Before establishment of the ‘genetically sterile’ male Ae aegypti line in the mass rearing laboratory and subsequent open releases, regulatory approvals were obtained from the appropriate Brazilian national regulatory body: the Brazilian National Biosafety Technical Commission (CTNBio). Releases were preceded by community engagement with consent and support obtained from regional (Bahia health secretary) and local community leaders (Town Mayor, health secretary and vector control authorities). Prior to sampling, informed consent was received from the landowners.

The field site is located in Itaberaba, a suburb of the city of Juazeiro, Bahia, Brazil (Latitude: -9° 26' 59", Longitude: -40° 28' 53") (Fig 1A). The site is located in a semi-arid part of Brazil and consists mainly of low-socioeconomic status residential housing. The majority of houses at the study site were single-story brick and concrete buildings with unscreened windows. The habitat across the sampled region was a homogenous urban environment.

The ‘genetically sterile’ line used was OX513A and was reared according to methodologies given in Carvalho et al. (2014) [19].

A total of 19,164 ‘genetically sterile’ male Ae. aegypti formed three releases. Individuals from each release were marked with the same coloured fluorescent powder (www.dayglo.com). Release 1 (red release) and release 2 (blue release) were performed on 21 February 2011 and consisted of 5,349 and 5,465 individuals released from points 1 and 2 (Fig 1B) respectively. Release 3 (yellow release) was performed on 25 February 2011 with 8,350 individuals released from point 1 (Fig 1B).

Aspiration sampling was used to recapture marked adults. Sampling was conducted using locally made battery powered hand-held aspirators. After obtaining consent from the respective property owner, each building was sampled for a set period of 15 minutes. Sampling locations were distributed across the study site (Fig 1B) and sampling was conducted for up to nine days post release (S1 Data). Sampling locations were chosen by randomly selecting one household from each of the 47 (60m by 60m) grid squares each day (with the exception of day 5 for the red and blue release and day 1 for the yellow release where, for logistical reasons, the number of households sampled was lower). Each mosquito collected was assessed to determine the i) origin (‘genetically sterile’ or wild as indicated by the presence/absence of fluorescent powder respectively), ii) sex and iii) genus (Aedes or non-Aedes). Weather variables (daily maximum temperature and maximum humidity) were recorded from a local weather station (situated approximately 10.2km north-west of Itaberaba).

A secondary analysis of the MRR data from a previously published study in Malaysia [12] was undertaken to obtain a comparative second estimate of the ‘genetically sterile’ insect’s dispersal ability and associated density kernel. In this study a total of 6,045 marked ‘genetically sterile’ males were released at a forested site in Pahang, Malaysia. For a detailed description of the study site and MRR methods please see Lacroix et al. (2012) [12].

All multivariable analyses were performed within a GLM framework. In instances where the number of recaptures is small relative to the total releases, the Poisson regression model may be used as an appropriate approximation [20].

We assume that the count response variable (recaptures) is Poisson distributed with mean μ and variance μ Yi∼Pois(μi). (1)

The response must be ≥ 0. Therefore a log link function is used to link the mean to the explanatory variables ln(μi)=xiβ, (2) where x_i β is a linear predictor xiβ=β0+β1xi1+…+βpxip, (3) where β denotes the unknown parameters to be estimated and x_i, the explanatory variables. Parameter estimates were obtained by maximising the log-likelihood (l) of the data: l(β|Y)=∑i=1n(yilnμi−μi−ln(y!))=∑i=1n(yixiβ−exp(xiβ)−lnyi!). (4)

In the situation where the response variable is overdispersed (variance>>mean) a Poisson GLM, where the variance is assumed to equal the mean, would be misspecified. In this instance, the negative binomial GLM, detailed below, may be used [21–23].

We assume that the count response variable follows a negative binomial distribution. A Poisson model is used for the count, conditional on the mean value, Z_i, which is assumed to have a gamma distribution, with mean, μ_i, and constant scale parameter, θ Yi∼Poisson(Zi),Zi∼gamma(μi,θ). (5)

Therefore the expected value of Y and the variance of Y are as follows E(Yi)=μi,Var(Yi)=μi+μi2θ (6)

The mean response, μ_i, may be linked to a linear combination of explanatory variables using the log link function (Eq (2)) and linear predictor (Eq (3)). Parameters were estimated by maximising the log-likelihood (l) of the model: l(β,θ|Y)=∑i=1i=nθ(ln(θ)−ln(θ+μi))+ln(Γ(θ+yi))−ln(yi!Γ(θ))−yi(ln(μi)−ln(θ+μi)), (7) where the gamma function, Γ, is Γ(n)=(n−1)! (8)

Considerable inconsistencies abound regarding different interpretations of the term ‘dispersal kernel’ [18,24,25]. Two kernel definitions, often used interchangeably, are i) the probability density function (pdf) of the dispersal distance of each disperser. Referred to as the distance kernel [18] or the distance pdf [25] and ii) the density of probability of a given bearing and dispersal distance from the source. Referred to as the location kernel [18] or the density pdf [25].

We adopt the terminology of Cousens et al. [25], henceforth referring to kernel type 1 as the distance pdf and type 2 as the density pdf. Both kernel types are true pdfs, integrating to 1 (the density pdf is integrated over the whole 2d space). Both kernel types are closely related. The distance pdf can be derived by multiplying the density pdf by 2πd where d is the distance from the source [18] (assuming radial symmetry). Examples of these kernel types are shown in Fig 2.

The density pdfs, assuming radial symmetry, used in this analysis are defined by the following functions Negativeexponentialkernel=12πa2e(−da)a>0, (9) Exponentialpowerkernel=b2πa2Γ2be(−dbab)a,b>0, (10) where d is the distance (metres), a and b are kernel parameters and Γ the gamma function (Eq (8)) [18]. The negative exponential kernel is characterised by the exponential power kernel when b = 1. The associated MDT functions are NegativeexponentialMDT=2a (11) ExponentialpowerMDT=a(Γ3bΓ2b). (12)

Estimates of FR₅₀ and FR₉₀ are made by assessing the cumulative distribution of the distance pdf at the 50% and 90% levels.

The outcome variable was the number (count) of marked ‘genetically sterile’ male Ae. aegypti recaptured. Potential explanatory variables included in the Brazil analysis were: i) a spatial measure which could either be the measured distance (m) between release and recapture or the density as calculated by a parameterisation of a given density pdf, ii) the number of days post release, the effect of which is assumed to be linear, iii) the number of wild Aedes species collected iv) the number of non-Aedes wild mosquitoes collected, v) the maximum temperature (°C) on the day of collection, vi) the maximum humidity (%) on the day of collection and vii) the directional quadrant, North, South, East or West (relative to release point) that the collection was made in.

Due to the relatively low recapture number, data from all three MRR experiments were combined for analysis.

For the Malaysia analysis the outcome variable was the number (count) of marked ‘genetically sterile’ male Ae. aegypti recaptured. Potential explanatory variables included in the analysis were: i) a spatial measure which could either be the measured distance (m) between release and recapture or the density as calculated by a parameterisation of a given density pdf, ii) the number of days post release, the effect of which is assumed to be linear iii) the number of wild Aedes species (specifically: aegypti, albopictus and togoi) collected, iv) the number of wild Culex collected and v) a categorical variable indicating if the recapture location was uphill or downhill from the release site.

Three models were evaluated to compare different transformations of the distance explanatory variable. Model 1 incorporated all explanatory variable including distance, Model 2 all explanatory variables including distance density (negative exponential kernel) and model 3 all explanatory variable including distance density (exponential power kernel).

For model 1 the full model was fitted using maximum likelihood techniques, utilising the GLM and Negative binomial GLM function of the statistical software package R [26] with the MASS package [21]. All explanatory variables were included in the initial model as well as an interaction term between distance and day post release. Model selection by minimising the Akaike information criterion (AIC), a penalised likelihood score, was then performed using the MASS package [21]. The AIC is calculated as AIC=2k−2ln(L) (13) where k is the number of parameters and L the maximised likelihood value.

For models 2 and 3 fitting was performed using the following process. First, the distance density was estimated using the assigned kernel. The GLM was then fitted using the same process as for model 1, as a function of the transformed distance explanatory variable. These steps were then optimised over the kernel parameter space allowing identification of the optimal combination of explanatory variables and kernel parameters as indicated by the AIC. The best overall model was judged to be the one with the minimum AIC value.

For comparison, the estimated survival of released insects in the Brazil study predicted using the GLM was also calculated using a non-linear regression approach [27], that was also used in the original analysis of the Malaysia data [12].

Following model estimation, 95% confidence intervals were calculated for the maximum likelihood kernel parameter estimates using the profile likelihood method. The maximised log-likelihood with respect to β, α and b, i.e. that corresponding to the maximum likelihood estimates (MLEs) of β, α and b, is defined as l(β^,a^,b^|Y).

First kernel parameter a was increased or decreased in small increments whilst β was held at the MLE (β^) and kernel parameter b was optimized conditional on β^ and the assumed value of α (a₀), giving the log-likelihood: l(β^,a0,b˜|Y), (14) where b˜=MLE(b|a0,β^).

Secondly, kernel parameter b was increased or decreased in small increments whilst β was held at the MLE (β^) and kernel parameter α was optimized conditional on β^ and the assumed value of b(b₀), giving the log-likelihood: l(β^,a˜,b0|Y), (15) where a˜=MLE(a|b0,β^).

After each change the log-likelihood of the model was recalculated and a corresponding test statistic assessed. For example, evaluating for a, the G² statistic was calculated G2=2(l(β^,a^,b^|Y)−l(β^,a0,b˜|Y)). (16)

The G² statistic was compared to the χ² distribution (with 1 degree of freedom) for the (1-α) percentile. Thus for 95% confidence intervals the critical G² value is 3.84. The log-likelihood surface was calculated with respect to kernel parameters of the optimal model for exploration and visualisation of the parameter space for both the Brazil and Malaysia analyses.

Recaptures for the three MRR experiments are summarised in Table 1. The locations of recaptures are shown in Fig 3. The mean count of recaptured marked males (the response), per sample, per day was 0.077 (variance = 0.73). Over the recapture period the maximum daily temperature ranged between 25.4°C-34.6°C and the maximum relative humidity between 66%-92%.

A summary of model performance using the untransformed- and transformed-distance explanatory variable is shown in Table 2. Combining all available data (from the red, yellow and blue releases) the best fitting model (lowest AIC) incorporated the exponential power kernel. The maximum likelihood exponential power density pdf has an associated MDT of 52.8m (95% CI: 49.9m, 56.8m), FR₅₀ of 52.4m (95% CI: 50.6m, 54.7m) and FR₉₀ of 83.0m (95% CI: 74.8m, 93.9m). The MLE kernel, log-likelihood surface and examples of kernels drawn from 95% CI parameter values are shown in Fig 4.

For all three models the distance or distance density and the number of days post release were strongly associated with recapture number. There was no evidence of an interaction between distance (untransformed or transformed) and the number of days post release in any of the models considered. Assuming no emigration, the estimated mortality rate of released insects of 0.62 (95% profile likelihood CI: 0.84, 0.42) would equate to a mean average lifespan of 0.62⁻¹ = 1.61 days (95% CI: 1.19 days, 2.38 days). Estimates of the mean average lifespan calculated using the non-linear regression approach [27] were 1.00 days (95% bootstrapped CI: 0.63 days, 1.48 days).

Other significant explanatory variables were the number of non-Aedes mosquitoes recorded from the sample and the maximum humidity. There was evidence of a lack of radial symmetry in dispersal from the release point as the quadrant explanatory variable was also associated with recapture number. A summary of the parameter estimates from the optimal model, using the exponential power transformation of distance as an explanatory variable is shown in Table 3.

The MRR performed in Malaysia was associated with consistently higher recaptures than the MRR experiments in Brazil. Of 6,045 released ‘genetically sterile’ males 3,034 (50.2%) were recaptured over the 15-day course of the experiment. The count of recaptured, marked males (the response) was very overdispersed (mean = 2.6 per sample per day, variance = 523), and therefore a negative binomial GLM was fitted. A summary of the model performance using the untransformed and transformed distance explanatory variable is shown in Table 4.

Again, the optimal model, as determined by AIC, used the exponential power density pdf, although the negative exponential density pdf produced only marginally inferior fit (AIC = 668.0 and 668.4 for the exponential power and negative exponential kernels respectively). The MLE exponential power pdf estimates a MDT for the ‘genetically sterile’ release of 58.0m (95% CI: 51.1m, 71.0m), FR₅₀ of 51.8m (95% CI: 47.9m, 58.7m) and FR₉₀ of 105.7m (95% CI: 86.5m, 141.1m). The MLE kernel, log-likelihood surface and examples of kernels drawn from 95% CI parameter values are shown in Fig 5.

The coefficient summaries from the negative binomial model using the exponential power transformed distance explanatory variable are shown in Table 5. This second analysis again indicates that distance is an important significant predictor of the expected count of recaptures. The number of days post release was also significantly associated with recapture number. Assuming no emigration, the estimated mortality rate of released insects of 0.46 (95% profile likelihood CI: 1.03, 0.13) would equate to a mean average lifespan of 0.46⁻¹ = 2.17 days (95% CI: 0.97 days, 8.85 days). Unlike the analysis for Brazil there was evidence of an association between the distance and the number of days post release explanatory variables.

For a direct comparison the distance pdf and density with respect to distance for the optimal kernels estimated from the Itaberaba and Malaysia MRR experiments have been overlaid (Fig 6).