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ABSTRACT |
 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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0.01/trimester). Our model predicted that domestic populations are not sustainable, and up to 90% of the individuals found in houses are immigrants. We discuss the potential of different strategies to control the transmission of Chagas disease by non-domiciliated vectors. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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The World Health Organization (WHO) recommends systematic blood screening and population control of triatomine vectors. In the last decade, intra-domiciliary insecticides have been massively sprayed in South and Central America with varying success. The Southern Cone Initiative proved successful, and house infestation and colonization by Triatoma infestans have been considerably reduced in several regions and countries, leading to a strong reduction in disease incidence in children.1 However, in Central America and in the Andean Pact region, 5–6 million people are still infected and 25 million are at risk of contracting Chagas disease. The common belief for the lower success of insecticide spraying in these areas is that vectors species are not strictly domiciliated (permanent domestic colonies) and can re-infest dwellings from peri-domestic and/or sylvatic habitats.2,3 The transmission risk posed by these non-domiciliated vectors is identified as a major problem and one of the new challenges for Chagas disease control.1 In such situations, the TDR (Special Program for Research and Training in Tropical Diseases) recommends devoting research efforts to adapt control strategies to local entomologic settings. The first need in this direction is a proper understanding of the dynamics of house infestation by non-domiciliated vector populations, especially to disentangle the importance of the local demography from the immigration process on the occurrence of vectors inside houses.
In the Yucatan peninsula, Mexico, Triatoma dimidiata is the main vector species of Chagas disease, and spatial and temporal patterns of variations in domiciliary bug abundance have been documented through field studies over several consecutive years.4–6 Domestic populations showed marked seasonal variations (with a strong increase in abundance during April to June), a high proportion of adults (> 80% of adults all year round), and a small average population size per house (typically about 5–20 insects/house/yr; please note that population size evaluated here correspond to the sum of all the individuals collected over a year, which is the simplest way to represent the total number of bugs present during the year and includes all its temporal variations). These data together with recent population genetics studies7 indicate that temporal variations of T. dimidiata domestic populations in the Yucatan peninsula are best explained by the seasonal infestation by dispersing peri-domestic and sylvatic adult bugs. The difficulty to control house infestation associated with this vector species has also been established in the field, because only 4 months were required for T. dimidiata populations to reach their original level after insecticide spraying.8 Hypothetically, other seasonal factors (such as variations in abundance of predators or some abiotic factors) could explain seasonality in the variation of vector population size. However, these factors are less likely to explain all the observed features of T. dimidiata population dynamics. In the absence of data supporting such alternative hypotheses, we thus focused on dispersal, which, conversely, has been well documented from the several field studies referred to above.
The main goal of this paper was to evaluate the potential of seasonal dispersal to explain the pattern of domestic vector population dynamics observed in Yucatan and to identify the demographic and dispersal characteristics allowing to reproduce the field data. Because of the low population size, usual methods to accurately quantify dispersal, survival, and fecundity in the wild are difficult to apply. In such challenging situations, a legitimate approach is to proceed to a "partial life cycle analysis,"9 coupling a simple population dynamic model with available data on demography and dispersal.
Several models of Chagas transmission to humans have been developed previously.10–13 However, few studies included a specific description of vector population dynamics (with or without considering a stage structure),14–16 and none of them described the source-sink situation described above with an explicit description of the stage structure needed to reproduce the patterns observed in Yucatan. We developed a specific model from the classic matrix formalism in which partial life cycle analysis has been developed.9
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MATERIALS AND METHODS |
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We searched for demographic and dispersal conditions required to reproduce the observed patterns of T. dimidiata domestic populations in the field.4 We fitted our model to the observed variations in bug abundance, to evaluate the values of dispersal, fertility, and survival rates providing the best fit. We performed two types of sensitivity analysis: one to identify which of the parameters had the strongest influence on the population dynamics and a second one to identify the range of demographic parameters and dispersal rates around the values that lead to variations in abundance and population structure similar to the observed field patterns.
Population dynamics model. Matrix model describing demography in the domestic habitat. Because survival probabilities of eggs and larval stages were similar in laboratory colonies of T. dimidiata,17–20 we considered survival as constant through all these stages. This allowed pooling the egg and all larval stages into a single stage. This stage can easily be divided into immature age classes of equal duration according to the choice of the time step of the model, thus reducing the size of the matrix model.
We developed a population model where the life cycle can be described in terms of the common survival probability for the 
immature age classes (SI), a constant adult survival probability (SA), and an adult fecundity F (female offspring per female), where represents the age at maturity expressed as a number of time steps. Multiplying by the time step duration, the age at maturity expressed in time units is obtained. The time step we chose with respect to field data is defined below.
The population dynamics is given by the linear system:
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where N(t) and N(t – 1) are vectors representing the population sizes of each age i at times t and t – 1, and A is a square matrix (called the population projection matrix) of dimension + 1. In matrix notation:
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We chose a 3-month time step for the model to match model predictions to field data, which were carried out every trimester.4,5 As the simplest assumption,9 we considered that all individuals remain a fixed time in the different immature age classes of the model. Under this assumption, we only need to match the age at maturity ( multiplied by the time step) in the model with the experimental average development time from egg to adult. For T. dimidiata, this development time ranges from 162 to 328 days.17–20 Because we selected a 3-month time step for the model, we tested development times of 180, 270, and 360 days, corresponding to = 2, = 3, and = 4 immature classes of 3 months (the time step of our model), respectively. Changing this parameter did not affect the conclusions of the paper, and results are shown only for an intermediate duration of 270 days (the development duration closest to the average (247 days) between the experimental values of 162 and 328 days). Accordingly, individuals of the first, second, and third immature classes were 0–90, 90–180, and 180–270 days of age, respectively, and each class included various immature developmental stages according to the average development time of each stage. The matrix we used was therefore of form A and dimension + 1 = 4:
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Primitive and irreducible matrices such as matrix A in Eq. 2 or 3 are known to produce either an exponential population decrease or increase, with a population growth rate given by its dominant eigenvalue and an asymptotically stable age structure given by the left eigenvector associated to this dominant eigenvalue.9 The population dynamics are summarized by these two quantities. Although building this matrix model was needed at first to describe the stage structured demography of T. dimidiata, it resulted in an unrealistic constant population growth. We introduced seasonal immigration and focused on the situation where domestic populations are not self-sustainable but rebuilt by non-domiciliated vectors immigrating from the peri-domestic habitat or the wild.
Matrix model with seasonal immigration. Seasonal immigration was modeled by adding a certain number of female adults (m) to the domestic population at the second time step of every year (equivalent to April to June, when an increase in population size was observed in the Yucatan peninsula).4 When seasonal immigration was introduced, the population dynamics was given by the system:
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where M(t) is the vector of immigrants at time t. Because t is expressed in trimester units and immigration occurs only in April to June, M(t) is different from 0 only every four time steps. M(t) includes no individuals in the three immature age classes and m adult female immigrants, i.e., M(t) = (0,0,0,m).
This no longer corresponds to a linear system, but its qualitative behavior could still be anticipated. In demographic conditions where the population can grow exponentially in the absence of immigration, the introduction of seasonal immigration has no impact on the asymptotic population growth rate and stage structure. On the contrary, when the dominant eigenvalue of matrix A is < 1, we expect population size to asymptotically fluctuate with a stable periodic pattern of alternation between a decrease in population size because of its non-sustainable dynamics and an increase because of immigration. We assessed this pattern of periodic cycles by recording in the model the population size and the proportion of adults just before immigration. Because our model is deterministic in nature (no source of stochastic variations have been modeled), the population size at any given time of the year is exactly the same year after year. We called the population size at a fixed time in the year "equilibrium population size," although the simulated population is inherently cyclic. Also, because these properties are asymptotic ones, they do not depend on the initial population. They were evaluated starting from an arbitrary population composed of one adult and one individual of each immature age class.
Model analysis.
Fit of the model to field data.
We first fitted our model to field data from the Yucatan peninsula to identify the parameter values that conformed best to the field pattern. Bug collections were available per trimester for 20004 and 2001,5 so T. dimidiata populations were calculated on a per house and per trimester basis for comparison with the field data. We estimated the values of five parameters (Init [the initial number of individuals], SI, S, F, and m), minimizing the sum of squares between the field and the model values. Modeled values were adjusted to consider a sex ratio of 60% adult females as observed in the field.4
Sensitivity analysis.
We performed two types of sensitivity analyses: first to evaluate the relative sensitivity of the domestic population to the model’s parameters (Init, SI, S, F, and m) and second to evaluate the range of parameter values allowing for patterns of temporal variation in population abundance and percentage of adults similar to the ones observed in the field.
Relative sensitivity of the domestic population to the model’s parameters data.
We assigned 500 values to the parameters Init = initial population/house, SI, S, F, and m, and used the Winding Stairs technique.21 The Winding Stairs is a sampling scheme designed to provide an economic way to select parameter values, reducing the total number of model evaluations by more than one half. We performed the sensitivity analysis using the Sobol method,22 which measures the model’s sensitivity to the parameters by partitioning the total variance of the output variable Y (total T. dimidiata population) in main effects and interaction effects among parameters.23 We calculated the first-order Sobol sensitivity index for the ith parameter (SobMi), which measures the effect of parameter xi on the output variable Y, and the total Sobol sensitivity index (SobTi), which takes into account the interactions (of order 2 or higher) between the ith parameter and the rest of the parameters. The total sensitivity index (SobTi) can be thought of as the expected fraction of variance that would be left if only the parameter xi were to stay undetermined.
Range of demographic and dispersal parameters consistent with the field data. We calculated population size at equilibrium and class structure of T. dimidiata populations while varying SI, SA, F, and m within a biologically relevant range of values. SI (the survival probability for the immature classes) was varied systematically from 0.1 to 1/trimester with equal steps of 0.1, which included values observed in laboratory colonies (0.6–0.8/trimester).17–20 Adult lifespans of 4, 6, 12, 18, and 24 months (the maximal value roughly corresponding to the 22-month lifespan provided by Zeledón and others)24 led to SA (adult survival probability) values of 0.25, 0.5, 0.75, 0.833, and 0.875/trimester, respectively, using the geometric model that assumes a constant survival probability.9 F (adult fecundity) was varied systematically from 0 to 1,000 female offspring/female/trimester (offspring represents female individuals of immature age class 1).24 The number of immigrants m used was 1, 10, and 100 individuals/year. This range included the different estimates of migration rates for T. dimidiata in different regions, including Yucatan.7,25,26
Population size at equilibrium was considered consistent with the field data when they were between 1 and 100, because the average population size per house is typically ~5–20,4,5 up to 100 insects/house/yr.27 The theoretical proportion of adults (with or without immigration) was considered to conform to the data when the number of adults as part of the whole population was > 80%.4
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Range of demographic and dispersal parameters consistent with the field data. As anticipated, in demographic conditions where the population can grow exponentially in the absence of immigration, the introduction of seasonal immigration has no impact on the asymptotic population growth rate and stage structure. We focused on demographic conditions where population was not self-sustainable. Under those conditions, the introduction of seasonal immigration resulted in a population persistence with a seasonal increase in bug abundance during April to June and a decrease during the rest of the year. More interestingly, we assessed if our implicit source-sink model was able to produce a proportion of adults and a population size consistent with field data.
Figure 2 shows that the lower the immature survival (SA) and the lower the fecundity (F), the higher the proportion of adults, although the shape of these relationships depended on adult survival SA. The increase in the proportion of adults was exponential when SA was high (Figure 2B), whereas it approached asymptotically its maximum value when SA was low (Figure 2A). Furthermore, the proportion of adults was typically much lower for high SA (Figure 2B) than for small SA (Figure 2A). In both cases, the proportion of adults depended little on the number of immigrants (results not shown). Overall, the most striking result was that, whatever the immature and adult survivals, fecundity must be very low (typically < 1–2 female offspring/female/trimester when SA= 0.25/trimester, and virtually null when SA= 0.875/trimester) to obtain a proportion of adults similar to that observed in the field (i.e., > 0.80).
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shows the population equilibrium before immigration of 1 (on the left y-axis) or 10 (on the right y-axis) individuals for low (SA = 0.25/trimester) and high (SA = 0.875/trimester) adult survival probability (Figures 3A and 3B, respectively). All other intermediate SA values resulted in intermediate population equilibrium values. Obviously, population size at equilibrium increased with SI, SA, and F (Figures 3A and 3B). Small population sizes consistent with the field data (i.e., between 0 and 100 bugs/house) were obtained for any of the immature and adult survival considered. However, whatever the combination of values of these two survivals, again, fecundity had to be kept < 12 female offspring/female/trimester to result in a population size 100. This range of fertility (0–12 female offspring/female/trimester) was even further reduced for larger numbers of immigrants or higher survival probabilities.
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These results confirmed that immigration from the peri-domestic or sylvatic habitats can explain satisfactorily the patterns of variation in the domestic population abundance of this vector species as observed in the Yucatan peninsula. However, dispersal alone was not sufficient to reproduce field data, because obtaining a proportion of adults > 80% and a small population size also required a very low fecundity, typically < 1–2 female offspring/female/trimester. When such a condition was fulfilled, immature and adult survivals were of little importance in determining a stage structure and population size consistent with field data.
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DISCUSSION |
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Field collections4,5 and population genetics data7 previously established that small domestic populations associated with a strong seasonality, and an important proportion of adults as observed in Yucatan, relied on significant immigration of triatomines from non-domestic habitats. We provide theoretical evidence that confirm that the observed pattern4 can be explained satisfactorily with such immigration.
This conclusion derives from the behavior of the mathematical model, and it is confirmed by the best fit of the model and the Sobol sensitivity analysis. The sensitivity analysis clearly identifies the number of immigrants as the key parameter in determining the total domestic population of T. dimidiata. The best fit of the model was obtained for an optimal number of immigrants equal to 4.52 bugs/yr, which corresponds to ~9–18 migrants/generation if we consider a generation time of 6 months to 1 year as observed from laboratory colonies of T. dimidiata.17–20 Such an estimate is in close agreement with population genetics studies of T. dimidiata in Yucatan, which reported 5–25 migrants/generation between domestic and non-domestic biotopes,7 as well as with other estimates of T. dimidiata dispersal rate in Colombia and Guatemala, which reported 2–326 and 9.725 migrants/generation between sylvatic biotopes and adjacent villages, respectively. The similarity of these estimates obtained using different approaches and different populations of T. dimidiata suggest that they are robust.
Although our model shows the need for immigration and provides an accurate estimate of T. dimidiata immigration rate, it also gives unexpected new insights into the demographic requirements to explain the level of abundance and the age structure observed in the field: an extremely low fecundity (typically < 1–2 female offspring/female/trimester) as indicated by both the best fit of the model and the sensitivity analysis. Furthermore, the best fit of the model also gave an extremely low immature survival, although a wide range of values were still allowed to reproduce field data. Interestingly, the per capita population growth rate (
) resulting from the model without immigration using these optimal parameter values is equal to 0.20 (an intrinsic rate of population growth of r = –1.61). This is the first quantitative evidence that domestic T. dimidiata populations in Yucatan are not self-sustainable, which confirms that it is a non-domiciliated vector of Chagas disease whose domestic population has to be periodically rebuilt by immigration. Under these conditions, up to 90% of the individuals found in houses would be immigrant adults.
Although our model mimics well the field cyclic behavior of T. dimidiata populations based on seasonal immigration, it could be argued that other factors such as natural enemies or abiotic factors also acting seasonally can theoretically be responsible for the same cyclic behavior. However, modeling of those alternative hypotheses would also have to produce predictions consistent with other observed features such as the large difference between the minimal and maximal observed population size and the proportion of adults. Seasonal predation is likely to be ruled out because most of the potential predators of triatomines (spiders, egg parasitoids, mites) have not been found in our study area in important numbers, despite being so conspicuous, and T. dimidiata has developed a marked habit of camouflaging itself to escape such predators.18 In addition, it would require a complex synchronized pattern of differential predation on larval stages and adults bugs at different times during the year to result in the stage structure variations observed in the field. On the contrary, the very good fit obtained suggests that seasonal dispersal on its own can explain not only the seasonality, but also the observed range of all year round variations in the population size and in the age structure. It is also worth noticing that such a fit was obtained for a rate of dispersal (number of adults per generation), which is remarkably consistent with independent estimates of the same dispersal rate based on population genetics and morphometric field studies. For all these reasons, we are confident that the immigration factor used in the model is the dominant one able to explain the observed data, although additional field and modeling studies would be needed to discard alternative hypotheses such as predation.
The demographic parameter values required to explain field data contrast sharply with those obtained in laboratory colonies of T. dimidiata from a wide range of different origins ranging from Yucatan and Chiapas, in Mexico to Costa Rica and including several cryptic species.28 Indeed, these studies indicate an immature survival between 0.6 and 0.8/trimester,19,20 an adult survival of 0.86 (equivalent to an adult lifespan of 22 months),23 and potential fecundity of the order of several hundreds of eggs per female per year.18,24 Most likely, these values are overestimated in the laboratory compared with natural conditions, because reduced or poor-quality blood meals in the field would reduce survival and/or lower the number of eggs an adult female triatomine can lay.29–32 Other hypotheses (e.g., higher mortality, leaving the houses, and not laying eggs because of maladaptation to the domiciliary conditions) could also explain the almost null fecundity, extremely low immature survival, and low adult survival required to fit our field data.
Adult survival as defined in the model corresponds to individuals remaining inside houses from one time step to another. The corresponding loss of individuals may in fact be interpreted as true mortality or emigration from houses into the non-domestic habitat. Thus, "true" adult survival may have been underestimated in the model. Population genetics data showed that T. dimidiata migrations occurred both ways: toward the houses and also from the houses toward non-domestic areas.7 Further modeling including a finer description of bug dispersal may lead to more precise estimates of adult survival.
The almost null fecundity and low immature survival may be caused by a number of factors including mating difficulties in the domiciles, females not being able to feed, the lack of appropriate oviposition sites for the females to lay their eggs, or a high mortality of immatures, particularly of the younger stages. Mating difficulties or inadequate oviposition sites seem less likely. Indeed, there is a small but repeatedly observed increase in (young) larval stages in the trimester after the arrival of the adults,4 indicating that at least some of the bugs are able to mate and/or lay eggs, and young larval stages can be detected in the domiciles. The ovaries of domestic females also contain an average number of eggs similar to laboratory-reared T. dimidiata, suggesting comparable ovogenesis and thus appropriate feeding (E. Dumonteil and others, unpublished data). On the other hand, there is molecular evidence that, within a single house in Guatemala, the T. dimidiata population is mostly composed of genetically unrelated individuals rather than siblings and that families of siblings are small.27 This is also in agreement with population genetics data, indicating that 10–50% of domestic T. dimidiata are actually coming from peri-domestic and sylvatic areas.7 Indeed, because these estimates are based on genetic or morphometric differentiation of bugs, they may provide an estimation of the minimum immigration detectable with these tools. All these data support the concept of a domestic population composed mostly of genetically unrelated immigrants having few descendants and thus favor the idea that survival of younger immatures stages (eggs and first larval stage) may indeed be low. As we have suggested before,4 this would indicate that there are attempts at colonizing the domiciles by T. dimidiata, but these remain unsuccessful. Nonetheless, further field studies would be required to clearly establish feeding, reproductive, and survival conditions of T. dimidiata in the domiciles and how these may contribute to such lack of colonization. Also, it would be of interest to determine whether immigration from sylvatic/peri-domestic areas into houses is an adaptive process or a simple by-product of a seasonal population increase and a higher dispersal activity in these areas. In this context, the low fecundity and survival of bugs inside houses (as predicted in this study) do not support the adaptive dispersal hypothesis, because they would not produce a selective advantage to the migrants or a successful colonization.
Our model also provides important insights for the optimization of vector control strategies. Traditionally, Chagas disease vector control programs have targeted domiciliated triatomine demography, using insecticides or housing improvement to reduce survival and fecundity, respectively, and ultimately leading to reductions in house infestation. In the case of T. dimidiata, our model clearly showed that there is little effect to be expected from modifying its demographic parameters in the domiciles because not only are these already low (and even extremely low for fecundity and larval stage survival), but our sensitivity analysis also indicates that these parameters contribute little to the size of T. dimidiata populations. This has indeed been observed in a field trial in which domestic populations were rebuilt only a few months after insecticide spraying, most likely by new immigrants.8 On the other hand, by confirming the dominant role of immigration on domestic infestation by T. dimidiata, our analysis unambiguously indicates that control interventions should target to reduce effectively triatomine immigration to the houses. In that respect, impregnated betnets33 and curtains34 seem to be effective control strategies where vectors have widespread sylvatic ecotopes.
Finally, our theoretical approach may also be applied to triatomine vectors from other regions of the Andean and Central America Countries. For instance, we used the matrix model to Zeledón’s data of T. dimidiata in a house of Costa Rica18 and also concluded that population cannot persist without immigration, with immigration being the dominant parameter, followed by fecundity. The approach proposed in this paper could be applied to other populations of non-domiciliated vectors to infer the field demographic parameters values required to suit the control strategies to the local entomologic setting. This should definitively help to better understand and control the transmission of Chagas disease associated with non-domiciliated vectors.
Received June 7, 2007. Accepted for publication September 14, 2007.
Financial support: Financial support has been provided by the program "Action des coopérations scientifiques avec l’Argentine" ECOS SUD (A04B02, Resp. C. Bernstein) to JR and FM and Grant 20020404 from SISIERRA/CONACYT to ED. This study received financial assistance from UNICEF/UNDP/World Bank/WHO special Programme for Research and Training in Tropical Diseases (TDR) Grant A60640 to SG.
* Address correspondence to Sébastien Gourbière, Laboratoire de Mathématiques, Physique et Systèmes (EA 4217), Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France. E-mail: gourbier{at}univ-perp.fr 
Authors’ addresses: Sébastien Gourbière and Raissa Minkoue, Laboratoire de Mathématiques, Physique et Systèmes (EA 4217), Université de Perpignan, Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France, Telephone: 33-4-68-66-17-63, Fax: 33-4-68-66-17-60, E-mail: gourbier{at}univ-perp.fr. Eric Dumonteil, Laboratorio de Parasitología, Centro de Investigaciones Regionales "Dr. Hideyo Noguchi," Universidad Autónoma de Yucatán, Ave. Itzaes 490 x 50, 97000 Mérida, Yucatán, Mexico, Telephone: 52-999-924-5910, Fax: 52-999-923-6120, E-mail: oliver{at}uady.mx. Jorge Rabinovich, Centro de Estudios Parasitológicos y de Vectores, Universidad Nacional de La Plata, La Plata, Argentina, Telephone: 54-221-423-3471, Fax: 54-221-423-2327, E-mail: jorge{at}ecopaedia.com.ar. Frédéric Menu, Laboratoire de Biométrie et Biologie Evolutive, UMR CNRS 5558, Université de Lyon, Université Lyon 1, Villeurbanne F-69622, France, Telephone: 33-4-72-43-29-03, Fax: 33-4-72-43-13-88, E-mail: menu{at}biomserv.univ-lyon1.fr.
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REFERENCES |
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| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
