Vector migration and dispersal rates for sylvatic Trypanosoma cruzi ...

Ecological Complexity 14 (2013) 145–156

Original Research Article

Vector migration and dispersal rates for sylvatic Trypanosoma cruzi transmission

Britnee A. Crawford *, Christopher M. Kribs-Zaleta

University of Texas at Arlington, United States

A R T I C L E I N F O

Article history:

Received 16 May 2012

Received in revised form 4 October 2012

Accepted 22 November 2012

Available online 26 January 2013

Keywords:

Trypanosoma cruzi

Migration

Dispersal

Distribution

A B S T R A C T

The spread of vector-borne diseases are greatly increased by a vector’s ability to migrate. Recent studies

of sylvatic Trypanosoma cruzi transmission have motivated the study of vector migration across

geographic regions. Due to the natural mechanisms in which vector-borne diseases are transmitted

between vectors and hosts, vector dispersal among different host populations is a critical factor in the

ability of the parasite to be spread across large regions. In this study we develop a general framework for

deriving large-scale, discrete-space migration rates from small-scale, continuous-space dispersal data.

We identify three defining characteristics of vector migration: distance, preferred direction of dispersal,

and strength of preference for a particular direction. We consider several migration scenarios in which

vectors may have no preference for dispersal in a particular direction or may disperse with a preferred

direction, such as northeast. We examine what effect preferred direction has on the migration rate, as

well as use the local to global framework to calculate numerical estimates for vector migration rates for

the primary vectors in the southeast U.S. and northern Mexico, Triatoma sanguisuga and Triatoma

gerstaeckeri, based on biological and experimental data. Results from this study can be applied to

metapopulation models for species that migrate.

� 2013 Elsevier B.V. All rights reserved.

Contents lists available at SciVerse ScienceDirect

Ecological Complexity

jo ur n al ho mep ag e: www .e lsev ier . c om / lo cate /ec o co m

1. Introduction

A key element in the description of the spatial ecology ofinfectious diseases is the effect of dispersal and migration on thetransmission of the disease. For vector-borne diseases, themovement of vectors is often a primary factor in geographicalspread. However, collection of field data on large-scale vectormovement over the course of many days is prohibitively difficult,and typically only short-term data is available, describing onlylocal dispersals over the course of a single day. Connecting thesedata to long-range migration across large regions through whichan infection spreads is difficult: any one vector may travel arelatively short distance, but aids in transporting the disease tonew hosts, from whom new vectors may transport it further.Several mathematical frameworks have been proposed to modelmovement of species, but many of them assume a given migrationrate as a model input rather than deriving it from small-scale dataon the individual vectors. This study proposes a new frameworkfollowing the latter approach.

Reaction-diffusion equations have been used to model dispersal(in continuous time and space) due to the extensive theorydeveloped for such equations (Mollison, 1972; Cantrell and Cosner,

* Corresponding author at: Dallas Baptist University, Dallas, TX 75211,

United States.

E-mail addresses: [email protected], [email protected] (B.A. Crawford).

1476-945X/$ – see front matter � 2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.ecocom.2012.11.003

2003). Other types of dispersal models are based in discrete time,such as dispersal modeled with integrodifference equation by Kotet al. (1996), in which the focus is to incorporate dispersal into apopulation model using a redistribution (distribution) kernel.Results show that broad-tailed dispersal distributions in discrete-time models can exhibit accelerating invasion speeds rather thanconstant-speed traveling waves.

Linking local dispersal to global migration across discreteregions is of particular importance in understanding spread ofvector-borne diseases, in which vector dispersal is often theprimary epidemiological connection between distinct transmis-sion cycles in adjacent regions (which can then be treated aspatches). The modeling approach in this study can be used formany types of diseases involving vector migration; here, weconsider Chagas’ disease spread by the parasite, Trypaonsoma cruzi.The disease is maintained primarily in sylvatic transmission cyclesincluding Triatoma vectors and several reservoir hosts. Communi-cation among transmission cycles occurs mainly as a result ofvector movement among the different sylvatic host populations, sostudying the spatial spread of T. cruzi across discrete sylvaticregions is of importance.

Mathematical models specific to studying dispersal of Triatoma

vectors include models focusing on Triatoma invasion of domesticareas. Slimi et al. (2009) develop a spatio-temporal model based oncellular automata at a small geographical scale (a village), wheredispersal is described by a parameter defined as the averagenumber of vectors entering and leaving the village per unit time.


mailto:[email protected]

mailto:[email protected]

http://www.sciencedirect.com/science/journal/1476945X


B.A. Crawford, C.M. Kribs-Zaleta / Ecological Complexity 14 (2013) 145–156146

Results of this model suggest alternative control strategies toprevent yearly infestation of villages. Barbu et al. (2010), alsointerested in the invasion of domestic areas, consider grid-basedmodels. Dispersal is measured by the number of adult vectorsdispersing in each cell inside the village. This study comparedseveral different spatially explicit models and determined thatvectors that invade villages are immigrating from not only peri-domestic habitats, but sylvatic habitats as well. Thus controlmeasures should not only be focused on the peri-domestic areas,but also on the forests surrounding villages. In each of thesestudies, dispersal on a small geographical scale was studied. In thisstudy, we use small-scale continuous space dispersal informationto generate large-scale, discrete-space migration rates, compatiblewith classical metapopulation models. The space becomesdiscretized by decomposing it into patches. In this study, thepatches are defined based on sylvatic T. cruzi transmission cycles.Allen et al. (2009) uses similar structure in constructing a 3-patchmodel with a classical metapopulation structure in which 2 speciesof rodents migrate between isolated habitats and a shared overlappatch.

The following sections therefore begin with further backgroundon the application context, starting with a description of thepatches – the transmission cycle defining each patch as well as thegeographical boundaries and landscape. We then estimatedispersal rates for T. cruzi vectors and apply those rates totranslate local dispersal to global migration across a largegeographical scale. We first consider whether vectors have apreference for direction of dispersal, and then the simplest case ifthere is no preference of direction of dispersal. Finally, we use theframework to calculate migration rates for the vector speciesmentioned here and compare rates for varying preferred directionsand degree of preference for a particular direction. The frameworkderived here can also be applied to dispersal and migration of otherspecies.

2. Background: T. cruzi transmission cycles and dispersalfrequency

2.1. Spatial ecology

The parasite T. cruzi, endemic to the Americas, is maintainedprimarily in sylvatic cycles, although the majority of research onthe transmission and control of the parasite has been devoted todomestic and peridomestic cycles in which the parasite can betransmitted to humans causing Chagas’ disease. Direct transmis-sion from human to human is not possible (except through bloodtransfusion and vertical transmission) and human to vectortransmission is too inefficient for the parasite to maintained indomestic cycles alone (Gourbiere et al., 2008); thus a thoroughunderstanding of sylvatic T. cruzi transmission cycles is critical toany long-term disease control plans.

The sylvatic transmission of T. cruzi is complex due to itspresence and circulation in multiple hosts and vector species aswell as large variation in strain types. The hosts, vectors, andstrains of T. cruzi vary throughout North and South America. Insylvatic settings in North America, the parasite is transmittedbetween Triatoma vectors and reservoir hosts, mainly raccoons(Procyon lotor), opossums (Didelphis virginiana), and several speciesof woodrat, namely the southern plains woodrat (Neotoma

micropus) (Roellig et al., 2009). In the U.S., hosts are associatedwith two strains of T. cruzi, types I and IIa (recently reclassified asstrain IV (Zingales et al., 2009)). In this study, we are particularlyinterested in the sylvatic transmission of T. cruzi IV in cycles thatrange from northern Mexico to throughout the south-southeasternUnited States because of the overlap of distinct transmissioncycles. In the wild, the infection cycles with type IV are maintained

by the Triatoma species, Triatoma gerstaeckeri and Triatoma

sanguisuga, and hosts, raccoons and woodrats. The raccoon isfound throughout most of North America, but primarily inhabitswooded areas, and is commonly found near water. The woodrathabitat is characterized by areas supporting cactus growth andthorny desert shrubs. Little information is given in the literatureregarding the habitats of T. gerstaeckeri and T. sanguisuga. However,one study (Kjos et al., 2009) gives an account of the biogeographyof these species in Texas. Based on their information and otherstudies, T. gerstaeckeri is found in dryer regions with a dense butscrubby vegetation (Villagran et al., 2008), while the habitat of T.

sanguisuga covers a broader range of vegetation. T. sanguisuga hasbeen found in Texas and other southeastern U.S. states. For furtherinformation on vectors, Gourbiere et al. (2012) provides a recentreview of triatomine biology and genetics.

In this region, there are several known distinct transmissioncycles. In northern Mexico and the southernmost part of Texas, themain T. cruzi vector is T. gerstaeckeri, associated in sylvatic settingsalmost exclusively with the southern plains woodrat (Kjos et al.,2009; Pippin, 1970; Eads et al., 1963). In the eastern portion ofTexas into the southeast United States, the predominant vector is T.

sanguisuga, part of the lecticularia complex (Ibarra-Cerdena et al.,2009; Sarkar et al., 2010) and is commonly associated withraccoons and opossums (Pung et al., 1995). A recent study done inthe U.S. analyzed 107 isolates of T. cruzi, and determined thatraccoons (and other hosts) are primarily infected with type IV,while opossums were only infected with type I (Roellig et al.,2008), and in fact are immune to type IV (Roellig et al., 2009). Sincewe are concerned here with the transmission of type IV, we do notinclude opossums as a host in our study. In Texas and Mexico, thetwo cycles overlap due to T. sanguisuga also being found inassociation with woodrats (Eads et al., 1963).

In order to describe the distinct transmission cycles geographi-cally, we define three distinct regions (or patches). We define twoouter patches based on the T. gerstaeckeri–woodrat cycle and the T.

sanguisuga–raccoon cycle, with an overlap patch containing bothvector species and both hosts. The patch boundaries in this studyare based on the Omernik ecoregion system, political boundaries,and host and vector distribution maps found in literature. TheOmernik system was derived in 1987 in collaboration with the U.S.Environmental Protection Agency. The Omernik system is hierar-chical in structure and consists of 4 levels. Ecoregions are areaswith similar ecosystems, and the boundaries are determined bypatterns of vegetation, climate, geology, wildlife, water quality,soils, and human land use. To determine the patches used in thismodel, we use the level 3 ecoregion system, consisting of 194regions describing North America.

Since the geographical scale of Chagas disease spread rangesfrom northern Mexico through southeast United States, weassume communication between the patches is through themigration of the T. cruzi vectors, rather than migration of hosts.Hosts are primarily territorial, and move within a fixed range,and do not typically encounter others of their species exceptmates. Then, since the geographical scale is so large and covers awide range of terrain, we do not assume hosts are as likely tocross into unsuitable habitat, thereby crossing patch boundaries.Thus we do not assume migration of hosts plays a significant rolein spreading the parasite across regions. The term migration usedhere defines movement of vectors that cross patch boundaries.There is little to no information regarding dispersal capabilitiesof T. sanguisuga and T. gerstaeckeri. However, there have beenseveral experimental studies on the dispersal capabilities of theSouth American species, Triatoma infestans and Triatoma sordida.The movement specific experimental studies are primarily lighttrap collections (Vazquez-Prokopec et al., 2006, 2004) and mark-release experiments (Schofield et al., 1991, 1992). The light trap

B.A. Crawford, C.M. Kribs-Zaleta / Ecological Complexity 14 (2013) 145–156 147

collections are designed to determine seasonal variations oftriatomine dispersal rather than to determine actual flight range,while the mark-release experiments are designed to determineflight range under natural conditions. It is assumed uponmaturation vectors initiate dispersal in search for a host. Vectorsmay also disperse due to death of host or if a host fails to return tothe nest (in search for food). Results of these experiments suggestthat flight initiation is usually associated with low nutritionalstatus and high temperatures (Vazquez-Prokopec et al., 2004),which is evidence for the fact that dispersal mainly occurs insearch for a host.

2.2. Patches

Patch 1 extends from south Texas to northern Mexico(including portions of Coahuila, Tamaulipas, and Nuevo Leon).Patch 1 is the region in which the primary T. cruzi hosts and vectorsare woodrats and T. gerstaeckeri (Ikenga and Richerson, 1984;Burkholder et al., 1980). Patch 3 extends from central Texas to theeastern half of the United States, including Louisiana, Mississippi,Alabama, Georgia, the Carolinas, and the Florida panhandle. Theprimary vector is T. sanguisuga associated with raccoons (Punget al., 1995). Patch 2 is the region in which the two transmissioncycles overlap due to T. sanguisuga’s association with woodrats aswell as raccoons in this region. Patch 2 includes the southwestportion of Texas and part of Coahuila, Mexico. The patches can beseen in Fig. 1.

2.2.1. Patch 1

Patch 1 is defined by the T. gerstaeckeri-woodrat transmissioncycle and includes portions of south Texas and certain northernstates of Mexico. The primary ecoregion in Patch 1 is the SouthernTexas Plains which has a generally semi-arid climate dominatedby scrub, thorny brush vegetation. The thorny brushland isdominated by mesquite and certain species of cactus, among otherdesert scrub vegetation. Curto de Casas et al., 1999 determine ageographic distribution for T. gerstaeckeri based on other scientificpapers recording findings of this species in Mexico, whichincludes the states of Coahuila, Tamaulipas and Nuevo Leon.Recently Cruz-Reyes and Pickering-Lopez (2006) conducted a

Fig. 1. Pat

review of literature regarding T. cruzi in Mexico, and determined T.

gerstaeckeri to be found in the same states mentioned previously.In their study, they do include a small northeast portion of San Luisde Potosı as potential habitat for T. gerstaeckeri. Recently, in astudy on the risk assessment of Chagas disease in Texas, Sarkaret al. (2010), a risk assessment predicts high suitable habitat for T.

gerstaeckeri in northeast Mexico to central and east Texas, whichaligns closely with Patches 1 and 2. Since our focus is on T.

gerstaeckeri and T. sanguisuga, we do not include this state in Patch1, due to the fact that this region coincides with the native area ofthe dominant vector in Mexico, Triatoma dimidiata. Because thePatch 1 climate and vegetation are consistent with the southernplains woodrat preferred habitat, we are certain that this speciesis found through Patch 1. According to the IUCN Redlist ofThreatened Species (Linzey et al., 2009), the southern plainswoodrat has also been found in northeast Coahuila, Tamaulipas,and Nuevo Leon.

Because Patch 1 does not include the T. sanguisuga–raccoontransmission cycle, the northern boundary of Patch 1 is primarilybased on the geographic extent of T. sanguisuga. Kjos et al. (2009)collected samples of T. sanguisuga, which was not found south ofthe southern borders of Webb, Duvall, Jim Wells, and Klebergcounties in Texas. Furthermore, Ibarra-Cerdena et al. (2009)determined a very similar boundary in Texas for the southernrange of the complex Triatoma lecticularia, which includes thespecies T. sanguisuga. We mention here that raccoons may be foundstatewide in Texas. However, based on known county records, theraccoon geographic distribution diminishes in the south tosouthwest portion of Texas. Beasley et al. (2007) show thatraccoons prefer forested areas over all other landcover types(grassland, shrubland, agriculture areas), with the least preferredhabitat being shrubland. Henner et al. (2004) found that raccoonsprefer trees as their den site 91% more than ground den sites. Theyalso conclude that raccoons consistently selected den sites thatallowed access to free water. Thus, due to the lack of abundant treecover and free water in Patch 1, we do not include raccoons as ahost of T. cruzi in this region. We further mention that no studies todate have associated the vector T. gerstaeckeri with the raccoon,thus agreeing with the conclusions that raccoons would not be asignificant host for T. cruzi in this region.

ches.


2.2.2. Patch 2

Patch 2 contains the most diverse climate and vegetation due tothe overlap of host and vector species in the model. This patchcontains the entire Edwards Plateau, portions of the SouthernTexas Plains, and the southernmost parts of the Texas BlacklandPrairies, East Central Texas Plains, and Western Gulf Coastal Plain.The vegetation is dominated in the south by the shrubland of theSouthern Texas Plains. The Edwards Plateau has a similarvegetation dominated by scrub forest. Mesquite occurs throughoutthe entire region with ash, juniper, and Texas oak dominant in thesouthern and eastern regions of the plateau which extend intothe Texas Blackland Prairies and East Central Texas Plains. TheBlackland Prairies and East Central Texas Plains (also called thePost Oak Savanna) are predominantly covered by prairie grassesand savannas (a transition between grassland and forest). Trees inthis region include Post oak and blackjack oak, with mesquiteinvading the southernmost parts of these regions (Linzey et al.,2009). This region also includes a portion of the Western GulfCoastal Plain with a primary grassland vegetation; however, muchof the region has been invaded by mesquite trees, oaks, and pricklypear cactus.

Patch 2 is defined as the region in which all four species can befound with T. sanguisuga feeding on both host species, woodratsand raccoons. In Patch 2, woodrats are predominantly found inthe regions dominated by mesquite and cactus. Studies havefound that N. micropus densities are directly linked to availabilityof cactus (Thies et al., 1996; Brown et al., 1972). In fact, Thieset al. (1996) found that cacti was the most abundant material inconstruction of the woodrat nests (over 50% of the material), aswell as a main source of food. Thus, in general, the southernplains woodrat can be found in most of Patch 2, albeit with lowerpopulation density than in regions completely dominated bythorny brushland (i.e., Patch 1). In Patch 2, raccoons are mostlyfound in regions with trees and near water. This corresponds toraccoons most likely preferring habitat in the southern part of theEdwards Plateau into the Texas Blackland Prairies and EastCentral Texas Plains. In this region, both vectors T. gerstaeckeri

and T. sanguisuga can be found. We further mention here that inthis region, T. sanguisuga feeds on both woodrats and raccoons(Pippin, 1970; Burkholder et al., 1980; Pung et al., 1995; Ikengaand Richerson, 1984). We assume that due to local vectordispersal T. sanguisuga sometimes move between raccoon densand woodrat nests so that they will be considered a singlepopulation.

The southern portion of Patch 2 is determined by thegeographic extent of T. sanguisuga (as mentioned previously).The eastern boundary of Patch 2 is primarily based on thepreferred habitat of the woodrat, since the woodrat prefers tomakes its home in regions with cactus growth. The regions eastof Patch 2 are dominated by forest and have extensive treecover, less favorable for cactus growth. According to countyrecords, the southern plains woodrat has a distinct easternboundary in Texas (Davis and Schmidley, 1997), which we use asthe eastern boundary of Patch 2. The northern boundary of Patch2 is the northern boundary of the ecoregion, Edwards Plateau.Although the southern plains woodrat habitat extends furthernorth than this boundary, according to the work done by Kjoset al. (2009), the vector T. gerstaeckeri has not been located anyfurther east or north than the boundary of Patch 2. The westernboundary is determined by the Southern Texas Plains ecoregion.The regions west of Patch 2 are predominantly desert (as part ofthe Chihuahuan Desert ecoregion). Although the woodrat maybe found in this region, we do not have any informationregarding T. gerstaeckeri or T. sanguisuga being significantvectors in this region. We note that in the risk assessment bySarkar et al. (2010), data collection of T. sanguisuga from two

museum collections in the western United States predictssuitable habitat in areas west of this boundary, although theymention the need for more data collection to test this prediction.Other species such as Triatoma rubida and Triatoma protracta

may be the dominant vectors in this region (Peterson et al.,2002).

2.2.3. Patch 3

Patch 3 contains the T. sanguisuga–raccoon T. cruzi transmissioncycle and covers a great area of land. The vegetation in this region isdominated by forest and woodland, including the South CentralPlains and Southeastern Plains as the main ecoregions. Wemention here that there are other hosts of T. cruzi in this region,most notably opossums. However, we only consider raccoons asthe primary host since we are concerned with the transmission ofT. cruzi type IV, to which opossums are immune (Roellig et al.,2009).

The southern plains woodrat is not native to this region. Thewestern boundary of Patch 3 is determined by the eastern rangeof the southern plains woodrat (Davis and Schmidley, 1997) andthe western boundary of T. sanguisuga (Kjos et al., 2009) Thesouthern boundary of Patch 3 is determined by the northernboundary of the woodrat as well as T. gerstaeckeri in the easternpart of Texas (Kjos et al., 2009; Davis and Schmidley, 1997). T.

sanguisuga and the raccoon are found in many parts of theeastern half of the U.S. covering a broad expanse of climate andvegetation. We note, however, that the southern boundary ofPatch 3 extends not further than the Florida panhandle. Theecoregion south of Patch 3 is the Southern Coastal Plain, with abroad range of vegetation, but mostly consists of coastal lagoons,marshes, and swampy lowlands. Although raccoons may inhabitthis type of land, research suggests that the complex of thetriatomine species, T. lecticularia, which includes T. sanguisuga, isnot found in this region. There are reports of T. sanguisuga foundnorth of the northern boundary of Patch 3. They are isolatedreports, and T. sanguisuga is not believed to be as dense north ofthis boundary as they are in the the southern part of the U.S.; thuswe do not include the regions to the north of Tennessee andNorth Carolina.

2.3. Dispersal rates

To estimate the parameters that describe the movement ofvectors across patch boundaries, we will distinguish two types ofrates, dispersal and migration rates. Dispersal refers to anymovement of vectors, while migration will be used to denotemovement only across patch boundaries. We consider the patchesin continuous 2-D space so that at every point exists some densityof vectors. Although there may potentially be several reasons avector may disperse, we will assume here that vectors disperseupon maturation, and define m as the dispersal rate due to vectormaturation. We will then calculate the cross-patch migration rateas the product of the dispersal rate, m and the migrationproportion (the proportion of dispersals that cross patchboundaries).

We first separately calculate the rate m at which vectors in anylocation will disperse. Since we assume that vectors begin to travelat maturity, we consider the 2-stage model with constant localvector population including per capita birth, death and maturationrates. The focus of the model presented here is to show howdispersal rates can be taken from spatially homogenous populationmodels. We do not give a detailed description of the vector lifecycle, but rather focus on keeping the model simple in order tocalculate the vector maturation rate. The model depicts birth,maturation, and natural mortality rates for each vector stage. Inthis model, J represents the local population of juveniles (nymphs)

Fig. 2. Interaction between global and local coordinate systems.


at time t, and A represents the local population of adults, andJ + A = N. The model is given by

J0 ¼ rA 1 � A þ eJ

K

� �� gJ � mJ J;

A0 ¼ gJ � mAA:

The parameter, r, represents the intrinsic growth rate of thevector population and K is the carrying capacity of the vectorpopulation. We define e as the factor in [0, 1] capturing how muchof the carrying capacity is used by a juvenile relative to an adult.The parameters, mJ and mA are the per capita natural mortalityrates for juveniles and adults, respectively. The parameter grepresents the rate at which juveniles become adults (ormaturation rate, units 1/year). We further note that the loss ofadult vectors due to migration is not considered in this model(since we do not assume adult vectors migrate).

We can now determine the dispersal rate m for each species dueto maturation as the per capita maturation rate g times theproportion of juveniles at equilibrium. We define the proportion ofjuveniles at equilibrium as x* = J*/N*, so then (1 � x*) = A*/N*. Todetermine x*, we rewrite the equilibrium condition (A0 = 0) as

gx�N� � mAð1 � x�ÞN� ¼ 0;

and determine for N� 6¼ 0, x� ¼ mA=ðmA þ gÞ. Thus, for each vectorspecies,

m ¼ gx� ¼ gmA

mA þ g

� �:

Using the parameter estimates by Kribs-Zaleta (2010), for T.

sanguisuga (g = 1/(2.25 year)), mA = 1/(1.44 year), we obtainmS = 0.271/year. For T. gerstaeckeri (g = 1/(1/year), mA = 1/0.78year), mG = 0.562/year.

3. Preferred direction

To derive the migration proportion (the proportion of dispersalsthat cross the patch boundary), we wish to sum up all of the localvector dispersals originating in one patch that cross the patchboundary into the neighboring patch. As shown in Fig. 2, we wishto sum up all of the dispersals originating from each point (x, y) thatend up on the other side of the patch boundary. In order to derivethe migration rate, we consider two different coordinate systems.The local coordinate system, used to sum all of the patch-crossingvectors originating from a given point, will be polar, while theglobal coordinate system summing over all points of origin will berectangular. To simplify calculations, we will assume that thepatch boundary is piecewise linear.

We consider that vector dispersal is described by threeproperties: dispersal distance, preferred direction of dispersal,and degree of preference for a particular direction. Each of these

Fig. 3. Varying direction with bo

properties is inherent to vector dispersal and independent of thecoordinate systems. We model vector dispersal as a piecewiseconstant distribution in two-dimensional continuous space. Toaccount for the preferred direction, we define this distributionusing a sequence of nested ellipses, where each ellipse in thesequence represents a particular level of dispersal. The dispersal isconstant on rings bounded by the ellipses, with the appropriateproportion of vectors dispersing within each ring.

Dispersal properties are reflected in the distribution parameters.We define the sequence {bi} as the threshold dispersal distancesperpendicular to the preferred direction. The values u0 and e

represent the preferred direction and degree of preference,respectively. These values are global parameters and do not dependon any coordinate systems. The value u0 is defined counterclockwiserelative to the outward normal to the patch boundary. The degree ofpreference parameter e gives the eccentricity of each ellipse. Wenote that 0 < e < 1, so that for values of e very close to 1, the degree ofpreference is very strong, while if e = 0, we assume the vectors haveno preferred direction of dispersal, and the model reduces toconcentric circles, which will be described in detail in the followingsection. In Figs. 3 and 4, we illustrate vector dispersal with sets ofnested ellipses for varying values of u0 and e.

Next, we will derive a framework for a preferred direction ofdispersal. We first define (local) dispersal distribution(s) usingnested ellipses in which dispersal originates from a given point.Then we derive the global migration proportion from each point ofdispersal in the originating patch.

3.1. Local migration framework

To determine the total number of patch crossing dispersalsoriginating from a given point, we will consider a local coordinate

ld points of origin (e ¼ 0:7).

Fig. 4. Varying eccentricity with bold points of origin (u0 ¼ �p=4).

Fig. 5. Preferred direction framework. Shaded region represents the area for which

dispersals have crossed patch boundary originating from a distance d away from the

patch boundary.


system defined by polar coordinates (r, u). We assume that vectordispersal originates at the pole. The variable r defines the dispersaldistance from the pole and u is measured counterclockwise. Thereference value u = 0 is measured p/2 clockwise from the outwardnormal to the patch boundary (see Fig. 2).

The preferred direction of migration will be modeled using asequence of nested ellipses with a common focus at the pole. Theneach ellipse ri is given by the equation

riðuÞ ¼ bi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � e2p

1 � e sinðu � u0Þ:

As stated previously, u0 and e are global parametersdepending only on the nature of vector dispersal and indepen-dent of the coordinate systems used. We also note here that thedefinition of a patch is a region in which the conditions aresufficiently homogenous in which to assume the effects ofclimate or environment on migrating individuals would be thesame for all individuals in the patch. The ellipse ri(u) is defined sothat the major axis points toward the direction of u0. Asmentioned previously, the sequence {bi} defines the dispersaldistances perpendicular to the preferred direction, u0. Based ondispersal data, we define a function, C(r, u), that measuresproportion of vectors per square kilometer. Because Triatoma

dispersal data is scarce we will define C(r, u) to be piecewiseconstant on elliptical rings RiðuÞ ¼ fðr; uÞ : ri�1 < r � rig, where allri have the same e and u0, but differing widths, given by bi. Wehave made the decision to truncate the distributions based onlimited data and suggestions that vector flight capacity is limited.Then

Cðr; uÞ ¼

f 1; 0 < r � r1ðuÞf 2; r1ðuÞ < r � r2ðuÞf 3; r2ðuÞ < r � r3ðuÞ...

f n; rn�1ðuÞ < r � rnðuÞ

8>>>>><>>>>>:

where each constant function fi is the proportion of vectors persquare kilometer dispersing from the focus at the pole to the ithring Ri(u) (measured in kilometers). We further assume no vectorsdisperse further than rn(u).

To construct fi, we use the proportionalized population densityfor each elliptical ring, where ci is the proportion of vectorsdispersing from the given point to Ri, so then

Pici ¼ 1. Then the

proportion of vectors per square kilometer is

f i ¼ ci=AðRiÞ;

where A(Ri) is the area of the elliptical ring bounded by ri and ri�1.Then AðRiÞ ¼ pðaibi � ai�1bi�1Þ, where ai is the length of the semi-major axis. Since ai can be given in terms of the length of the semi-minor axis bi as ai ¼ bi=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � e2p� �

, we have

AðRiÞ ¼pðb2

i � b2i�1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � e2p ;

and consequently

f i ¼ci


pðb2i � b2

i�1Þ; (1)

where bi and bi�1 are half the length of the minor axis for theouter and inner ellipses, respectively. Furthermore, we note thatR R

Cðr; uÞ dA ¼P

i f iAðRiÞ ¼P

ici ¼ 1.We assume that vectordispersal originates at a focus of the ellipse located at the poleat a distance, d, away from the patch boundary. To compute thevector dispersals crossing the patch boundary, we wish todetermine the two u values, u1 and u2 (see Fig. 5), such that theoutermost ellipse rn(u) intersects the patch boundary. Theparameters d, u1 and u2 are constant over r, u, but variable over x,y. Thus, for the point of dispersal, d km from the patch boundary,we solve

d

sinðuÞ ¼bi


1 � e sinðu � u0Þ(2)

for u, obtaining two u values, u1(d) and u2(d), which become thelower and upper limits, respectively, of the du integral. We notethat u1 and u2 depend on e and u0 as well. But, since these are globalparameters, we do not write u1 and u2 as explicit functions of e andu0.

By solving (2), we determine by using the identitysinðA � BÞ ¼ sin A cos B � cos A sin B,

de cos u0 þ bn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � e2p� �

sin u � de sin u0 cos u ¼ d: (3)

Since we can write a sum of two sinusoidal functions as a singlesinusoidal function, then a1 sin u þ a2 cos u can be written as

Fig. 6. Global coordinate system.


B sinðu þ fÞ, where B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 þ a22

qand f ¼ arctanða2=a1Þ. Then in

Eq. (3),

B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

de cos u0 þ bn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � e2p� �2

þ ðde sin u0Þ2r

and

f ¼ arctan�de sin u0

de cos u0 þ bn


!:

It then follows that u1 ¼ arcsinðd=BÞ � f and u2 ¼p � arcsinðd=BÞ � f:

We define Mð�dÞ (for a negative argument indicating locationrelative to the patch boundary) as a unitless quantity representingthe proportion of vectors crossing the patch boundary originatingfrom the pole. Thus,

Mð�dÞ ¼Z u2ð�dÞ

u1ð�dÞ

Z rnðuÞ

0Cðr; uÞr dr du;

where we define Mð�dÞ ¼ 0 for d such that rn(u) does not intersectd/sin(u). A visual representation can be seen in Fig. 5. Furthermore,we define the maximum distance a vector can be from the patchboundary and still disperse to the patch boundary, as the point onthe outermost ellipse such that there is only one intersection withthe patch boundary. This maximum distance, denoted rmax, iscalculated by finding the unique solution to u1ð�dÞ ¼ u2ð�dÞ. Ifu1ð�dÞ ¼ u2ð�dÞ, it follows that �d ¼ B, so then

rmax ¼ bneffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � e2p cos u0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ effiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � e2p cos u0

r� �: (4)

3.2. Global migration framework

To sum patch-crossing vector dispersals over all points of originin the given patch, we define now a rectangular global coordinatesystem for each boundary segment. The origin of this system isdefined to be the counterclockwisemost point (with regard to theoriginating patch) on the piecewise linear boundary segmentbetween the two patches (see Fig. 6 for an illustration) and theangle between consecutive segments is no greater than 908 ineither direction. Each point of dispersal is then described by therectangular coordinates, (x, y), with y defined by the verticaldistance to the patch boundary, where y < 0 represents theoriginating patch and y > 0 represents the destination patch. Thusthe patch boundary is given for each segment by the line y = 0. Wefurther note that rmax differs for each segment because u0 differs foreach segment.

We note that as mentioned in Section 2.3, only a proportion ofvectors are capable of dispersal. We wish to calculate theproportion of vectors in the entire patch that cross the patchboundary. Because MðdÞ is the proportion of vectors originatingfrom the pole that cross the patch boundary, we integrate MðyÞ(since in the global coordinate system, d ¼ �y) for each point x, y

within the maximum dispersal range. This integral will give us thearea times the proportion of all vectors dispersing. Thus, todetermine the cross-patch migration rate, we divide by the area A

of the patch from which the dispersal originates and multiply bythe dispersal rate, m, to obtain

m j ¼m

A

Z l j

0

Z 0

�rmax

MðyÞ dy dx;

where lj represents the length of the jth linear boundary segment.

Furthermore, since the integral does not depend on x, m j

becomes

m j ¼m � l j

A

Z 0

�rmax j

MðyÞ dy (5)

for each linear segment of boundary. Then m ¼P

jm j.This method does have some limitations and constraints. First,

in order for the rectangular coordinate systems described above toremain entirely within the patches of origin, the angle turnbetween any two consecutive segments be no greater than 908.Second, the method considers dispersals originating from a givenpoint to cross a single linear segment of a patch boundary. Inpractice, dispersals which originate sufficiently close to the cornerwhere two such segments meet may cross boundary segment linesbeyond the segments’ endpoints, in which case the methodmiscounts them (overcounts or undercounts). To address the issue,we also require that min jl j > amax, i.e., the shortest boundarysegment must still be longer than the longest dispersal distance, inorder for the dispersal ellipse generated from any point to cross nomore than two boundary segments. Then we introduce correctivecalculations to resolve the errors induced near corers.

For each type of corner (concave or convex), these computa-tions cover regions in which the originating dispersals areovercounted (counted as crossing a patch boundary, when theyshould not be) or undercounted (failure to count dispersals thatcross patch boundaries). A supplemental document availableonline provides detailed definitions and computations for eacherror type. The numerical approximations for the migration ratesin Sections 4.2 and 5 have been adjusted for the error, which, forthe patches in this study, sum to no more than 3%.

4. No preferred direction

In the simplest case, we assume vectors have no preferreddirection of dispersal. We then consider the eccentricity e is 0, andthus the nested ellipses reduce to concentric circles, each withradius bi. Since when e = 0, each ri becomes bi, C depends only onthe sequence {bi}. In this case, vector dispersal originates from thecenter of the circles. Here,

Cðr; uÞ ¼

f 1; 0 < r � b1

f 2; b1 < r � b2

f 3; b2 < r � b3

..

.

f n; bn�1 < r � bn

8>>>>><>>>>>:

(6)

Table 1Dispersal data from Schofield et al. (1992, 1991) for a single flight.

Schofield et al. (1992) Schofield et al. (1991) Range

0.552 0.469 0–5 m

0.034* 0.015 60 m

0.018* 0.008 75 m

0.018* 0.008 90 m

0.378 0.504 >100 m

Table 2Averages of proportions of vectors dispersing within 0–100 m for a single flight.

Proportion Interval

c1 = 0.5105 0–5 m

c2 = 0 5–50 m

c3 = 0.0375 50–75 m

c4 = 0.0125 75–100 m

c5 = 0.4395 >100 m


and

f i ¼ci

pðb2i � b2

i�1Þ: (7)

4.1. Estimation of dispersal kernel

In order to derive numerical values for C with no preferreddirection, we utilize two studies in which Triatoma vector dispersalis the focus. Both studies were mark-release experimentsperformed in the vector’s natural climatic conditions in which apreferred direction of dispersal could not be determined. Due tothe lack of dispersal information regarding T. gerstaeckeri and T.

sanguisuga, we use two studies on two South American Triatoma

species, T. infestans and T. sordida (Schofield et al., 1992, 1991).Although these species are not naturally located in the patches inthis model, we assume that the dispersal capabilities of each ofthese species are at least an upper bound for the dispersalcapabilities of T. gerstaeckeri and T. sanguisuga.

For each study, vectors were released from a central locationand captured during the night. To determine distances flown,circles were drawn out with fluorescent paint at regular intervals(in meters). For each study, the proportion of vectors travelingwithin a specific range of distances was reported. We give the rawdata and an average of those proportions in Tables 1 and 2. We notethat in Schofield et al. (1992), specific values for 60, 75, and 90 mwere not given, but rather values for <100 m. In order to use thisdata with the same intervals provided in Schofield et al. (1991), wedecomposed the data in Schofield et al. (1992) for values <100 mby using the same proportions in Schofield et al. (1991). Thesevalues are marked with an ‘*’ in Table 1. In each study, thereappears to be a distinction between trivial flights (0–5 m) andlonger distance flights (>50 m). Thus, for vector dispersal weassume that all flights were either trivial or long range, i.e., weassume no flights occur for 5–50 m.

In each study, only a proportion of the vectors were actuallyrecovered, and it is assumed that the remainder of the bugs flewfarther than 100 m. After taking the averages of each study, wedetermine that 44% of the bugs flew farther than 100 m. Since wedo not have a maximum dispersal distance, we will derive thenumerical values for C using three possible values (low-mid-highestimates) for the maximum dispersal distance. We further notethat we will use gi as to represent the range of distances traveledfor one flight, while the bi used in the C function will be the overalltotal distances for the entire period of flight.

Because we have 5 intervals determined by the experiments, todetermine b5 assuming a minimum dispersal capability we firstnote that in Eq. (7), ci represents the proportion of bugs travelingbetween gi�1 and gi meters for one flight. Thus, c5 = 0.44 representsthe proportion of bugs traveling between 100 and g5 meters andf 5 ¼ 0:44=pðg2

5 � 1002Þ. If g5 is to be a minimum and thedistribution is to be unimodal, then f5 = f4. The remainder of theci values are given in Table 2. Thus,

f 4 ¼ f 5;c4

pððg4Þ2 � ðg3Þ

2Þ¼ c5

pððg5Þ2 � ðg4Þ

2Þ;

0:125

pð1002 � 752Þ¼ 0:44

pðg25 � 1002Þ

:

(8)

Solving Eq. (8) for g5, we determine g5 = 405 m.To determine a maximum value for g5, we consider results

published by Lehane and Schofield (1978). In this paper it isdetermined that Triatoma vectors are capable of flights up to1350 m. Thus, we choose a maximum value for g5 to be 1350 m. We

note that because of limited data, in all future calculations, we willuse the high dispersal range of 1350 m.

Based on experimental results by Pippin (1970) regarding thevectors in the model, we estimate that on average a vector willtravel to find a host within 2 weeks after feeding, but will starve (orno longer have the energy to fly) within 3 weeks of flying. Then, themaximum number of flying days is estimated to be 35. We notethat because of the limited amount of data, this number is only anestimate determined by one study. Then, in order to modeldispersal over more than one flight, we need a distribution of thesum of the 35-day flight distances. As we increase the number ofdays flown to 35, by the Central Limit Theorem, the 35-daydistribution will become approximately normally distributed withmean, 35 times the mean for the original 1-day distribution andstandard deviation

ffiffiffiffiffiffi35p

times the standard deviation of theoriginal distribution.

To develop the 35-day distribution, we determine the propor-tion of vectors flying per meter for each of the ranges in Table 2. Weuse these values as estimates for the heights of a 1D piecewiseconstant function (say P), with intervals of length 1 m. To obtainthe 1D distribution for the sum of the flight distances for 35 days offlying, we will repeatedly apply the convolution to P 35 times. Forcomputational simplicity, we approximate this normal distribu-tion with a piecewise constant function, P(r), with units proportionof vectors per meter, where

PðrÞ ¼ h j for bð j�1Þ< r � b j with b0 ¼ 0: (9)

We chose P to have 9 pieces for optimal processing time. Theinterval for the final piece is chosen to be the entire right tail ofthe normal distribution. The remaining breakpoints and values forthe function are then determined by a least-squares optimizationroutine. Finally, we multiply each value of P by its appropriateinterval to obtain the proportion of vectors traveling within thegiven distance interval using the equation

c j ¼ h j � ðb j � bð j�1ÞÞ: (10)

Recall, the distribution function C is a function of 2 variables, r

and u, where

Cðr; uÞ ¼ f j for bð j�1Þ< r � b j: (11)

In order to derive fi estimates for C, we use the cj values (as theproportion of vectors dispersing to the jth elliptical ring) in Eq. (1).

To provide some context for this method and the resultingdistribution, we compare to the approach of Ramirez-Sierra et al.(2010) who studied the dispersal of another T. cruzi vector,

Table 3Geographical parameter values for T. cruzi transmission regions.

Patch area Boundary lengths

A1 ¼ 1:594 � 1011 m2 lAB ¼ 193; 600 m

A2 ¼ 1:783 � 1011 m2 lBC ¼ 137; 707 m

A3 ¼ 1:133 � 1012 m2 lDE ¼ 99; 691 m

lEF ¼ 103; 427 m

Table 4Estimated 35-day vector dispersal distribution, extrapolated from single-flight data

in Tables 1 and 2.

j bj(m) hj

1 2485.05 9.96 � 10�8

2 4970.11 2.40 � 10�6

3 7455.16 2.34 � 10�5

4 9940.21 9.26 � 10�5

5 12,425.26 1.51 � 10�4

6 14,910.32 1.02 � 10�4

7 17,395.37 2.81 � 10�5

8 19,880.42 3.18 � 10�6

9 47,216 3.58 � 10�10


T. dimidiata in the Yucatan peninsula of Mexico. In this study, theyestimated dispersal capacity by collecting vectors in infestedhouses and inferring travel distances from the houses to theperiphery of the village. Results for flight distance capabilityconclude the maximum range of dispersal distance for a singleflight for T. dimidiata is between 95 and 120 m. Some subsequentstudies (Barbu et al., 2010, 2011) then constructed dispersalkernels based on the data of Ramirez-Sierra et al. Travel distancesin this context may be limited by the fact that vectors were lookingfor the nearest available house. The data in studies by Schofieldet al. (1991, 1992) applied in the present study use the release andrecapture approach in a clear field in which vectors had to travelfurther in search of suitable hosts. It is therefore not surprising thattheir estimates were higher. We use the higher estimates bySchofield et al. (1991, 1992) because we are considering a sylvaticcontext with low host population density.

4.2. Calculation of migration rates for no preferred direction

In the case of no preferred direction, the intersection of thepatch boundary with the outermost circle, is found by solving

d

sinðuÞ ¼ b9;

where b9 is the radius of the outermost circle. Then,

u1ðdÞ ¼ arcsind

b9

� �and u2ðdÞ ¼ p � arcsin

d

b9

� �: (12)

To manage the complexity of the calculations we will break thepatch boundaries in Fig. 1 into four linear segments (two for Patch1/2 boundary and two for Patch 2/3 boundary) as seen in Fig. 7. Wewill calculate separate migration rates for each (appropriate)vector species between Patches 1 and 2 and between Patches 2 and3. We define l12 as the sum of the length of the two linear segmentsmaking up the boundary between Patches 1 and 2, while l23 is thesum of the lengths of the linear segments making up the boundarybetween Patches 2 and 3. But, we note that for e = 0, we do not needto consider multiple boundary segments, since there is nopreference for direction. Thus, rather than using separate termsfor each boundary segment, we use one term with the sum of thelengths of each boundary segment for each patch boundary.

To calculate the migration rate of T. gerstaeckeri from Patch 1 toPatch 2,

m12 ¼mG

A1ðl12Þ

Z 0

�rmax

Z u2ðyÞ

u1ðyÞ

Z r9

ysinu

Cðr; uÞr dr du dy;

Fig. 7. Patch boundaries linear segments.

where the u limits are given in Eq. (12), rmax ¼ b9, mG ¼ 0:562=year.The geographic parameters are given in Table 3.

The migration rate of T. gerstaeckeri from Patch 2 to Patch 1 is

m21 ¼mG

A2ðl12Þ

Z 0

�rmax

Z u2ðyÞ

u1ðyÞ

Z r9

y=sin uCðr; uÞr dr du dy:

Then, m12 ¼ 0:0042=year and m21 ¼ 0:0038=year.The migration rate of T. sanguisuga from Patch 2 to 3 and 3 to 2

can be calculated similarly. The geographical parameter values forno preferred direction are given in Table 3.

5. Numerical calculations for preferred direction

We now consider how preferred direction affects the migrationrate. In the case of preferred direction,

Cðr; uÞ ¼ f j for bð j�1Þ< r < b j; (13)

where

f i ¼ci


pðbiÞ2 � ðbi�1Þ2Þ

and

riðuÞ ¼ bi


1 � sinðu � u0Þ:

The specific values for ci are determined by the values in Table 4and Eq. (10). We note here that although we are using the samelateral b values (as in no preference for direction), the preferreddirection affects the fi values, so the results will be different fore > 0 as illustrated in Fig. 4.

Based on the description of calculating the migration rates for apreferred direction in Section 4.2, we calculate the migration ratesfor varying preferred directions. The migration rate from Patch 1 to2 is then defined to be

m12 ¼mG

A1

lAB

R 0�rmax

R u2ðyÞu1ðyÞ

R r9ðuÞy=sin u Cðr; uÞr dr du dy

þlBC

R 0�rmax

R u2ðyÞu1ðyÞ

R r9ðuÞy=sin u Cðr; uÞr dr du dy

0@

1A:

We note that when rmax appears in each integral, it is specific toeach patch boundary (since it depends on u0 (see Eq. (4))).

Table 5uj values for outward normals for boundary segments from Patch 1 to 2 and from

Patch 2 to 3.

Segment uj

AB 2158BC 08DE 808EF 1088

0π4

π2

3π4 π 5π

43π2

7π4 2π

θ0

rate

Ratefrompatch1topatch2,e=0.1

Fig. 9. Migration rate as u0 ranges from 0 to 2p compared with migration rate for no

preference of direction (solid line) (e ¼ 0:1).

Table 6Migration rates for e ¼ 0:5, Patches 2/3.

Preferred direction m23 m32

No preference 0.00101 0.000155

(N) 0.00116 0.000165

(NW) 0.000415 0.000373

(W) 0.000240 0.000497

(SW) 0.000344 0.000408

(S) 0.000935 0.000190

(SE) 0.00228 0.0000721

(E) 0.00324 0.0000405

(NE) 0.00260 0.0000599

Table 7Migration rates for e ¼ 0:5, Patches 1/2.

Preferred direction m12 m21

No preference 0.00427 0.00385

(N) 0.0111 0.000941

(NW) 0.0107 0.000976

(W) 0.00685 0.00220

(SW) 0.00267 0.00622

(S) 0.00114 0.0102

(SE) 0.00121 0.00997

(E) 0.00304 0.000547

(NE) 0.00745 0.00204


Furthermore, when u0 appears in each integral, it is defined to bethe angle counterclockwise from the outward normal of thespecific linear boundary segment to the major axis of the ellipse.One way to determine u0 is to first define the angle made by themajor axis of the ellipse, u (in the counterclockwise direction),where u0 ¼ 0 represents an ellipse with a northward preferreddirection. We then define the angle of the outward normal to thejth boundary segment, uj (in the counterclockwise direction),where uj = 0 is the angle for the outward normal for a horizontalboundary segment (running east to west). Then, we define u0 forsegment j relative to uj by using u0 ¼ u0 � u j. For example, if thepreferred direction is west, then for segment AB, u0 ¼ 90� andu0 ¼ 90� � 35� ¼ 55� (when the migration rate is calculated fromPatch 1 to 2). We note that for the migration rate from Patch 2 to 1,the outward normal would be u j þ 180�. Table 5 gives the valuesfor uj for each boundary segment (for rates from Patches 1 to 2 and2 to 3).

The other migration rates are calculated similarly. We note herethat we will use the same calculation for b9 as in Section 4.1(assuming the highest range of dispersal).

To determine how preferred direction affects migration, we willcalculate the varying cross-patch migration rates as u0 ranges from0 to 2p and compare with the migration rates for no preference fordirection. If we consider the migration rate as a function of u0, thefunction is even, with a maximum value when the preferreddirection is out of the patch (u0 = 0) and a minimum value when thepreferred direction is toward the inside of the patch from whichmigration originates (u0 = p). An example of this result can be seenin Fig. 8. Fig. 8 shows the migration rate from Patch 1 to 2 for e = 0.5and as u0 varies from 0 to 2p, compared with the migration ratefrom Patch 1 to 2 assuming no preference for direction.

We also compare the rates for e ¼ 0:1 and e ¼ 0:5 as seen inFigs. 8 and 9. In Fig. 8, we observe that if the preferred direction iscloser to being directly out of the origination patch, the increase inmigration rate (compared with no preference of direction) isgreater than the decrease if the preferred direction of migration isdirected into the origination patch. However, we notice that forlower values of e, as represented in Fig. 9, this effect is not asapparent. In fact, for e ¼ 0:1, the amplitude between the highestmigration rate (at u0 ¼ 0) and the rate with no preferred direction

0π4

π2

3π4 π 5π

43π2

7π4 2π

0.001

0.002

0.003

0.004

0.005

0.006

rate Rate from patch 1 to patch 2, e=0.5

θ0

Fig. 8. Migration rate as u0 ranges from 0 to 2p compared with migration rate for no

preference of direction (solid line) (e ¼ 0:5).

is only 1.2 times the amplitude between the lowest migration rate(at u0 ¼ p) and the rate with no preferred direction, while fore ¼ 0:5, the amplitude between the highest rate and rate with nopreferred direction is 2.5 times higher than the amplitude betweenthe lowest rate and the rate with no preferred direction. Thus, forweaker degree of preference, the amplitude of the effect of thepreferred direction becomes closer to being symmetric withrespect to direction out of the origination patch versus directioninto the origination patch.

A summary of migration rates for varying preferred directionswith e ¼ 0:5 is given in Tables 6 and 7. As expected, the migrationrates are highest when the direction of migration is directly intothe destination patch. For example, the highest migration rate fromPatch 2 to 3 is when the preferred direction is east, and the highestmigration rate from Patch 3 to 2 is when the preferred direction iswest.

6. Conclusions

One of the primary goals of population modeling is to describehow individual events on a local scale build into collective effectson a larger scale, with much study having been made of the


resulting emergent properties of systems; dispersal of infectiousindividuals has often been studied in continuous space in order tocapture spatially small-scale movement, but here we develop aframework for incorporating such small-scale dispersal intosimpler, spatially discrete models that take a more globalperspective in defining populations by common epidemiologicaland ecological characteristics. The framework established in thisstudy allows us to translate local dispersal to global migrationacross discrete regions, for migrating species whose dispersaldistance capabilities are known. Migration rates may be calculatedunder the assumption of no preferred direction of dispersal or anassumed (or known) preference for a particular direction. Themethodology provides a way to generate migration rates to be usedin epidemiological or ecological metapopulation models. Morespecifically, there are applications to vector-borne diseases, forwhich climate change has made the invasion of vector into newterritories a major public health concern.

We assume vector dispersal is based on three inherentproperties: dispersal distance, preferred direction of dispersal,and degree of preference for direction. The parameters describingthese properties have different effects on the migration rate.Increasing dispersal distance will increase the migration rate,while varying the preferred direction may increase or decrease therate depending on the actual geographical location of theorigination and destination patch and the actual preferreddirection (into or out of the origination patch). In our calculations,we observe asymmetry in the migration rates depending onwhether the preferred direction is out of or in toward theorigination patch. As the degree of preference increases, theincrease in migration rate if preferred direction is out of the patchis greater than the decrease in migration rate if the preferreddirection is into the patch, when compared with the rate with nopreferred direction. This means that preference in direction tendsto increase migration rates overall, since preference increasesmigration (patch-crossing) on the preferred side more than itdecreases it on the opposite, non-preferred side of a patch.

Vector dispersal and migration have a clear role to play intransmission dynamics and ecological mechanisms for vector-borne diseases. One recent study (Rascalou et al., 2012) highlightedthe importance of vector dispersal and migration on the spread ofvector-borne pathogens (including T. cruzi) in ‘sink’ vectorpopulations. Other mathematical studies focusing on migrationof T. cruzi vectors (Slimi et al., 2009; Barbu et al., 2010) concentrateon migration over a small scale (a village) into domestic areas,whereas in this study we derive a method to calculate the rate ofmigration by connecting local dispersal to migration over a largescale. Large-scale migration should be considered because of thespread of T. cruzi across the Americas through sylvatic settings.

The calculations in this chapter rely heavily on experimentaland field studies on Triatoma vectors. Because there are virtually nostudies on the dispersal capabilities on the North American speciesfocused on in this study, T. gerstaeckeri and T. sanguisuga, we utilizestudies on the more well known vectors from South America, T.

infestans and T. sordida. Since Triatoma vectors are sensitive totemperature and humidity for flight initiation (Curto de Casas andCarcavallo, 1995) we acknowledge the difference in climatebetween the South American regions and the patches consideredhere. Because of the lack of data on the North American vectors, wenote the need for experimental studies regarding distance andfrequency of vector dispersal. If, for example, vectors dispersemuch more frequently than the rate we give here, which is basedon maturation, the corresponding migration rates could besignificantly higher.

Generalizations to the model could include variation in habitatsuitability, where different migration rates could be calculated forspecies traveling from a region of higher habitat preference

(suitability) to a region of lower habitat suitability or vice versa.Future work, already in progress, will apply the rates derived inthis study to examine the effects of migration of T. cruzi vectorsacross patch boundaries model of three distinct transmissioncycles of T. cruzi in northern Mexico and southeastern UnitedStates.

Acknowledgements

This research was supported by a 2008 Norman HackermanAdvanced Research Program grant, and by the National ScienceFoundation under Grant DMS-1020880. Conclusions are those ofthe authors, and do not reflect the opinions of the funding sources.The authors also wish to acknowledge Christopher Hall andMichael Yabsley for discussions of the underlying biology, and GaikAmbartsoumian and James Grover for suggestions which improvedthe manuscript.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the

online version, at http://dx.doi.org/10.1016/j.ecocom.2012.11.003.

References

Allen, L.J., Wesley, C.L., Owen, R.D., Goodin, D.G., Koch, D., Jonsson, C.B., Chu, Y.,Hutchinson, J.M., Paige, R.L., 2009. A habitat-based model for the spread ofhantavirus between reservoir and spillover species. Journal of TheoreticalBiology 260, 510–522.

Barbu, C., Dumonteil, E., Gourbiere, S., 2010. Characterization of the dispersal ofnon-domiciliated Triatoma dimidiata through the selection of spatially explicitmodels. PLoS Neglected Tropical Diseases 4 (8), 1–11 e777.

Barbu, C., Dumonteil, E., Gourbiere, S., 2011. Evaluation of spatially targetedstrategies to control non-domiciliated Triatoma dimidiata vector of Chagasdisease. PLoS Neglected Tropical Diseases 5 (5), 1–12 e1045.

Beasley, J.C., DeVault, T.L., Retamosa, M.I., Rhodes Jr., O.E., 2007. A HierarachicalAnalysis of Habitat Selection by Raccoons in Northern Indiana. Wildlife DamageManagement, Internet Center for USDA National Wildlife Research Center-StaffPublications.

Brown, J.H., Lieverman, G.A., Dengler, W.F., 1972. Woodrats and cholla: dependenceof a small mammal population on the density of cacti. Ecology 53, 310–313.

Burkholder, J.E., Allison, T.C., Kelly, V.P., 1980. Trypanosoma cruzi (Chagas) (Proto-zoa: Kinetoplastida) in invertebrate, reservoir, and human hosts of the LowerRio Grande Valley of Texas. Journal of Parasitology 66, 305–311.

Cantrell, R.C., Cosner, C., 2003. Spatial Ecology via Reaction-Diffusion Equations.John Wiley & Sons Ltd., Chichester, West Sussex, England.

Cruz-Reyes, A., Pickering-Lopez, J.M., 2006. Chagas disease in Mexico: and analysisof geographical distribution during the past 76 years. Memorias do InstitutoOswaldo Cruz 101, 345–354.

Curto de Casas, S.I., Carcavallo, R.U., 1995. Climate change and vector-borne dis-eases distribution. Social Science and Medicine 40, 1437–1440.

Curto de Casas, S.I., Carcavalla, R.U., Galındez Giron, I., Burgos, J.J., 1999. BioclimaticFactors and Zones of Life. Atlas of Chagas’ Disease Vectors in the Americas, Riode Janeiro.

Davis, W.B., Schmidley, D.J., 1997. The Mammals of Texas, online edition. TexasTechnical University.

Eads, R.B., Trevino, H.A., Campos, E.G., 1963. Triatoma (Hemiptera: Reduviidae)infected with Trypanosoma cruzi in South Texas wood rat dens. SouthwesternNaturalist 8, 38–42.

Gourbiere, S., Dorn, P., Tripet, F., Dumonteil, E., 2012. Genetics and evolution oftriatomines: from phylogeny to vector control. Heredity 108, 190–202.

Gourbiere, S., Dumonteil, E., Rabinovich, J.E., Minkoue, R., Menu, F., 2008. Demo-graphic and dispersal constraints for domestic infestation by non-domicilatedChagas disease vectors in the Yucatan Peninsula, Mexico. American Journal ofTropical Medicine and Hygiene 78, 133–139.

Henner, C.M., Chamberlain, M.J., Leopold, B.D., Wes Burger Jr., L., 2004. A multi-resolution assessment of raccoon den selection. The Journal of Wildlife Man-agement 68, 179–187.

Ibarra-Cerdena, C.N., Sanchez-Cordero, V., Peterson, A.T., Ramsey, J.M., 2009. Ecol-ogy of north american triatominae. Acta Tropica 110, 178–186.

Ikenga, J.O., Richerson, J.V., 1984. Trypanosoma cruzi (Chagas) (Protozoa: Kineto-plastida: Trypanosomatidae) in invertebrate and vertebrate hosts from Brew-ster County in Trans-Pecos Texas. Journal of Economic Entomology 77, 126–129.

Kjos, S.A., Snowden, K.F., Olson, J., 2009. Biogeography and Trypanosoma cruziinfection prevalence of Chagas disease vectors in Texas, USA. Vector-Borneand Zoonotic Diseases 9, 41–50.



Kot, M., Lewis, M.A., van den Driessche, P., 1996. Dispersal data and the spread ofinvading organisms. Ecology 77, 2027–2042.

Kribs-Zaleta, C., 2010. Estimating contact process saturation in sylvatic transmis-sion of Trypanosoma cruzi in the U.S. PLOS Neglected Tropical Diseases 44, 1–14e656.

Lehane, M.J., Schofield, C.J., 1978. Measurement of speed and duration of triatominevectors. The Tranactions of the Royal Society of Tropical Medicine and Hygiene72, 438.

Linzey, A.V., Timm, R., Alvarez-Castaneda, S.T., Castro-Arellano, I., Lacher, T., 2009.Neotoma micropus. In: IUCN 2009. IUCN Red List of Threatened Species. Version2009.2, http://www.iucnredlist.org (accessed 10 March 2010).

Mollison, D., 1972. Possible velocities for a simple epidemic. Advances in AppliedProbability 4, 233–258.

Peterson, A.T., Sanchez-Cordero, V., Beard, C.B., Ramsey, J.M., 2002. Ecological nichemodeling and potential reservoirs for Chagas disease, Mexico. Emerging Infec-tious Diseases 8, 662–667.

Pippin, W.F., 1970. The biology and vector capability of Triatoma Sanguisuga TexanaUsinger and Triatoma Gerstaeckeri (Stal) (Hemiptera: Triatominae). Journal ofMedical Entomology 7, 30–45.

Pung, O.J., Banks, C.W., Jones, D.N., Krissinger, M.W., 1995. Trypanosoma cruzi in wildraccoons, opossums, and triatomine bugs in southeast Georgia, USA. Journal ofParasitology 81, 583–587.

Ramirez-Sierra, M.J., Herrera-Aguilar, M., Gourbiere, S., Dumonteil, E., 2010. Pat-terns of house infestation dynamics by non-domiciliated Triatoma dimidiatarevlea a spatial gradient of infestation in rural villages and potentital insectmanipulation by Trypanosoma cruzi. Tropical Medicine and International Health15, 77–86.

Rascalou, G., Pontier, D., Menu, F., Gourbiere, S., 2012. Emergence and prevalence ofhuman vector-borne diseases in sink vector populations. PloS ONE 7 (5), 1–15e36858.

Roellig, D.M., Brown, E.L., Barnabe, C., Tibayrenc, M., Steurer, F.J., Yabsley, M.J., 2008.Molecular typing of Trypanosoma cruzi isolates, United States. Emerging Infec-tious Diseases 14, 1123–1125.

Roellig, D.M., Ellis, A.E., Yabsley, M.J., 2009. Genetically different isolates of Trypa-nosoma cruzi elicit different infection dynamics in raccoons (Procyon lotor) and

Virginia opossums (Didelphis virginiana). International Journal for Parasitology39, 1603–1610.

Sarkar, S., Strutz, S.E., Frank, D.M., Rivaldi, C.-L., Sissel, B., Sanchez-Cordero, V., 2010.Chagas disease risk in Texas. PLoS Neglected Tropical Diseases 4 (10), 1–14e836.

Schofield, C.J., Lehane, M.J., McEwan, P., Catala, S.S., Gorla, D.E., 1991. Dispersiveflight by Triatoma sordida. The Tranactions of the Royal Society of TropicalMedicine and Hygiene 85, 676–678.

Schofield, C.J., Lehane, M.J., McEwen, P., Catala, S.S., Gorla, D.E., 1992. Dispersiveflight by Triatoma infestans under natural climatic conditions in Argentina.Medical and Veterinary Entomology 6, 51–56.

Slimi, R., El Yacoubi, S., Dumonteil, E., Gourbiere, S., 2009. A cellular automata modelfor Chagas disease. Applied Mathematical Modelling 33, 1072–1085.

Thies, K.M., Thies, M.L., Caire, W., 1996. House construction by the southern plainswoodrat (Neotoma micropus) in southwestern Oklahoma. The SouthwesternNaturalist 41, 116–122.

U.S. Environmental Protection Agency (EPA). Omernik’s Level III Ecoregions of theContinental United States, in National Atlas of the United States, January 2005.http://nationalatlas.gov (accessed 15 February 2010).

Vazquez-Prokopec, G.M., Ceballos, L.A., Kitron, U., Gurtler, R.E., 2004. Active dis-persal of natural populations of Triatoma infestans (Hemiptera: Reduviidae) inrural northwestern Argentina. Journal of Medical Entomology 41, 614–621.

Vazquez-Prokopec, G.M., Ceballos, L.A., Marcet, P.L., Cecere, M.C., Cardinal, M.V.,Kitron, U., Gurtler, R.E., 2006. Seasonal variations in active dispersal of naturalpopulations of Triatoma infestans in rural north-western Argentina. Medical andVeterinary Entomology 20, 273–279.

Villagran, M.E., Marin, C., Hurtado, A., Sanchez-Moreno, M., Antonio de Diego, J.,2008. Natural Infection and Distribution of Triatomines (Hemiptera: Reduvii-dae) in the State of Queretaro, vol. 102. The Royal Society of Tropical Medicineand Hygiene, Mexico, pp. 833–838.

Zingales, B., Andrade, S.G., Briones, M.R.S., Campbell, D.A., Chiari, E., Fernandes, O.,Guhl, F., Lages-Silva, E., Macedo, A.M., Machado, C.R., Miles, M.A., Romanha, A.J.,Sturm, N.R., Tibayrenc, M., Schijman, A.G., 2009. A new consensus for Trypa-nosoma cruzi intraspecific nomenclature: second revision meeting recommendsTcI to TcVI. Memorias do Instituto Oswaldo Cruz 104, 1051–1054.

http://www.iucnredlist.org/

Vector migration and dispersal rates for sylvatic Trypanosoma cruzi ...

Documents

Transcript of Vector migration and dispersal rates for sylvatic Trypanosoma cruzi ...