Downloaded from rsif.royalsocietypublishing.org on January 22, 2010

Rational spatio-temporal strategies for controlling a Chagas
disease vector in urban environments
Michael Z. Levy, Fernando S. Malaga Chavez, Juan G. Cornejo del Carpio, Daril A. Vilhena, F. Ellis
McKenzie and Joshua B. Plotkin
J. R. Soc. Interface published online 8 January 2010
doi: 10.1098/rsif.2009.0479

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J. R. Soc. Interface
doi:10.1098/rsif.2009.0479
Published online

Rational spatio-temporal strategies
for controlling a Chagas disease vectorin urban environments
Michael Z. Levy1,2,3,*, Fernando S. Malaga Chavez4,
Juan G. Cornejo del Carpio4, Daril A. Vilhena1,
F. Ellis McKenzie3 and Joshua B. Plotkin1
1

Department of Biology, University of Pennsylvania, 219 Carolyn Lynch Laboratories,
Philadelphia, PA 19104, USA
2
Department of Biostatistics and Epidemiology, Center for Clinical Epidemiology andBiostatistics, University of Pennsylvania School of Medicine, Philadelphia, PA 19104, USA
3
Fogarty International Center, National Institutes of Health, 31 Center Drive – MSC 2220,
Bethesda, MD 20892-2220, USA
4
Dirección Regional del Ministerio de Salud, Av. Daniel Alcides Carrión 505-Cercado,
Arequipa, Peru
The rational design of interventions is critical to controlling communicable diseases,
especially in urban environments. In the case of the Chagas disease vector Triatoma
infestans, successful control is stymied by the return of the insect after the effectiveness of
the insecticide wanes. Here, we adapt a genetic algorithm, originally developed for the travelling salesman problem, to improve the spatio-temporal design of insecticide campaigns
against T. infestans, in a complex urban environment. We ﬁnd a strategy that reduces the
expected instances of vector return 34-fold compared with the current strategy of sequential
insecticide application to spatially contiguous communities. The relative success of alternative control strategies depends upon the duration of the effectiveness of the insecticide, andit shows chaotic ﬂuctuations in response to unforeseen delays in a control campaign. We
use simpliﬁed models to analyse the outcomes of qualitatively different spatio-temporal strategies. Our results provide a detailed procedure to improve control efforts for an urban Chagas
disease vector, as well as general guidelines for improving the design of interventions against
other disease agents in complex environments.
Keywords: disease ecology; vector-borne disease; control theory; Chagas disease;
Triatoma infestans; urban

1. INTRODUCTION

2002; Applegate 2007), but such algorithms are yet tobe adapted to the control of important infectious
diseases, such as Chagas disease.
Chagas disease kills more people in the Americas
than any other parasitic disease (World Health Organization 2003); recent estimates of the number ofindividuals infected with Trypanosoma cruzi, the etiologic agent of the disease, fall between 8 and 10
million individuals (Organizacion Panamericana de la
Salud 2006; Remme et al. 2006). Triatoma infestans,the principal insect vector of Chagas disease in SouthAmerica, is the target of an elimination programme
known as the Southern Cone Initiative. This initiative
has been remarkably successful, eliminating T. infestansfrom affected areas across southern South America
through the household application of insecticide. As aresult of the initiative, three countries, Chile (Lorca
et al. 2001), Brazil (Silveira & Vinhaes 1999) and
Uruguay (World Health Organization 1994), haveInterventions intended to control infectious diseases areoften ephemeral and fail to provide long-lasting protection. Even transient interventions, however, cansigniﬁcantly reduce disease transmission if employed
in a rational manner. The problem of optimizing thedesign of control efforts in time and space is difﬁcult
because the number of possible designs can be astronomical. A similar difﬁculty arises in the ‘travelling
salesman problem’, which consists of minimizing the
distance of a route taken by a salesman visiting numerous locations. A rich library of algorithms has beendeveloped to address the travelling salesman problem(Reinelt 1994; Larranaga et al. 1999; Gutin & Punnen*Author for correspondence (mzlevy@gmail.com).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2009.0479 or via http://rsif.royalsocietypublishing.org.
Received 30 October 2009
Accepted 14 December 20091

This journal is q 2010 The Royal Society

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2 Controlling a Chagas disease vector

M. Z. Levy et al.

been declared free of Chagas disease transmission by
T. infestans. Triatoma infestans control campaigns are
currently ongoing in southern Peru, where, in contrast
to other areas (Dias et al. 2002), the vector has
become an urban problem (Levy et al. 2006, 2007).
Even in rural areas, T. infestans readily re-infests
treated households after the residual effect of the insecticide has waned (Cecere et al. 2004; Dohna et al. 2007).
Each instance of vector re-infestation incurs signiﬁcant
expense to a control campaign because ﬁeld teams must
revisit and retreat re-infested households. Vectors may
re-infest a treated community from unnoticed orunder-treated sites within the community, through passive transport by humans or animals, or by active
dispersal from untreated communities (Cecere et al.
2006; Gurtler et al. 2007). In a city, where communities
are very close to one another and vector dispersal rates
are high (Levy et al. 2008), a large proportion of
the instances of re-infestation are likely due to active
dispersal by the insect.
A campaign to control T. infestans in the city of Arequipa, Peru, is currently ongoing. Resource constraints
limit the rate of advance of this campaign; thus not
all infested communities can be treated simultaneously.
An optimal design for this campaign must balance thedual and conﬂicting aims of maximizing control of thevector and minimizing costs (Forster & Gilligan 2007;
Rowthorn et al. 2009). The campaign, like manyurban vector control campaigns, follows a hub-andspoke model (Davis 1999) in which control personnel
begin and end each day in a central location, where
they receive insecticide, and are then transported
to the community in which they will be intervening.
Changing the order of treatment of affected communities, therefore, does not incur any additional cost to
the programme and may help decrease the riskthat untreated communities pose to those under
insecticide control.
The ordering of the application of insecticide in
T. infestans control campaigns is currently guided by
the rule of continuity (in time) and contiguity (inspace). This rule is considered the best practice to prevent vectors from returning to treated areas from
untreated areas (World Health Organization 2002).
Here, we develop a mathematical model that calculates
the expected number of instances of vector reinfestation owing to active dispersal. We use thismodel to evaluate vector control strategies based on
the rule of continuity and contiguity. We then showthat existing, best-practice strategies can be signiﬁcantly improved by a genetic algorithm originallydeveloped for the travelling salesman problem. We
study the properties of 1000 algorithm-generated control strategies and analyse the conditions under which
tactics present in these strategies improve control in
simpliﬁed systems. Unexpected delays in campaigns
to control disease agents are a constant in low-resource
settings with multiple public health priorities. We
consider how the timing of such delays affects therelative success of alternative control strategies. Our
results help to elucidate general principles for the successful control of disease agents and other unwanted
organisms in complex environments.
J. R. Soc. Interface

2. MATERIAL AND METHODS
2.1. Example study area
The district of Socabaya in the city of Arequipa has
grown dramatically owing to recent immigration. Population inﬂux has resulted in clusters of pueblos jovenes(new communities) separated from each other bytitled land. In preparation for the Chagas disease insecticide control campaign, trained specialists visited
Socabaya and surveyed 30 per cent of households inthe 53 communities in the district (ﬁgure 1) to estimate
the prevalence of T. infestans infestation (IID, following
local notation). Surveys were ‘rapid assessments’, lasting approximately 20 min per household. Control
personnel also produced detailed maps of community
borders, deﬁned by the predominant political unit of
the city, the ‘localidad’, carefully eliminating uninhabited areas. There were 13 699 households in the
district, arranged on rectangular city blocks. We wereunable to collect geographical coordinates on all
houses; therefore, we produced a grid within each community, with the number of nodes of the grid equal to
the number of houses in the community. Our calculations were done on the household level; we used thenodes of the grid to represent the position of eachhousehold. Triatomines infected with Trypanosoma
cruzi, the etiologic agent of Chagas disease, had been
repeatedly reported in a community at the northernend of the district of Socabaya. This community was
given preference by the Ministry of Health in the control
campaign and, therefore, we set it in the ﬁrst position in
all the alternative spray orderings we considered.
2.2. Vector dispersal model
Our model of vector dispersal is compartmental andsimilar to models of infectious disease transmission—
speciﬁcally, our model is an infectious – recovered–susceptible (IRS) model, a variant on the well-describedsusceptible–infected–recovered (SIR) model (Anderson &May 1991). Prior to vector control, a community is considered infested and ‘infectious’—meaning it can serve
as a source of dispersing insects. The community
becomes uninfested and uninfectious upon completionof treatment with insecticide, and it is ‘immune’ tore-infestation for the duration of the insecticide’s
residual effect. The community then becomes ‘susceptible’ to vector re-infestation and remains susceptible
for the remainder of the control campaign.
The period of time, measured in days, during whichhouses in each community are infectious, immune andsusceptible is determined by the ordering of communities in the vector control campaign, F. We translate
the ordering of communities into a schedule of days,ti, at which the spraying of half of the houses in each
community i is completed: 
j¼i1
1 X
Nfi
t i ðf Þ ¼
:
Nfj þ
r j¼1
2
Here, Nfj denotes the number of households incommunity j under the ordering F, and r is the rateof insecticide application (number of houses treatedDownloaded from rsif.royalsocietypublishing.org on January 22, 2010

Controlling a Chagas disease vector

M. Z. Levy et al. 3

houses in the two communities and the probability
that each house in the source community is infested
as estimated from entomologic surveys (IIDj ):XX
IIDj  Pab  f ðdab Þ:
Pij ¼

N

a in i b in j

E

W

Given an ordering of communities in a control campaign, F, the total expected number of instances ofvector re-infestation events owing to active dispersal is
the sum of the expected number of insects returning
to each community i from each source community
j during the period of time, denoted by Tij, thata house in i remains susceptible and a house in j
infectious:S

instances of vector returnjF ¼ SPij  Tij :Here, Tij, measured in days, depends on the average ofthe times at which spraying is completed in houses incommunities i and j:
Tij ¼ ðtj  ti Þ  L
Figure 1. Map of the 53 communities of the district ofSocabaya, Arequipa, Peru. Inset shows the outline of themetropolitan area of the city. Shading indicates the prevalence of T. infestans infested households. Infestation
prevalence: light pink, 0–0.2; light red, 0.2–0.4; dark red,
0.4– 0.6; brown, 0.6– 0.8; dotted line, river. Scale bar, 1 km.

per day). No community is left partially treated; once
spraying begins in a community, it is completed
before the campaign can advance to the next
community.
We use a Cauchy function derived from experimental
trapping data (Levy et al. 2008) to model the rate ofvector dispersal, given the Euclidean distance between
source and destination houses, dab. The function has
two parameters. The ﬁrst parameter, a, determines theshape of the curve; we estimated that a ¼ 28 throughtrapping experiments in urban Arequipa (Levy et al.
2008). The second parameter, b, represents the intrinsic
rate of vector movement. The expected number of
instances of vector return scales linearly with b, andthis term cancels out when comparing alternative strategies in the optimization routine. The complete function
2af ðdab Þ ¼ b 2
2a þ dabis graphed in ﬁgure S1 in the electronic supplementarymaterial.
The probability of vector dispersal to a given destination house is decreased in the presence ofalternative destination houses. We model the probability that a vector initiates dispersal to a given
house, rather than any other house, as the rate of dispersal to that house proportional to dispersal rates toall possible destination houses:f ðdab Þ:
all i f ðdib Þ

Pab ¼ P

We calculate the expected number of instances of vector
return per day from community j to community i as the
product of the summed rate of dispersal between all
J. R. Soc. Interface

ðor 0 if tj , ti þ LÞ;

where L denotes the duration of insecticide effectiveness, which we set at 100 days based loosely on a
study from Arequipa (Palomino et al. 2008). When
more than one spray brigade is available, Tij can be
decreased through concurrent treatment of successive
communities in the spray ordering. We calculated
Tij for the T. infestans control campaign in Socabaya
for two spray brigades of ﬁve individuals. From ourexperience in previous campaigns, each brigade
treats an average of 18 households per day. Following
practice in ﬁeld situations, brigades work in parallelin two different communities at all times in themodel.
2.3. Alternative vector control strategies
We considered ﬁve alternative codiﬁed strategies bywhich to apply insecticide to the district of Socabaya.
We ﬁrst generated an ordering for the control campaign
for Socabaya following the rule of contiguity (the contiguous strategy). The ordering was speciﬁed by eye,
without the aid of a computer, in the manner inwhich spray campaigns are routinely planned in Peru
and elsewhere (ﬁgure 2a). To counter the subjectivityof our analysis, we also formalized two versions of therule of contiguity. In the ﬁrst version, the spray campaign proceeds from each treated community to its
nearest geographical neighbour (the nearest-neighbour
strategy, ﬁgure 2c). In the second version, the campaign
proceeds to the untreated community that is nearest, on
average, to all treated communities (the average
distance strategy, ﬁgure 2d ).
We next considered two alternative strategies thatwere not based on the rule of contiguity. In the ﬁrst
alternative, the campaign moves outwards in concentric
circles, spraying communities in order of their distancefrom the ﬁrst community (the concentric circle strategy,
ﬁgure 2e). In the second alternative, we used the vector
data and dispersal model to deﬁne a spray orderingin which the campaign always proceeds to theuntreated community that is the greatest source ofDownloaded from rsif.royalsocietypublishing.org on January 22, 2010

4 Controlling a Chagas disease vector

M. Z. Levy et al.

(a)

(b)

(c)

(d )

(e)

(f)

Figure 2. Alternative spatio-temporal designs for the vector control campaign. Six alternative orderings of insecticide application
are shown: (a) the contiguous strategy; (b) the best route of 1000 generated by a genetic algorithm; (c) the nearest-neighbour
strategy; (d) the average distance strategy; (e) the concentric circle strategy; ( f ) the greedy algorithm strategy. The progressof control campaigns following these orderings, and the expected instances of vector re-infestation for each, is shown inmovies in the electronic supplementary material.

redispersing insects to all treated communities (thegreedy algorithm (Cormen & Cormen 2001) strategy,
ﬁgure 2f ).
2.4. Optimization algorithm
We used a genetic algorithm to minimize the expected
number of re-infestation events of treated communities
owing to active dispersal from untreated communities.
We initiated the algorithm with 80 replicates of the contiguous strategy in order to study how the algorithm
might alter the current best-practice design. The expected
number of instances of the return of T. infestans from
untreated areas within the district was calculated foreach ordering as described above. The reciprocal ofthis number was the ‘ﬁtness’ of the ordering, andserved as the objective function maximized by thealgorithm. In successive iterations, 80 pairs of parental
ordering were selected from the population by tournament. In each tournament, ﬁve individuals (sprayorders) were selected at random from the population.
The individual with the highest ﬁtness among the ﬁve
was selected as the maternal ordering. The process
was repeated, excluding the maternal ordering fromthe population, and a second ordering was selected as
the paternal ordering (Patel et al. 2005). The maternaland paternal ordering were then ‘mated’ in a process
mimicking biological reproduction. The process ofmating retained aspects of each parental order while
J. R. Soc. Interface

creating diversity in the speciﬁc ‘offspring’ arrangement(Patel et al. 2005). We used two functions to introduce
diversity in the mating process, crossover and mutation,
which are described in ﬁgure S2 in the electronic supplementary material. We produced a library of 1000
algorithm-generated orderings of insecticide application.
We show the best algorithm-generated ordering inﬁgure 2b. In each realization, we ran the algorithm until
1000 successive generations occurred without an improvement in the ﬁtness of the most-ﬁt order in the population.
The algorithm was written in the R statistical package(R 2.6; www.r-project.org).
2.5. Sensitivity analysis
We evaluated the sensitivity of the relative effect of the
ﬁve orderings generated from codiﬁed best-practicestrategies, and 1000 orderings generated by the genetic
algorithm. We tested the sensitivity to parametersdescribing the duration of insecticide effectiveness, L(ﬁgure 3), and the shape of the function relating
vector dispersal to distance, a. We recalculated the
expected number of instances of re-infestation for all
strategies varying L from 0 to 200 days, and varyinga from 0 to 100. We also tested the sensitivity of
the alternative strategies to unforeseen delays in thespray campaign. We considered the effects of 30-day
delays and varied the timing of these delays from the
beginning to the end of the control campaign.

Downloaded from rsif.royalsocietypublishing.org on January 22, 2010

Controlling a Chagas disease vector
(b)

B

best

best

(a)

M. Z. Levy et al. 5AGrankingranking

B
C
D
E

F
0

A

worst

worst

D

50

100

150

200

C, D, E, F
0

50

100

150

200

day of begin of delay

duration of insecticide effectFigure 3. Sensitivity of the relative success of alternative strategies to model parameters. The ranked ﬁtness of alternative orderings of the control campaign over a range of parameter values for the duration of insecticide activity (a) and in the presence of
30-day delays beginning at different points in the control campaign (b). Shown are: the 1000 algorithm-generated control strategies (shades of grey), the contiguous strategy (A, red), the best algorithm-generated strategy (B, blue), the greedy algorithm
strategy (C, magenta), the concentric circle strategy (D, green), the average distance strategy (E, brown), the nearest-neighbour
strategy (F, orange) and the most robust algorithm-generated strategy (G, yellow, in (a) only). The effects of 30-day delays
beginning after the 200th day of the spray campaign were minimal and are not shown.

In order to better understand the behaviour of alternative control strategies in the IRS system and vector
diffusion model described above, we considered two
simpliﬁed model systems. We asked under which conditions treating distal communities prior to more
proximate ones is beneﬁcial to vector control. We
ﬁrst considered three aligned communities of sizes
N1, N2 and N3, the ﬁrst of which has already received
insecticide treatment (treated; ﬁgure 4a). We ﬁxed the
distance between the treated and the proximate communities (d1) at 100 m, and we allowed the distance
between the treated and the distal communities (d2)to vary. We compared the number of expected
instances of vector return within the system for thetwo possible spray orders, sequential (treated –proximate –distal) and non-sequential (treated–distal–proximate). We added one simpliﬁcation to the modelby assuming equal prevalence of infestation in all communities. We also assumed that vector dispersal from acommunity was proportional to the number of housesin the community. Otherwise, parameter values andmethods were as described above. We then examinedthe effects of geography, keeping the size of communities constant at 2000 houses in a four-community
system (ﬁgure 5a). Finally, we applied successful tactics to a seven-community subset of the district ofSocabaya (ﬁgure 6).

3. RESULTS

(a)
N1

N2

(b)

N3
d2

d1
3000
beneficial
size of distal community(number of houses)

2.6. Simpliﬁed model systems

detrimental

2400
1800
1200

neutral

600
0
0

500
distance between proximate
and distal communities (m)1000

Figure 4. A simpliﬁed, three-community system. (a) Schematicof the system. The largest community, of size N1, is treatedﬁrst. (b) Conditions under which treating a distal community
prior to a proximate community is beneﬁcial (ﬁlled areas), detrimental and neutral to vector control. The size of the treatedcommunity (N1) is set at 1000 households. The distance
between the treated and proximate communities (d1) is set at
100 m. The size of the proximate community (N2) is set atthree example levels: 50 (black), 100 (dark grey) and 150(light grey). The duration of the insecticide is 100 days.

3.1. Vector surveys
Vector infestation rates varied greatly (0–83%) acrosscommunities in the district of Socabaya. Communities
near the centre of the district generally had higher infestation rates than those at the northern and southern
J. R. Soc. Interface

extremes (ﬁgure 1). Entomologic data were not available
at the community level for 19 of the 53 communities; weassumed that the prevalence of infestation in thesecommunities was equal to the district mean.

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6 Controlling a Chagas disease vector(a)M. Z. Levy et al.

(b)
D
50 m

treatedA

Table 1. The relative number of instances of vector
re-infestation owing to vector redispersal for alternative
control strategies.

100 m

B
C

Figure 5. A simpliﬁed four-community system. (a) Schematicof the system. Community A is always treated ﬁrst, and community D is always treated last. The size of each community is
set at 2000 households. The positions of communities A, B
and D are set (ﬁlled squares), while the position of community
C is varied across every point in a 200  200 grid. (b) The
positions of community C which favour treating in the orderof (A –B –C –D) are shown in black; the positions of community C which favour treating in the order of (A –C –B –D) arein grey.

(a)

(b)

(c)

(d )

Figure 6. Alternative tactics for ordering insecticide application
to a seven-community subset of Socabaya. The route ofinsecticide application is shown by dotted lines; each community is shaded in proportion to the expected number of
instances of vector return. (a) The contiguous strategy, (b) theconcentric circle strategy, (c) a route employing the tactic of
postponing treatment of a community contiguous to multiple
communities and (d) a route employing the same tactic aswell as spraying a larger, highly infested, community earlier.

3.2. Comparison of alternative vector
control strategies
Strategies to apply insecticide to Socabaya following
formalized rules of contiguity performed more poorly
than the contiguous ordering generated by eye. Under
our model, the nearest-neighbour strategy would leadto an 18.8-fold increase in the number of returning
insects compared with the contiguous strategy, andthe average distance strategy would lead to a 3.5-foldincrease (table 1). The concentric circle strategy
J. R. Soc. Interface

control strategy

instances of re-infestation(relative to contiguous strategy)

A. contiguous
B. algorithm-generated
C. nearest neighbour
D. average distanceE. concentric circle
F. greedy algorithm

10.10 (range 0.03 –0.24)
4.47
3.50
0.26
18.8

performed better than the contiguous strategy: spraying Socabaya by this strategy would lead to an
expected 3.9-fold decrease in the number of expected
instances of re-infestation. The greedy algorithm strategy, however, would lead to a 4.5-fold increase inreturning insects.
Each of the 1000 algorithm-generated strategies was
better than all the codiﬁed strategies. On average, thealgorithm-generated ordering had a 12-fold decrease in
the number of instances of active re-infestation compared with the contiguous strategy (5th and 95th
percentiles: 5.9 and 19.6). The best of the algorithmgenerated ordering (ﬁgure 2b) had 34-times fewer
expected instances of vector return. Movies
showing the progress and expected outcomes for thecontiguous and best algorithm-generated strategies areshown in ﬁgures S4– S9 in the electronic supplementarymaterial.
Any application of the rule of contiguity ‘by eye’ isbound to be imperfect and leave room for improvement.
However, rather than merely identifying better contiguous routes by which to spray Socabaya, the genetic
algorithm produced solutions that deviated fromsequential treatment of geographically proximate communities. We tabulated the frequency at which each
community occurred at each position in the 1000 algorithm-generated spray orders; these are shown in thetwo-dimensional histogram in ﬁgure 7. The position ofcommunities near the centre of the district deviatedmost often from the contiguous route. These communities were particularly susceptible to vector return
when Socabaya was treated following the contiguous
strategy (ﬁgure 2a). The histogram was more peakedfor communities that were very infectious, or highly
susceptible to vector re-infestation, reinforcing theimportance of careful placement of these communities
in the ordering of an intervention campaign. The histogram was ﬂatter for less susceptible and less infectious
communities, suggesting that the algorithm could more
freely shufﬂe the position of these communities in orderto improve the overall outcome of the spray order.
3.3. Sensitivity analysis
The relative effect of alternative control strategies was
sensitive to the duration of effectiveness of the insecticide, L, especially as this parameter was decreased
from the value used in the optimization routine (100days). The contiguous strategies fared better when theDownloaded from rsif.royalsocietypublishing.org on January 22, 2010

Controlling a Chagas disease vector

M. Z. Levy et al. 7

(b)
(a)
most
susceptible

leastsusceptible

position incontiguous
strategytreated earlier

treated later

en

cy

(c)

f

u
req

most
infectious

position in 1000 algorithm-generated orders

least
infectious
treated earlier

treated later

Figure 7. Frequency of position of communities in 1000 algorithm-generated strategies for the vector control campaign. (a) Communities are ordered by their position in the contiguous strategy used to initiate the algorithm (dotted line). (b) Communities arealigned to show deviation from their original position in the contiguous strategy, and ordered by their susceptibility to re-infestation
owing to vectors returning from all other communities and (c) their infectiousness to all other communities. Communities in the ﬁve
initial and ultimate positions in the contiguous ordering are not included in (b,c), so that the trends in positioning of the remaining
communities are more visible.

duration of insecticide was low; for L , 50 days, thebest algorithm-generated ordering was surpassed by
the contiguous strategy. Some algorithm-generatedorderings, however, were more robust to changes inthe duration of the effectiveness of the insecticide.
The most robust ordering, which had the highest
average ranking across the range of parameter values
considered, remained superior to the contiguous strategy for all values of L over 20 days. This most robust
strategy, however, was not robust to changes of othermodel parameters.
The ranking of alternative ordering was only minimally sensitive to the parameter describing therelationship between vector dispersal rate and distance,a (ﬁgure S3 in the electronic supplementary material).
Most of the sensitivity to the dispersal parameter was
focused at the very low end of the range considered;in all cases each algorithm-generated ordering was
superior to all of the ordering derived from the current
best-practice strategy.
We observed chaotic behaviour in the performance ofalternative control strategies in response to the timingof unforeseen delays. A strong dependency of the rankings of alternative strategies on the precise start point of
the delay is visible as vertical striations in the plot in
J. R. Soc. Interface

ﬁgure 3. In general, sensitivity to delays was greater
when the delays occurred at, or slightly before, themidpoint of the intervention campaign.

3.4. Simpliﬁed model systems
In a three-community system, there were a limited set ofparameter regimes that conferred a premium on treating
communities out of geographical order (ﬁgure 4). Two
conditions were necessary for the premium: (i) the
distal community must be much larger than the proximate community and (ii) the combined size of theproximate and distal communities must exceed
the number of households that can be sprayed during
the immune period of the ﬁrst treated community, but
not by too much (ﬁgure 4). The ﬁrst condition reﬂects
the advantage conferred by depleting a larger source of
vectors early. This advantage is countered by thesecond condition, which limits the amount of time thata smaller, closer, source of vectors can be allowed tothreaten a treated community. These competing conditions deﬁne the region in which deviating fromgeographical order is beneﬁcial—when the beneﬁt of
the ﬁrst condition exceeds the cost of the second.

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8 Controlling a Chagas disease vector

M. Z. Levy et al.

Analysis of a four-community system revealed a very
important tactic for vector control: when a proximate
community is susceptible to two distal communities,
it can be advantageous to postpone its treatment
until the vector population in at least one of the distal
communities is controlled (ﬁgure 5).
The seven-community subset of the district of Socabaya we considered (ﬁgure 6) exhibited many instancesof re-infestation under the contiguous strategy. These
communities were repeatedly treated out of geographical order by the algorithm-generated strategies. In
this subset of Socabaya, postponing the treatment of
a community contiguous to two others decreased the
expected number of instances of vector return(ﬁgure 6c). Employing this tactic in conjunction withtreating a large, highly infested, community earlier
further reduced the return of vectors (ﬁgure 6d).

4. DISCUSSION
The design of interventions against infectious diseases
can determine whether or not transmission is eventuallycontrolled. Reintroduction of disease agents followingintervention can be disastrous to a control campaign
and has frustrated efforts to control the Chagas disease
vector, T. infestans, in particular. The World HealthOrganization (2002) currently recommends applying
insecticide to communities affected by T. infestans following the ‘rule of contiguity’ in order to prevent
redispersal by the insect. In a complex urban environment, this rule cannot always effectively be translated
into practice. We show here that a genetic algorithm,
originally developed for the travelling salesman problem, can improve the ordering of blanket vector
control campaigns and diminish the risk of return ofvectors to areas under insecticide control.
Strategies based on the rule of contiguity, and several codiﬁed alternatives, all shared a limitation: they
were short-sighted and overextended the control campaign. The contiguous strategy left untreatedcommunities by the wayside, allowing a path of treatedcommunities to become increasingly exposed along its
ﬂank. Other codiﬁed strategies treated some communities prematurely and allowed them to remain
surrounded by untreated communities long after theeffectiveness of the insecticide had waned. Althoughthese strategies were successful early, as the campaigns
progressed the areas at risk accumulated, eventually
making the protection of all treated communities anintractable problem.
In contrast to the rule of contiguity and alternative
codiﬁed rules, the genetic algorithm explored many
more potential solutions, allowing it to overcome theshort-sightedness of the ﬁxed strategies. The control
campaigns generated by the algorithm often treatedcommunities that were not faced with an immediate
re-infestation threat but which would have faced athreat later in the campaign. Even in simpliﬁed systemsof three or four communities, there were some advantageous tactics that improved control by treating
communities out of strict geographical order. These
included clearing a path for the campaign by treating
J. R. Soc. Interface

large distal communities prior to smaller proximate
communities, and postponing the treatment of a proximate community if it was highly susceptible to multiple
distal communities until one or more of the distal communities were treated. In more complex systems, suchas the district of Socabaya, there are likely many moretactics that would decrease the risk of vector returnby treating communities out of geographical order.
The ﬂexibility of the optimization algorithm allowed
it to take advantage of different tactics at different
points in the control campaign, and thereby better
tailor the campaign to the particular geography andvector prevalence of the intervention area.
The duration of the effectiveness of an intervention
was a key factor in the success of a control campaign.
When an intervention had little or no residual effect,
contiguous strategies worked well; however, even aminimal lasting effect of insecticide created conditions
that favoured non-contiguous strategies. In otherwords, the contiguous strategy was the best option
when the intervention was poorly effective. The outcome of algorithm-generated spray ordering was moresensitive to an overestimation of the duration of theeffectiveness of the insecticide than to an underestimation. Conservative estimates of the duration of theresidual effect of insecticide against T. infestans will
provide more robust designs for a control campaign.
Unexpected delays in campaigns to control diseaseagents are to be expected. The success of a treatment
schedule against T. infestans depended strongly on
the unforeseeable timing of such delays. The dependence was chaotic—in some cases, the outcome of acontrol campaign was minimally affected by a delay ininsecticide application if the delay began on a certain
day, but greatly decreased if the delay began a daylater. No design was robust to all unforeseen delays.
The contiguous strategy did not perform better in theface of the uncertainty created by unforeseen delays.
The unpredictable detrimental effects of a delay are
best countered by rerunning the optimization algorithm, following the interruption of insecticide
application, to ﬁnd an optimal route by which totreat the remaining communities.
In practice, the design of control campaigns must be
guided by economic considerations (Forster & Gilligan
2007; Rowthorn et al. 2009). While urban vector control
campaigns follow a hub-and-spoke model, with theadvantage of a central location for storage of insecticideand biosafety equipment, rural campaigns often follow a
point-to-point model (Davis 1999) in which control personnel must travel directly from one community to thenext. Treating communities out of geographical order
may incur additional travel costs in such rural campaigns. To adapt the methods presented here to rural
campaigns, a utility function (McPake et al. 2002)
that weighs the beneﬁt of decreasing the number of
instances of vector re-infestation against the cost ofincreasing the distance travelled by a control teamcan be maximized by the genetic algorithm.
Our analysis was limited in a number of ways andincluded simpliﬁcations. We did not have accurate
information on the prevalence of vector infestation in
19 communities. Our prevalence data were taken atDownloaded from rsif.royalsocietypublishing.org on January 22, 2010

Controlling a Chagas disease vector
the level of the community, while the process of vector
dispersal occurs on a household scale, leaving our
results prone to errors owing to the modiﬁable unit problem (Openshaw 1983). Our estimates of vector
dispersal came from a single study; improved estimatesof dispersal rates over different distances, and in different environments, would help quantify the trade-off
between increased operational expenses and improved
vector control (Gurevitz et al. 2006; Marcet et al.
2008). We also could not take into account vector
return from parts of the city outside of the district ofSocabaya. Vector dispersal rate and insecticide activity
are also affected by temperature and rainfall (VazquezProkopec et al. 2006). While in Arequipa seasonalchanges are relatively minor given the high desert
environment, incorporating seasonal weather ﬂuctuations into the redispersal risk model may allow forfurther improvement of the spatio-temporal orderingof insecticide application, especially in other areas
with greater yearly weather ﬂuctuation. We limited
our focus to improving an existing vector control
campaign based on blanket application of insecticide.
Alternative methods of vector control, especially thosecomplementing insecticide application with greatercommunity involvement, may be more optimal andcost-effective strategies (Vazquez-Prokopec et al. 2009).
Many control measures against infectious diseases,
including drug treatments, vaccines and behaviouralinterventions, are impermanent and depend on careful
implementation for their success. The present work
addresses the spatio-temporal ordering for the control
of an existing problem; adapting these, or other, travelling salesman algorithms for more traditional SIR(Anderson & May 1991) models may provide a valuable
tool to curb the emergence of new disease transmission
cycles. The problems related to controlling dengue andother urban vector-borne infections are the most directly
analogous to those of controlling Chagas vectors. In
addition, the methods presented here may also be
useful in controlling directly transmitted infections.
We thank Robert Gilman, Caryn Bern, Eleazar Cordova,
Cesar Naquira, Hector Garcia, Lance Waller, Scott DukeSylvester, David Smith, Natalie Bowman, Cecile Viboud,
Rajan Patel, Megan Christenson, Nicole Salemno, thecroquisistas and all who helped on the ground in Arequipa,
and members of the Plotkin lab. Funding for this study
came from NIH K01 AI079162-02, NIH P50 AI074285-03
and NIH-NIDDK-08-007. J.B.P. also acknowledges funding
from the Burroughs Wellcome Fund, the James
S. McDonnell Foundation, the Alfred P. Sloan Foundationand the David and Lucile Packard Foundation.

REFERENCES
Anderson, R. M. & May, R. M. 1991 Infectious diseases ofhumans: dynamics and control. Oxford, UK: Oxford
University Press.
Applegate, D. L. 2007 The traveling salesman problem: a computational study. Princeton, NJ: Princeton University
Press.
Cecere, M. C., Vazquez-Prokopec, G. M., Gurtler, R. E. &Kitron, U. 2004 Spatio-temporal analysis of reinfestation

J. R. Soc. Interface

M. Z. Levy et al. 9

by Triatoma infestans (Hemiptera: Reduviidae) following
insecticide spraying in a rural community in northwestern
Argentina. Am. J. Trop. Med. Hyg. 71, 803– 810.
Cecere, M. C., Vasquez-Prokopec, G. M., Gurtler, R. E. &Kitron, U. 2006 Reinfestation sources for Chagas disease
vector, Triatoma infestans, Argentina. Emerg. Infect.
Dis. 12, 1096–1102.
Cormen, T. H. & Cormen, T. H. 2001 Introduction toalgorithms. Cambridge, MA: MIT Press.
Davis, G. B. 1999 The Blackwell encyclopedic dictionary ofmanagement information systems. Malden, MA: Blackwell
Business.
Dias, J. C., Silveira, A. C. & Schoﬁeld, C. J. 2002 The impactof Chagas disease control in Latin America: a review. Mem.
Inst. Oswaldo Cruz 97, 603–612. (doi:10.1590/S007402762002000500002)
Dohna, H. Z., Cecere, M. C., Gurtler, R. E., Kitron, U. &Cohen, J. E. 2007 Re-establishment of local populationsof vectors of Chagas disease after insecticide spraying.
J. Appl. Ecol. 44, 220–227. (doi:10.1111/j.1365-2664.
2006.01243.x)
Forster, G. A. & Gilligan, C. A. 2007 Optimizing the control
of disease infestations at the landscape scale. Proc. Natl
Acad. Sci. USA 104, 4984–4989. (doi:10.1073/pnas.
0607900104)
Gurevitz, J. M., Ceballos, L. A., Kitron, U. & Gurtler, R. E.
2006 Flight initiation of Triatoma infestans (Hemiptera:
Reduviidae) under natural climatic conditions. J. Med.
Entomol. 43, 143–150. (doi:10.1603/0022-2585(2006)
043[0143:FIOTIH]2.0.CO;2)
Gurtler, R. E., Kitron, U., Cecere, M. C., Segura, E. L. &Cohen, J. E. 2007 Sustainable vector control and management of Chagas disease in the Gran Chaco, Argentina.
Proc. Natl Acad. Sci. USA 104, 16 194–16 199. (doi:10.
1073/pnas.0700863104)
Gutin, G. & Punnen, A. P. 2002 The traveling salesman problem and its variations. Dordrecht, The Netherlands:
Kluwer Academic Publishers.
Larranaga, P., Kuijpers, C. M. H., Murga, R. H., Inza, I. &Dizdarevic, S. 1999 Genetic algorithms for the travelling
salesman problem: a review of representations and operators. Artif. Intell. Rev. 13, 129–170. (doi:10.1023/
A:1006529012972)
Levy, M. Z., Bowman, N. M., Kawai, V., Waller, L. A.,
Cornejo del Carpio, J. G., Cordova Benzaquen, E.,
Gilman, R. H. & Bern, C. 2006 Periurban Trypanosoma
cruzi-infected Triatoma infestans, Arequipa, Peru.
Emerg. Infect. Dis. 12, 1345 –1352.
Levy, M. Z. et al. 2007 Targeted screening strategies to detect
Trypanosoma cruzi infection in children. PLoS Negl. Trop.
Dis. 1, e103. (doi:10.1371/journal.pntd.0000103)
Levy, M. Z. et al. 2008 Impregnated netting slows infestation by
Triatoma infestans. Am. J. Trop. Med. Hyg. 79, 528–534.
Lorca, M., Garcia, A., Contreras, M. C., Schenone, H. &Rojas, A. 2001 Evaluation of a Triatoma infestans elimination program by the decrease of Trypanosoma cruzi
infection frequency in children younger than 10 years,
Chile, 1991– 1998. Am. J. Trop. Med. Hyg. 65, 861– 864.
Marcet, P. L., Mora, M. S., Cutrera, A. P., Jones, L., Gurtler,
R. E., Kitron, U. & Dotson, E. M. 2008 Genetic structureof Triatoma infestans populations in rural communities ofSantiago del Estero, northern Argentina. Infect. Genet.
Evol. 8, 835– 846. (doi:10.1016/j.meegid.2008.08.002)
McPake, B., Kumaranayake, L. & Normand, C. E. M. 2002
Health economics: an international perspective. London,
UK: Routledge.
Openshaw, S. 1983 The modiﬁable areal unit problem.
Norwich, UK: Geo Books.

Downloaded from rsif.royalsocietypublishing.org on January 22, 201010

Controlling a Chagas disease vector

M. Z. Levy et al.

Organizacion Panamericana de la Salud. 2006 Estimacion
cuantitativa de la enfermedad de Chagas en las Americas.
Montevideo, Uruguay: Organizacion Panamericanade la Salud. OPS/HDM/CD/425-06.
Palomino, M., Villaseca, P., Cárdenas, F., Ancca, J. & Pinto,
M. 2008 Eﬁcacia y residualidad de dos insecticidas piretroides contra Triatoma infestans en tres tipos deviviendas. Evaluacion de campo en Arequipa, Perú. Rev.
Peru Med. Exp. Salud Publica 25, 9– 16.
Patel, R., Longini Jr, I. M. & Halloran, M. E. 2005 Finding
optimal vaccination strategies for pandemic inﬂuenza
using genetic algorithms. J. Theor. Biol. 234, 201–212.
(doi:10.1016/j.jtbi.2004.11.032)
Reinelt, G. 1994 The traveling salesman: computational
solutions for TSP applications. Berlin, Germany:
Springer-Verlag.
Remme, J. H. F. et al. 2006 Tropical diseases targeted for
elimination: Chagas disease, lymphatic ﬁlariasis, onchocerciasis, and leprosy. In Disease control priorities indeveloping countries (eds D. T. Jamison, J. G. Breman,
A. R. Measham, G. Alleyne, M. Claeson, D. B. Evans,
P. Jha, A. Mills & P. Musgrove), pp. 433–449.
New York, NY: The World Bank and Oxford University
Press.
Rowthorn, R. E., Laxminarayan, R. & Gilligan, C. A. 2009
Optimal control of epidemics in metapopulations.

J. R. Soc. Interface

J. R. Soc. Interface 6, 1135–1144. (doi:10.1098/rsif.2008.
0402)
Silveira, A. & Vinhaes, M. 1999 Elimination of vector-borne
transmission of Chagas disease. Mem. Inst. Oswaldo
Cruz 94(Suppl. 1), 405–411. (doi:10.1590/S007402761999000700080)
Vazquez-Prokopec, G. M., Ceballos, L. A., Marcet, P. L.,
Cecere, M. C., Cardinal, M. V., Kitron, U. & Gürtler, R. E.
2006 Seasonal variations in active dispersal of natural populations of Triatoma infestans in rural north-western
Argentina. Med. Vet. Entomol. 20, 273–279. (doi:10.1111/
j.1365-2915.2006.00637.x)
Vazquez-Prokopec, G. M., Spillmann, C., Zaidenberg, M.,
Kitron, U. & Gurtler, R. E. 2009 Cost-effectiveness of
Chagas disease vector control strategies in northwestern
Argentina. PLoS Negl. Trop. Dis. 3, e363. (doi:10.1371/
journal.pntd.0000363)
World Health Organization. 1994 Chagas disease: eliminationof transmission. Wkly Epidemiol. Rec. 69, 38 –40.
World Health Organization. 2002 Control of Chagas disease:second report of the WHO expert committee. Geneva,
Switzerland: World Health Organization.
World Health Organization. 2003 Annex 2, Deaths by cause,sex and mortality stratum in WHO regions, estimates for2002. In The world health report 2003. Geneva,
Switzerland: World Health Organization.