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ABSTRACT |
 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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This study uses entomologic thresholds based on a mosquito simulation model (CIMSiM) and a dengue simulation model (DENSiM) described previously.5 These models suggest that if dengue-infected persons are introduced into a community of susceptible individuals, an epidemic will not emerge unless a minimum number of mosquitoes (i.e., a threshold) are present to carry out dengue virus transmission. An epidemic is defined as a 10% increase of dengue seroprevalence within a year.
Use of pupal demographic surveys as indicators of risk for dengue transmission has two main limitations. First, theoretical vector population thresholds (i.e., pupae/person) have to be validated in the field, taking into account various factors that affect their values, such as frequency of new virus introductions, air temperature, herd immunity to each dengue serotype, and human density. 5,6 Model-generated vector thresholds and field observations suggest that a low number of mosquitoes is required for dengue transmission when herd immunity is low and ambient temperature is high. 5,7 Because of variations in herd immunity and temperature, greater precision is required to validate vector thresholds, which can be achieved by using reliable vector indicators related to dengue transmission. A second limitation of using demographic surveys as indicators of risk for dengue transmission is the large sample size needed for reliable pupal surveys from which precise estimates can be made. Large sample sizes result from the highly aggregated or clumped spatial dispersion of pupae per container, house, or person. For example, a study conducted in Puerto Rico showed that a sample size of 4,000 households was required to achieve a precision of 15% error of the true mean pupae/person.8 A similarly high sample size was observed in another study in India.9 Both studies used the negative binomial distribution (NBD) to describe and calculate the expected sample size. A recently derived finite correction for the NBD has helped reduce the sample size needed for a precision of 15% error of the true mean pupae/person to ~1,000 households in the Puerto Rican study. 10 Nevertheless, even this reduced sample size is still a large, impractical sample size for Ae. aegypti and dengue control. Unreliable estimates of pupae/person may result in uncertainty about when and where to practice vector and dengue control.
The main purpose of this study was to develop and assess two approaches for simplifying the assessment of dengue vector populations by means of pupal demographic surveys. The first method attempted to find a generalized statistical model that describes the distribution of Ae. aegypti pupae and that is valid for most pupal surveys. For this method, the task was to determine whether the statistical distributions of pupae/person could be described by a common aggregation parameter (k) of the NBD. 11 If a common k existed, a simplified sequential sampling program (SSP) based on the NBD could be developed. 12 SSPs are efficient sampling schemes used to classify the vector populations below or above the population thresholds. The theoretical pupal demographic thresholds were used to set the upper thresholds above which vector control should be applied to avoid dengue transmission.5 SSPs have been successfully applied in various mosquito studies for effective reduction of the sample size required for reliable appraisals of population levels.8
The second approach for simplifying sampling used the presence or absence of Aedes spp. pupae instead of pupae/person and thus avoided the time-consuming process of finding and counting all pupae in a sampling unit (e.g., household). Once a pupa was found in a household, the inspector could move on to a different household and take another sample. Inference was based on an empirical model of the relationship between the proportion of habitat units infested and the number of pupae per habitat unit. 13–15 This model enabled calculations of threshold proportions of infested households above which dengue transmission could occur. As before, an SSP was developed, but this one used a regular binomial distribution model for presence/absence of pupae per household.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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). Randomly selected households were inspected for the presence of containers holding water, and each detected container was sampled to extract and count the number of Ae. aegypti pupae present. Random sampling was accomplished by enumerating each household in the study area and using a random number generator to draw households to be sampled (SPSS, Chicago, IL). To obtain the number of pupae/person, the total number of pupae found in all containers within a household was divided by the mean number of persons per household in that community; this mean was derived by using year 2000 US Census data.
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= 0.05) was used. If the observed value of U exceeds 2 SE of Û, the null hypothesis is rejected. 16 The null hypothesis was not rejected for any of the 10 pupal surveys (Table 1
). Method 1: The NBD and determination of a common k. The NBD describes counts of individuals per habitat unit and is characterized by having a large proportion of zeros; that is, many samples have no individuals, and only a few samples have many individuals. This statistical distribution is generally adequate to describe insect counts for which the variance is much larger than the mean. 11 Fitting a theoretical NBD to a series of samples (i.e., a pupal survey) requires two parameters: the average random variable (mean pupae/person) and the parameter k (aggregation parameter). Once k is known, the average can be set as the Ae. aegypti’s threshold density, and the expected distribution of the number of pupae/person can be generated. To be able to generalize these results to a particular region, we need to determine if the vector populations have a common k.
A successive approximation, the zero-intercept regression method, was used to determine whether the number of pupae/person in the 10 Puerto Rican surveys had a common k.11 Successive approximations to a common k are made after adjusting k for heterogeneity of variance. Three approaches were used to test for agreement with a common k: 1) overall 
2 test of the homogeneity of k-values (2A <2g-2,0.05; g = number of surveys), 2) significance test of the slope of the zero-intercept regression line (1/k; 2B >21,0.05), and 3) significance test of an intercept component (2C <21,0.05) that measures the difference between two straight lines: one fitted with and the other without the constraint of a zero intercept. 11
An SSP based on the NBD with a common k. An SSP tends to minimize the number of samples required to determine whether the vector population level is above or below a pre-established threshold. 12 For this method, two hypotheses are set out: H1 establishes a lower, safe threshold (i.e., a maximum) below which vector control is unneeded, and H2 establishes an upper threshold (i.e., a minimum) above which vector control should be applied. Random samples of households are sequentially contrasted with the thresholds each time a sample is taken to determine whether the cumulative number of mosquitoes has passed above or below the thresholds. If few, if any, pupae are observed after several samples are taken, the overall Ae. aegypti pupal density is most likely low; conversely, if, after taking several samples, we observe large numbers of pupae, the overall density is likely large. What is uncertain is how much longer sampling needs to continue to enable researchers to reach a reliable decision about the Ae. aegypti’s pupal level. When the true pupal density is well above or below thresholds, few samples will be needed to make a decision. However, if the true value is close to either threshold or between threshold limits, a large number of samples may be required to determine the pupal level reliably. Also, the larger the difference between H1 and H2, the smaller the sample size needed. We can calculate the maximum number of samples to be taken while trying to reach a decision, but if a large effort is required, the pupal density is most likely not well above the critical threshold. In such cases, follow-up sampling should be conducted a few days later.
Sequential sampling thresholds are characterized graphically by a pair of parallel lines (Figure 1):
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In these equations, Y is the expected, accumulated number of pupae observed after taking n household samples, b is a common slope, and a is the y-intercept. To calculate these equations for the NBD, we need to know the mean number of pupae/person and also k, and we need to set the Type I () and II (β) errors. In this study, we use the common kc previously derived, and values for and β were 0.05 and 0.10, respectively. 16,17
Figure 1 shows H1, the lower threshold of mean pupae/person, to be 
0.19 and H2, the upper threshold of mean pupae/person, to be 
0.53. The value 0.53 was calculated by using methods previously described.5 This value represents the threshold value expected at 28°C, with 0% dengue prevalence, after the introduction of one viremic person on Day 90 of the year. The value 0.19 is an arbitrary low or safe level at which dengue transmission is unlikely under the conditions for which H2 was calculated. To show the decision process, this study used datasets from three localities (Playa/Playita, Las Mareas, and Coqui) with different means (2.02, 0.99, and 0.19 pupae/person; Table 1). These sites were randomly sampled until a decision about pupal level was reached ( Figure 1 ). This process was conducted twice for each pupal survey (Figure 1) to show that the sample size required to make a decision varies each time, even if the same database is used. For the locality with the highest mean (2.02 ± 0.35 pupae/person), we determined that the mean number of pupae/person was > 0.53, based on collection of pupae from 4 (first random samplings) and 21 households (second random samplings), respectively. For this locality, vector control would be advised. For the locality with a mean of 0.99 ± 0.33 pupae/person, 9 and 83 samples, respectively, were needed to conclude that the mean was > 0.53 and that vector control would be advised. For the locality with a mean of 0.19 ± 0.05 pupae/person, 56 and 109 samples were needed to conclude that the mean number of pupae/person was lower than the lowest threshold of 0.19 and that vector control would not be advised. Based on Type I and II probability levels ( = 0.05, β = 0.10), the probability of erroneously concluding that control is not needed is 5/100 surveys; the probability of wrongly recommending vector control when it is not needed is 10/100 surveys. The lower the probabilities, the larger the samples sizes required to make a determination regarding needed intervention.
Contrary to fixed-sampling programs, the number of samples required to make a decision in sequential sampling is a random variable for which the expected or average value can be calculated. Krebs 16 summarized the methodology for calculating the expected number of samples required to reach a decision in SSPs given the factors of type of statistical distribution, Type I and II probability levels, and the critical lower (H1) and higher (H2) threshold values. Those values were calculated for this study to estimate the expected sample sizes for the range of threshold vector densities and proportions of infested households.
Method 2: A sampling program based on presence or absence of pupae. The main purpose of developing a sampling scheme based on presence or absence of pupae is to reduce the need to count and identify every pupa found in a sampling unit (i.e., household). For the previous method, the pupal demographic thresholds were derived for pupae/person. For the sampling method based on presence/absence of pupae, the threshold levels had to be adjusted based on proportions of infested households.
To determine whether a significant relationship existed between pupae/person and the proportion of pupa-infested households, we explored use of the empirical model developed by Gerrard and Chiang. 13 The formulas for this model are as follows:
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In this model, a and b are parameters estimated by using least squares linear regression (SPSS). Once the parameters are calculated, the expected proportions of infested sampling units corresponding to established thresholds of pupae/person can be calculated by replacing the mean number of individuals per sampling unit.
The precision (D) of the estimates derived by using this model can be explored in terms of the sample sizes (nD) required to achieve a given ratio SE for a mean (D = SE/mean). 13–15 The formula to predict sample sizes for certain precision levels is as follows:
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SSPs based on proportions of pupa-infested households. The probability of the presence or absence of pupae in a sampling unit as derived above can be treated as a binomial variable. Threshold values used in the SSP are the extrapolated proportions derived from the Gerrard and Chiang model. 13 The following contrasting hypotheses are represented by the equations:
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In these equations, p is the observed proportion of infested households, and 
1 and 2 are the lower and upper threshold estimates of the population proportions, respectively. The SSP and the expected sample sizes required to make a decision about needed intervention were calculated using the formulae and programs described previously. 16,17
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RESULTS |
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2A = 6.1 < 28,0.05 = 15.5), significant slope of the regression line (1/kc; 2B= 67.2 > 21,0.05 = 3.8), and nonsignificant y-axis intercept (2C= 0.002 < 21,0.05 = 3.8). Method 1: An SSP based on the NBD with a common k. We illustrated the application of sequential sampling decision equations by considering the case in which the vector density threshold is 2.92 pupae/person with conditions of 24°C, 0% herd immunity, and one viremic person introduced on Day 90 of the year.
First, Type I ( = 0.05) and Type II (β = 0.10) errors were defined; then the two hypotheses were defined as follows:
H1: mean pupae/person 1.00 (safe level not requiring vector control)
H2: mean pupae/person 2.92 (critical dengue transmission vector threshold)
The lower line, H1, represents a safe level at which dengue transmission probably will not occur, so its value should be lower than but not too close to the upper threshold represented by H2. We were primarily interested in testing whether the pupal density of Ae. aegypti is 2.92 pupae/person, the upper threshold. We arrived at the following equations for the lower and upper lines:
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Tables instead of graphs are preferred for field work, and Table 2 was created by using these equations. Other potentially useful threshold values for Puerto Rico and other tropical areas include 1.42 pupae/person at 26°C, 0.53 pupae/person at 28°C, 0.13 pupae/person at 30°C, and 0.07 pupae/person at 32°C (tables not provided).4 Table 3 shows how the decision tables can be used. This table used the upper and lower thresholds of 2.92 and 1.00 pupae/person, respectively. The number of pupae resulting from the first sampled household was divided by the average number of householders in the community (producing the number of pupae/person) and was entered in the cell under the heading "Cumulative number of observed pupae/person" (third column), which corresponds to "Household number" 1 (first column). According to the table logic, a decision can now be made if the number of pupae collected in the first household was 100 (H2: µ 2.92 pupae/person). We cannot make a decision if the mean density is 1.00 pupa/person (H1: µ 1.00 pupa/person) until we have sampled at least 47 households. If the number of accumulated pupae/person in this first sample is lower than 100, another random household has to be sampled. The number of pupae/person found in the second household is added to the first one and entered in the second cell. We can now check to see whether this accumulated value is 102 pupae/person; if the value is lower than 102, we continue sampling and repeating the procedure. To decide whether the pupal density is 1.00 pupa/person, we need an accumulated number of pupae/person that is no more than the number of pupae/person indicated in the table, corresponding to the number of samples taken. For example, if we sample 48 households and the cumulative number of pupae is 1.5 pupae/person (Table 3, row 48), we can conclude that the mean density is < 1.00 pupa/person and we can stop sampling.
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Method 2: A sampling program based on presence/absence of pupae and the binomial distribution.
Mean pupae/person(loge) and proportion of infested households {loge [– loge (1–p)]} were fitted to a linear regression based on the model of Gerrard and Chiang. 13 In this study, the regression was significant (R2 = 0.86; P < 0.01), with regression coefficients a = 12.785 ± 3.831 (t = 7.17; P < 0.01) and b = 1.495 ± 0.211 (t = 7.09; P < 0.01). This equation was used to calculate the equivalent proportions of households infested with Ae. aegypti pupae corresponding to the upper and lower threshold densities shown in Table 2. Lower limits were arbitrarily set as before.
Table 2 shows decision equations generated for the extrapolated proportion of households infested with Ae. aegypti pupae. Based on these equations, decisions can now be made as before, except that, for this method, instead of using the cumulative number of pupae/person, we enter the cumulative number of positive (pupa-infested) households as they are sequentially and randomly sampled.
Decision equations for the extrapolated proportion of infested households when the density of pupae/person is expected to be < 1.00 (proportion = 0.166) or > 2.92 (proportion = 0.311) pupae/person are represented as follows:
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In these equations, Type I errors are defined as = 0.05 and Type II errors by β = 0.10. Table 4 shows a field table using these equations.
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). Relative precision (10–15%) was achieved in the pupal surveys with a larger number of samples (600–800 households).
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), and the corresponding expected sampling sizes were 107 and 63 households, respectively (Figure 3). For the proportion of pupa-infested houses at the same temperature, the maximum sample size required was between the minimum and maximum thresholds (0.166–0.311; Table 2; Figure 4), or ~54 samples. The number of samples required in each SSP was similar (Figures 3 and 4), although more time and resources can be saved by recording only the presence or absence of pupae (i.e., indicating whether household is infested), which does not require a detailed counting of every pupa, unlike when enumerating pupae/person.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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and 4). However, additional pupal surveys are needed to determine the general applicability of these results.
The second approach used to simplify pupal surveys was based on the model published by Gerrard and Chiang. 13 This method was implemented to reduce the number of required samples and the time required to analyze each sample because it examines only presence or absence of pupae. Whether the parameters observed in this study apply to other sites is unknown. The model’s parameters should be calculated by using data from a representative number of local pupal surveys to assess the model’s applicability. In this study, pupal surveys made after randomly sampling 600–800 households yielded precision levels of 10–15% (SE/mean; Figure 2).
For dengue disease prevention and control, the application of these methods is as valid as the threshold values used to calculate the sampling schemes. Because pupal demographic thresholds for Ae. aegypti have not been definitively validated, the tools discussed here are intended to be used for exploratory purposes. This study used only conservative threshold values to develop these sequential plans (0% dengue seroprevalence); however, actual thresholds would likely be higher because Puerto Rico is highly endemic for dengue. Recent evaluations of immunoglobulin G antibody prevalence in several locations in Puerto Rico have yielded values in excess of 90% (CDC, unpublished data). However, this result must be interpreted with caution because this composite measure of herd immunity does not consider the specific immunity for each of the four dengue virus serotypes.
Validating thresholds for Ae. aegypti can be made more efficient by using the sequential plans because many areas with and without active dengue transmission can be rapidly evaluated. The main limitation to validating entomological thresholds for local dengue transmission is an occasional lack of concurrence of above-threshold numbers of Ae. aegypti with dengue transmission. 18 Field validation involves sampling Ae. aegypti by using a reliable and epidemiologically relevant estimator (e.g., pupae/person) and evaluating the presence or absence of dengue virus transmission (e.g., virus in mosquitoes) in human communities where dengue circulation is expected. Entomologic threshold validation should take into account the variations in the minimum number of mosquitoes required for dengue transmission, depending on temperature, virus movement, and herd immunity.5 Comparisons between the actual numbers of Ae. aegypti and theoretical thresholds are starting to surface, thus helping researchers to know when dengue transmission is likely to occur. 19
Received November 26, 2008. Accepted for publication March 25, 2009.
Acknowledgments: The author thanks the Division of Entomology and Ecology Activity, Dengue Branch, Centers for Disease Control and Prevention, for the exemplary work that provided data from the 10 pupal surveys in Puerto Rico; he especially thanks the following personnel: Manuel Amador, Annette Diaz, Veronica Acevedo, Belkis Caban, Juan Medina, Jesús Flores, Gilberto Felix, Orlando González, and Andrew MacKay. Dr. Rhonda J. Ray reviewed and corrected the manuscript.
Financial support: Funding for this study was provided for by the Division of Vector Borne Infectious Diseases, Centers for Disease Control and Prevention.
Disclosure: The author does not have a commercial or other association that might pose a conflict of interest.
* Address correspondence to Roberto Barrera, Dengue Branch, Centers for Disease Control and Prevention, Calle Canada, San Juan, Puerto Rico 00920. E-mail: rbarrera{at}cdc.gov 
Author’s address: Roberto Barrera, Dengue Branch, Centers for Disease Control and Prevention, Calle Canada, San Juan, Puerto Rico 00920, Tel: 787-706-2399, Fax: 787-706-2496, E-mail: rbarrera{at}cdc.gov.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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