence of year-dependent changing efficiency on GPs’ influenza rates as the strongest factor for predicting GPs from highly vaccinating regions (OR = 4.31 [1.12–16.60]), followed by the patient’s vaccination refusal despite GP advice due to already receiving a vaccination from another physician (OR = 3.20 [1.89–5.43]) and vaccination information gathering through medical colleagues (OR = 2.26 [1.19–4.29]). Conclusions The results of this study suggest a correlation between GPs’ attitudes and regional vaccination rates. Beneath GPs’ individual attitudes, the regional attitude patterns of patients, colleagues and medical assistants surrounding those GPs seem decisive and should be integrated into future campaigns to increase vaccination rates at a regional level.
... For instance, with coauthors, I investigated the effect of lowering travel rates between locations [20] and of interconnection between a large urban centre and smaller satellite cities [22]. Other interesting issues studied include the effect of vaccination targeted at high risk areas [29], cooperation between governments on vaccination policy [123]. Geographically targeted vaccination has also been considered at smaller spatial scales; see, e.g., [26,29,101,118,121,135]. ...
Preprint
Full-text available
SARS-CoV-2 started propagating worldwide in January 2020 and has now reached virtually all communities on the planet. This short review provides evidence of this spread and documents modelling efforts undertaken to understand and forecast it, including a short section about the new variants that emerged in late 2020.
... This same logic also applies at the global scale, and suggests that a global coordinated response may be necessary for the control of antibiotic resistance (58). In the absence of coordination, countries have the incentive to free-ride off of the vaccination efforts of their neighbors without ensuring that their own coverage levels are at the social optimum (59). This notion has received a lot of attention particularly in the context of disease eradication due to the massive potential gains arising from the eliminated need to maintain vaccination. ...
Article
Full-text available
Despite vast improvements in global vaccination coverage during the last decade, there is a growing trend in vaccine hesitancy and/or refusal globally. This has implications for the acceptance and coverage of a potential vaccine against COVID-19. In the United States, the number of children exempt from vaccination for “philosophical belief-based” non-medical reasons increased in 12 of the 18 states that allowed this policy from 2009 to 2017 ( 1 ). Meanwhile, the overuse and misuse of antibiotics, especially in young children, have led to increasing rates of drug resistance that threaten our ability to treat infectious diseases. Vaccine hesitancy and antibiotic overuse exist side-by-side in the same population of young children, and it is unclear why one modality (antibiotics) is universally seen as safe and effective, while the other (vaccines) is seen as potentially hazardous by some. In this review, we consider the drivers shaping the use of vaccines and antibiotics in the context of three factors: individual incentives, risk perceptions, and social norms and group dynamics. We illustrate how these factors contribute to the societal and individual costs of vaccine underuse and antimicrobial overuse. Ultimately, we seek to understand these factors that are at the nexus of infectious disease epidemiology and social science to inform policy-making.
Chapter
SARS-CoV-2 started propagating worldwide in January 2020 and has now reached virtually all communities on the planet. This short review provides evidence of this spread and documents modelling efforts undertaken to understand and forecast it, including a short section about the new variants that emerged in late 2020.
Article
Full-text available
Background The development of public health policy is inextricably linked with governance structure. In our increasingly globalized world, human migration and infectious diseases often span multiple administrative jurisdictions that might have different systems of government and divergent management objectives. However, few studies have considered how the allocation of regulatory authority among jurisdictions can affect disease management outcomes. Methods Here we evaluate the relative merits of decentralized and centralized management by developing and numerically analyzing a two-jurisdiction SIRS model that explicitly incorporates migration. In our model, managers choose between vaccination, isolation, medication, border closure, and a travel ban on infected individuals while aiming to minimize either the number of cases or the number of deaths. Results We consider a variety of scenarios and show how optimal strategies differ for decentralized and centralized management levels. We demonstrate that policies formed in the best interest of individual jurisdictions may not achieve global objectives, and identify situations where locally applied interventions can lead to an overall increase in the numbers of cases and deaths. Conclusions Our approach underscores the importance of tailoring disease management plans to existing regulatory structures as part of an evidence-based decision framework. Most importantly, we demonstrate that there needs to be a greater consideration of the degree to which governance structure impacts disease outcomes.
Preprint
Full-text available
The global health community has earmarked a number of diseases for elimination or eradication, and these goals have often been praised on the premise of long-run cost-savings. However, decision-makers must contend with a multitude of demands on health budgets in the short- or medium-term, and costs-per-case often rise as the burden of a disease falls, rendering such efforts beyond the cost-effective use of scarce resources. In addition, these decisions must be made in the presence of substantial uncertainty regarding the feasibility and costs of elimination or eradication efforts. Therefore, analytical frameworks are necessary to consider the additional effort for reaching global goals, like elimination or eradication, that are beyond the cost-effective use of country resources. We propose a modification to the net-benefit framework to consider the implications of switching from an optimal strategy, in terms of cost-per-burden-averted, to a strategy with a higher likelihood of meeting the global target of elimination of transmission by a specified date. We illustrate the properties of our framework by considering the economic case of efforts to eliminate transmission of gambiense human African trypanosomiasis (gHAT), a vector-borne parasitic disease in West and Central Africa, by 2030. Significance Statement Various diseases are earmarked for elimination by the global health community. While the health economic implications of elimination have been discussed before, the combination of uncertainty, cost-effectiveness in terms of cases averted, and elimination in the face of rising per-case costs has not been tackled before. We propose an approach that considers the tension between the dual objectives of cost-effectiveness and elimination while incorporating uncertainty in these objectives. We apply our method to strategies against human African trypanosomiasis in three settings, but this method could be directly applied to simulation-based studies of the cost-effectiveness of other disease elimination efforts. The method yields common metrics of efficiency when stakeholders have different objectives.
Article
Full-text available
Study of meningococcal carriage is essential to understanding the epidemiology of Neisseria meningitidis infection. Methods. Twenty cross-sectional carriage surveys were conducted in 7 countries in the African meningitis belt; 5 surveys were conducted after introduction of a new serogroup A meningococcal conjugate vaccine (MenAfriVac). Pharyngeal swab specimens were collected, and Neisseria species were identified by microbiological and molecular techniques. Results. A total of 1687 of 48 490 participants (3.4%; 95% confidence interval [CI], 3.2%–3.6%) carried meningococci. Carriage was more frequent in individuals aged 5–14 years, relative to those aged 15–29 years (adjusted odds ratio [OR], 1.41; 95% CI, 1.25–1.60); in males, relative to females (adjusted OR, 1.17; 95% CI, 1.10–1.24); in individuals in rural areas, relative to those in urban areas (adjusted OR, 1.44; 95% CI, 1.28–1.63); and in the dry season, relative to the rainy season (adjusted OR, 1.54; 95% CI, 1.37–1.75). Forty-eight percent of isolates had genes encoding disease-associated polysaccharide capsules; genogroup W predominated, and genogroup A was rare. Strain diversity was lower in countries in the center of the meningitis belt than in Senegal or Ethiopia. The prevalence of genogroup A fell from 0.7% to 0.02% in Chad following mass vaccination with MenAfriVac. Conclusions. The prevalence of meningococcal carriage in the African meningitis belt is lower than in industrialized countries and is very diverse and dynamic, even in the absence of vaccination.
Article
Full-text available
The WHO declared the 2014 west African Ebola epidemic a public health emergency of international concern in view of its potential for further international spread. Decision makers worldwide are in need of empirical data to inform and implement emergency response measures. Our aim was to assess the potential for Ebola virus to spread across international borders via commercial air travel and assess the relative efficiency of exit versus entry screening of travellers at commercial airports. We analysed International Air Transport Association data for worldwide flight schedules between Sept 1, 2014, and Dec 31, 2014, and historic traveller flight itinerary data from 2013 to describe expected global population movements via commercial air travel out of Guinea, Liberia, and Sierra Leone. Coupled with Ebola virus surveillance data, we modelled the expected number of internationally exported Ebola virus infections, the potential effect of air travel restrictions, and the efficiency of airport-based traveller screening at international ports of entry and exit. We deemed individuals initiating travel from any domestic or international airport within these three countries to have possible exposure to Ebola virus. We deemed all other travellers to have no significant risk of exposure to Ebola virus. Based on epidemic conditions and international flight restrictions to and from Guinea, Liberia, and Sierra Leone as of Sept 1, 2014 (reductions in passenger seats by 51% for Liberia, 66% for Guinea, and 85% for Sierra Leone), our model projects 2·8 travellers infected with Ebola virus departing the above three countries via commercial flights, on average, every month. 91 547 (64%) of all air travellers departing Guinea, Liberia, and Sierra Leone had expected destinations in low-income and lower-middle-income countries. Screening international travellers departing three airports would enable health assessments of all travellers at highest risk of exposure to Ebola virus infection. Decision makers must carefully balance the potential harms from travel restrictions imposed on countries that have Ebola virus activity against any potential reductions in risk from Ebola virus importations. Exit screening of travellers at airports in Guinea, Liberia, and Sierra Leone would be the most efficient frontier at which to assess the health status of travellers at risk of Ebola virus exposure, however, this intervention might require international support to implement effectively. Canadian Institutes of Health Research. Copyright © 2014 Bogoch et al. Open Access article distributed under the terms of CC BY. Published by Elsevier Ltd. All rights reserved.
Article
Full-text available
The quick spread of an Ebola outbreak in West Africa has led a number of countries and airline companies to issue travel bans to the affected areas. Considering data up to 31 Aug 2014, we assess the impact of the resulting traffic reductions with detailed numerical simulations of the international spread of the epidemic. Traffic reductions are shown to delay by only a few weeks the risk that the outbreak extends to new countries. © 2014 European Centre for Disease Prevention and Control (ECDC). All rights reserved.
Article
Full-text available
Traditionally, the spread of infectious diseases in human populations has been modelled with static parameters. These parameters, however, can change when individuals change their behaviour. If these changes are themselves influenced by the disease dynamics, there is scope for mechanistic models of behaviour to improve our understanding of this interaction. Here, we present challenges in modelling changes in behaviour relating to disease dynamics, specifically: how to incorporate behavioural changes in models of infectious disease dynamics, how to inform measurement of relevant behaviour to parameterise such models, and how to determine the impact of behavioural changes on observed disease dynamics. Copyright © 2014 The Authors. Published by Elsevier B.V. All rights reserved.
Article
Full-text available
SUMMARY Measles vaccination is estimated to have averted 13·8 million deaths between 2000 and 2012. Persisting heterogeneity in coverage is a major contributor to continued measles mortality, and a barrier to measles elimination and introduction of rubella-containing vaccine. Our objective is to identify determinants of inequities in coverage, and how vaccine delivery must change to achieve elimination goals, which is a focus of the WHO Decade of Vaccines. We combined estimates of travel time to the nearest urban centre (⩾50 000 people) with vaccination data from Demographic Health Surveys to assess how remoteness affects coverage in 26 African countries. Building on a statistical mapping of coverage against age and geographical isolation, we quantified how modifying the rate and age range of vaccine delivery affects national coverage. Our scenario analysis considers increasing the rate of delivery of routine vaccination, increasing the target age range of routine vaccination, and enhanced delivery to remote areas. Geographical isolation plays a key role in defining vaccine inequity, with greater inequity in countries with lower measles vaccine coverage. Eliminating geographical inequities alone will not achieve thresholds for herd immunity, indicating that changes in delivery rate or age range of routine vaccination will be required. Measles vaccine coverage remains far below targets for herd immunity in many countries on the African continent and is likely to be inadequate for achieving rubella elimination. The impact of strategies such as increasing the upper age range eligible for routine vaccination should be considered.
Article
Before the development of mass-vaccination campaigns. measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was similar to300 000-500 000. We develop a model, the TSIR (Time-series Susceptible-Infected-Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era (1944-1966). The 60 cities analyzed span a size range from London (3.3 X 10(6) inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission ("true mass action"). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R, is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus. efficient in "regulating" the susceptible population-even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly "deterministic" through most of the growth and decline phase.
Article
The gains from cartel formation and the stability of a dominant cartel are investigated for the price-leadership model. We show that there is a general interest in the establishment of a cartel with the competitive fringe reaping a disproportionate share of the benefits. In contrast to results involving a continuum of firms, with a finite number of firms (each with the same cost curve) there is always a stable dominant cartel. /// A propos de la stabilité d'une structure de marché caractérisée par la collusion de firmes dominantes pour établir un leadership de prix. Le mémoire étudie les gains dérivés de la formation d'un cartel et la stabilité d'un cartel dominant dans le cadre d'un modèle de leadership de prix de la firme dominante. On montre qu'il y a un intérêt général à créer un cartel même si les firmes satellites à la périphérie du cartel ramassent une part plus que proportionnelle des bénéfices. Contrairement à ce que l'on observe quand on est en présence d'un continuum de firmes, quand leur nombre est fini -- chacune avec la même courbe de coûts -- il y a toujours un cartel dominant stable.