3.1 SIT efficacy

3.1.1 Hatching rates (Egg fertility)

Figure 3.1 shows the evolution of the hatching rate (i.e. number of hatched eggs divided by the total number of eggs) as measured in the laboratory, for each of the 30 ovitraps at each site.

Figure 3.1: Median number of hatched and total eggs by trap, date and site, with 25 and 75% quantile bands. Individual trap counts are represented lightly in the background. The intervention period, where sterile adult males were released is highlighted with a light orange background.

Figure 3.2 displays the predicted mean trend of the hatching rate for the control and release sites, with 95% confidence interval bands, from a thin-plate spline smoothing of the trap-specific rates.

Figure 3.2: Temporal trend of the mean and 95% CI hatching rate in the control and release sites predicted by a generalised additive model.

Figure 3.3 provides a different view of the same information, by looking at the empirical distributions of the numbers of hatched and unhatched eggs in the surveys of individual traps. Surveys are aggregated by whether they took place during the intervention period (with a 14-day lag) and by whether the corresponding trap was located in the Release or the Control sites.

Figure 3.3: Distribution of the numbers of hatched and unhatched eggs at each survey of a trap either during the intervention period (with a 14-day lag) in the Release and Control sites.

Figure 3.4 provide yet an alternative comparison in terms of the hatching rates (hatched / unhatched).

Figure 3.4: Hatching rates of individual surveys, by site type and (14-day lagged) period.

A single-number summary of the effect of the intervention on the hatching rates displayed in Figure 3.4 is the Induced sterility, which is the relative reduction of the hatching rate in the release area during the intervention with respect to the hatching rate in the control area. \[\begin{equation} I_s = 100 \, (1 - H_r / H_c). \end{equation}\]

Table 3.1 displays the overall hatching rates in the control and release sites (\(H_c\) and \(H_r\) respectively) during or outside of the intervention period, and the corresponding relative reductions (RR) of hatching rates.

Note that the value of RR during intervention represents the induced sterility, whereas the value of RR outside of the intervention is expected to be essentially zero.

Table 3.1 includes an adjusted stratified bootstrap percentile interval by period, which shows that the induced sterility (i.e., the RR during intervention) is much higher in magnitude than the estimation error due to random variation.

Table 3.1: Induced eggs sterility in- and off-intervention with Bootstrap Confidence Intervals. The corresponding hatching rates (HR) in the control and release sites used for the calculation are also displayed.

Period	HR Control	HR Release	Relative reduction	CI95 lo	CI95 hi	Histogram
Off-intervention	0.83	0.81	1.72	-0.48	4.14
Intervention	0.77	0.37	51.70	48.58	54.54

Figure 3.5: Global average hatching rates in the Control (grey) and Release sites (orange) either in or out of the intervention period. Posterior distributions with 66 and 95 % quantile intervals.

Figure 3.6: Relative reduction in hatching rates out of the intervention period (grey) and during intervention (i.e., induced sterility, orange). Posterior distributions with 50 and 95 % Credible Intervals.

Figure 3.7: Estimated hatching rates in each trap within sites, during the intervention period. Posterior meadian with 66 and 95 % quantile intervals.

Let \(Y_{ijk}\) be the number of hatched eggs observed in trap \(i\), in site \(j\) (control/release) and period \(k\) (in/out-intervention), out of \(n_{ijk}\) total laid eggs. We can assume, \[\begin{equation} \tag{3.1} Y_{ijk} \sim \text{Bi}(\pi_{ijk}, n_{ijk}), \end{equation}\] where \(\pi_{ijk}\) is the hatching rate at trap \(i\), site \(j\) and period \(k\).

Furthermore, we assume that the trap-specific hatching rate can be decomposed additively into a global average by site and period and a trap-specific deviation, in the logit scale. \[\eta_{ijk} = \text{logit}(\pi_{ijk}) = \alpha_{jk} + \alpha_i,\] where \(\alpha_i \sim \mathcal{N}(0, \sigma)\).

Figure 3.5 represents the posterior distributions for \(\text{expit}(\alpha_{jk})\) and confirms that the global average hatching rates in the Control and Release sites are practically the same out of the intervention period, with a marginal induced sterility due to sampling variation and to some residual effect of the intervention period after the 14-day lag.

Conversely, it also confirms that the estimated hatching rates and the corresponding induced sterility displayed in orange in Figure 3.6 can be entirely attributed to the intervention.

Moreover, Figure 3.7 displays the variability of hatching rates across traps. Even the highest trap-specific hatching rate in the release site is well below the smallest hatching rate in the control site.

3.1.2 Egg density

An analogous study can be made in terms of egg density. I.e., the average number of eggs laid by trap, depending on the site and the period. The objective is to see whether there is a appreciable reduction in egg density on the release area during de intervention period.

However, the descriptive plot in Figure 2.4, shows that the median number of eggs laid by trap remains fairly stable over time. In particular, there seems to be no appreciable effect of the intervention.

Table 3.2: Reduction in eggs density during intervention or not. The corresponding egg densities in the control and release sites used for the calculation are also displayed.

Period	Dens Control	Dens Release	Relative reduction (%)
Off-intervention	65.22	110.31	-69.14
Intervention	95.29	110.47	-15.93

With this perspective, the results presented in Table 3.2 are mostly noise and should not be interpreted beyond the observation that no reduction in egg density is apparent from the experimental data.

Let \(Y_{ijkl}\) be the logarithm of the total number of eggs laid in trap \(i\), in site \(j = 1, \ldots, 4\), week \(k\) and period \(l\). We can assume, \[\begin{equation} \tag{3.2} Y_{ijkl} \sim \mathcal{N}(\mu_{ijkl}, \sigma), \end{equation}\] where \(\mu_{ijkl}\) is the expected log-number of eggs laid at trap \(i\), site \(j\), week \(k\) and period \(l\).

Furthermore, we assume that the trap-specific log-density can be decomposed additively into a global average by site and period and week- and trap-specific deviations. \[\mu_{ijk} = \alpha_{jl} + \alpha_l + \alpha_i,\] where \(\alpha_k, \sim \mathcal{N}(0, \sigma_\text{W})\) and \(\alpha_i, \sim \mathcal{N}(0, \sigma_\text{T})\).

Note that in contrast to the model for hatching rates, this model uses a separate parameter for each of the three control sites. This was required here due to the relatively important difference in densities across sites. In addition, a week-specific varying intercept accounts for some of the temporal variation.

Figure 3.8: Global average egg densities in the Control (grey) and Release sites (orange) either in or out of the intervention period. Posterior distributions with 66 and 95 % quantile intervals.

Figure 3.9: Estimated densities in each trap within sites, during the intervention period. Posterior meadian with 66 and 95 % quantile intervals.

Figure 3.8 shows the estimated posterior global average densities for the control and release sites. The multi-modal posterior distributions at the control sites is due to differences across sites and weeks.

This shows that there is a clear difference between the average densities at the control and release sites outside of the intervention period. In contrast the difference is much smaller during intervention, primarily due to an increase in the density in one of the control sites.

Moreover, there the variability across traps is relatively much larger than the differences between the release and control sites as shown by Figure 3.9.

This supports the conclusion that the impact of the intervention on the egg density, is marginal, if any.

3.2 Trap attractiveness

Check whether some traps are located near hotspots, by looking at the spatial distribution of the number of eggs collected (hatched or not) over the full period.

For this analysis, we would ideally need the egg-survey field data disaggregated by trap. However, only the eggs that made it to the laboratory are grouped by trap.

Warning: Using the laboratory analyses of hatching rates can be potentially misleading due to confounding with the survival process.

Figure 3.9 gives already a good overview of trap attractiveness in terms of egg densities (i.e., average egg counts). In what follows we consider the share of egg counts within sites.

Figure 3.10: Eggs collected per trap, relative to total eggs collected on each site. The band represents the expected 95% confidence variation from a permutation procedure under homogeneous probability.

Figure 3.10 displays the relative egg catch (that made it to the lab) for each trap, with respect to the total eggs collected at each site, ordered by ranking.

The expected and 95% band of the variation of the rankings if the traps were exchangeable is displayed for reference, in order to help identifying which traps are located on more or less attractive spots, beyond random sampling variability.

Leveraging spatial coordinates of traps to interpret these catches in a map would help identifying hotspots that affect multiple traps in their vicinity.

3.3 Required number of ovitraps

In this section we determine the minimum number of ovitraps required to estimate the egg density with a given accuracy.

Egg density is the mean (i.e. expected) number of eggs collected in an ovitrap over a week. The individual trap counts represented in Figure 2.4 are the realised observations for all traps and weeks, for each site.

We assume that all traps are exchangeable (although this is contrary to the results in the section Trap attractiveness), and all weeks are exchangeable. However, we consider that the Control and Released sites might possibly have different egg densities.

In other words, that all of the observed trap counts can be seen as realisations from a random variable \(X_s\) such that \[ E[X_s] = \delta_s, \] where \(\delta_s\) is the egg density at site \(s\) (Control or Release).

The empirical distributions of these variables are as follows:

The empirical distribution of egg counts on the release site seems to be a bit shifted to the right, in correspondence with the results in table 3.2.

For each site \(s\), let \(x_{ij}\) be the eggs laid on trap \(i\) at week \(j\). Instead of modelling these values explicitly, we simply estimate the egg density \(\delta = E[X]\) as the average egg counts over traps and weeks. \[ \hat{\delta} = \frac1{n_k} \sum_{k = 1}^{n_k} x_{i[k]j[k]} \]

The standard error of the sample mean is \(\sigma / \sqrt{n}\), where \(\sigma^2 = V[X]\). From this, a Confidence Interval at level \(1 - \alpha\) is \(\hat{\delta} \pm z_{1 - \alpha/2}\, \sigma / \sqrt{n},\) where \(z_{1 - \alpha/2}\) is the \(1 - \alpha/2\) quantile of the Standard Normal distribution.

We define the precision \(e\) of the estimate as the relative semi-width of the CI with respect to the estimated value \[ e = z_{1 - \alpha/2}\, \sigma\, n^{-1/2} / \hat{\delta} \]

and we solve for \(n\), for a target precision \(e\) such that: \[ n = (z_{1 - \alpha/2}/e)^2\, (\sigma/ \hat{\delta})^2. \]

The only unknown is the variance \(\sigma^2\) of the trap counts. This can be conveniently modelled by Taylor’s power law (Taylor 1961), i.e. assuming a parametric power relationship between the mean and the variance: \[ \sigma^2 = a\, \delta^b, \] which gives \[ n = (z_{1 - \alpha/2}/e)^2\, a\, \delta^{b - 2}. \] The parameters \(a\) and \(b\) can be estimated empirically by grouping the observations \(x_{ij}\) by week, for instance, and regressing the sample means and variances in the log scale, since: \[ \log{\sigma^2} = \log{a} + b\, \log\delta. \]

Figure 3.11: Relationship between the partial sample means and variances of the weekly ovittrap counts in the log scale. The linear trend represents Taylor’s power law.

If the Taylor’s law is applicable, these values should approximately align. Indeed, Figure 3.12 shows that the relative standard error of the density estimates is fairly constant over time. Despite the larger densities and smaller variances observed in the Release site (Figure 3.11), the larger number of traps in Control sites account for a smaller overall standard error.

Figure 3.12: Temporal trend of the Relative Standard Error (SE/Mean) of the density estimates (average number of catches per trap).

In the current case, having monitored 30 ovitraps over 20 weeks on each site, we gathered 1870 observations over the 3 Control sites and 620 over the Release site.

This is sufficient for estimating the egg density with a precision of about 4% on the Control area and of 6% on the Release area (Table 3.3).

3.4 Impact on the wild population

Figure 3.14 shows that the number of adult males collected increased by an order of magnitude during the intervention period at the release site, whereas the captures in the control sites and the captures of females in all sites remained relatively constant over time.

Nevertheless, a decline of the female abundance is noticeable after the intervention period. However, due to the natural variation of the measure, a proper statistical model should be applied in order to quantify this decline and its uncertainty more precisely.

Figure 3.14: Temporal variation of the total number of adults collected by sex at the Control and Release sites.

Assuming the same probability of capture for sterile or wild males, this gives an idea of the size of the wild population. Which is instrumental for the calculation of the competitiveness of the sterile males (see next Section).

3.5 Mating competitiveness of sterile males

The competitiveness \(\gamma\) of the sterile male individuals is defined as their relative capacity to mate with a wild female, compared to a wild male. Thus, in a homogeneously mixed population with \(M_s\) sterile males and \(M_w\) wild males, the probability that a mating occurs with a sterile individual is \[ p_s = \frac{\gamma M_s}{M_w + \gamma M_s} = \frac{\gamma R_{sw}}{1 + \gamma R_{sw}}, \] where \(R_{sw} = M_s/M_w\) is the sterile-wild male ratio.

At a given sterile-wild male ratio \(R_{sw}\), we observe a fertility rate \(H_s\) in the field. Assuming a residual fertility rate \(H_{rs}\) for sterile males and a natural fertility rate \(H_w\) for wild males, the realised fertility rate \(H_s\) in the field is: \[\begin{equation} H_s = p_s H_{rs} + (1-p_s)H_w = \frac{\gamma R_{sw}}{1 + \gamma R_{sw}} H_{rs} + \frac{1}{1 + \gamma R_{sw}} H_w. \end{equation}\] Solving the above equation for \(\gamma\) we obtain the Fried index, which estimates the competitiveness based on observed fertility rates and the sterile-wild male ratio: \[\begin{equation} F = \frac{H_w – H_s}{H_s-H_{rs}} \bigg/ R_{sw} \end{equation}\] where \(H_w\) is the percentage egg hatch in the control area (i.e. the natural fertility); \(H_s\) is the percentage egg hatch in the release area (observed fertility under a sterile-wild male ratio \(R_{sw}\)) and \(H_{rs}\) is the residual fertility of sterile males. This requires the estimation of:

• The sterile-wild males ratio \(R_{sw}\).

  Since we don’t have marks for sterile males, we need to derive this from the observed difference of sex-ratios in the release and control sites during the intervention period (with a 7-day lag, Fig. 3.14).

  Let \(\rho_0 = M/F\) and \(\rho = (M_s + M_w) / F_w)\) be the sex ratios in the control and sit areas respectively. Assuming that the wild sex-ratio is similar in both sites, \(\rho_0 = M/F \approx M_w / F_w\), we have

  \[ R_{sw} = M_s/M_w = \rho\, F_w / M_w - 1 = \rho / \rho_0 - 1. \]

• The natural fertility \(H_w\). Estimated from egg-hatching rates in the control area.

• The observed fertility in the release area \(H_s\). Estimated from egg-hatching rates in the SIT area during the intervention period (with a 14-day lag).

• The residual fertility of sterile males \(H_{rs}\). This depends on the level of radiation used for sterilisation, and the influx of females from beyond the SIT area. This cannot be estimated from SIT data and needs to be fixed by the user. Using \(H_{rs} = 0\) reduces to the so-called CIS index. Here we use an estimate of \(H_{rs} = 0.01\) with a Gaussian variation with a SD of 0.5 in the logit scale.

Using captures from the intervention period only, we obtained the following stratified nonparametric bootstrap estimate of the sampling distribution of \(R_{sw}\)

Figure 3.15: Bootstrap estimate of the competitiveness. The orange big point marks the estimate yield by the observed data, whereas the black point-range shows the bootstrap mean and 95% percentile interval.

These parameters yield an estimated competitiveness of \[ \hat\gamma = 0.08\; (0.05, 0.12) \]

In other words, between 9 and 22 sterile males would be required to match the mating performance of a wild individual.

Nevertheless, note that experts (Bouyer and Vreysen 2020) argue that a sterile-wild male ratio exceeding 4 biases the results and the competitiveness is no longer reliable. It must be noted that a competitiveness below 0.2 is considered too low to apply SIT efficiently (Bouyer and Vreysen 2020).

5.1 On the required number of ovitraps

I’ve followed what seems to be the standard approach to this question (see e.g. Carrieri et al. (2011)). However, I have some reservations about it. Firstly, the target measure (i.e., egg density, or expected number of eggs per trap and week, across traps and weeks) does not seem a particularly interesting quantity to base sample size decisions upon. It would make much more sense to ensure sufficient sample size to identify changes in egg density during the intervention, or changes in hatching rates. This could be done by computing egg densities for different periods (which would multiply the required sample size) or better, by explicitly modelling trends in egg counts. Secondly, the estimate is very sensitive to the estimation of Taylor’s power law, which can be quite inaccurate. For instance, in Carrieri et al. (2011), estimated sample sizes varied wildly (by factors between 1/4 and 4) from 2007 to 2008 in different regions. Thirdly, modelling egg counts would be so much more informative. Explicitly accounting for the variation across traps and weeks would reduce residual noise and help identifying changes in trends of egg density due to the intervention, giving more power to the procedure and requiring smaller sample sizes as a consequence. I have introduced this line of work with the statistical models for hatching rates and egg densities, which account for the variation across traps (i.e. trap attractiveness) and also for the weekly variation in the latter case.

5.2 Needs in terms of sterile males per hectare per week

Addressing this question would require first, to define some target measure, and second, to model how these results scale with the number and frequency of releases. I do not have these elements at the moment. This is to be further discussed.