Goal

• assess the effect of boosted SIT on the standardized apparent density (SAD) of an Ae. aegypti population

  • H₀: constant SAD after starting boosted SIT

  • H₁: decreasing trend in SAD, related to boosted SIT

Challenges

• High variability in count data

• two study sites (Payet and Langevin), with ecological differences

• Boosted SIT only implemented in Payet, during a short time period (3 months)

• Strong seasonal - as well as trap-to-trap, variations

Strategy

• Choice of the outcome: apparent density of Ae. aegypti, standardized with respect to time period to eliminate other time trend than boosted SIT effect

• Choice of a model to assess the boosted SIT effect, while providing useful information on spatial variation sources: we used a spatial (mixed-effect) Poisson model (Moraga 2019, chap. 6)

• Bayesian modelling framework, with an integrated nested Laplace approximation (INLA) approach: Bayesian process approximated by analytical solutions, implemented in INLA (Rue et al. 2017)

• For period \(i \in 1,..., I\\,\,\) (\(I=8\,\,\)), the response \(y_i\,\,\) is a length-\(J\,\,\) vector of standardized apparent densities (SAD), with J,,$ the number of traps in the design (\(J=12\,\,\)). The \(J\,\,\) components of \(y_i\,\,\) are \(a_{i,j} / E_i\), i.e., the apparent density divided by the expected density \(E_i\); i.e., the average density during period \(i\,\,\) for the \(J\,\,\) traps

• We used a spatial Poisson model

\[y_i \sim \mathcal{P}(E_i \times m_i)\,\,\text{and}\,\, \log(m_i) = \eta_i = b \times x_i + u_j\]

• \(m_i\,\,\) the expectation of \(y_i\), \(x_i\,\,\) the number of exposure months to boosted SIT:1, 2; or 3 for traps in Payet from periods 6 to 8, and 0 elsewhere (traps in Langevin at any period, and in Payet before the 6th period); \(b\,\,\) is the fixed-effect coefficient for \(x_i\)

• \(u_j\,\,\) a \(J\)-length vector of random-effect coefficients (one per trap), representing the spatial variation around \(b\,\,\), with \(\sum_{j=1}^J u_j = 0\). We used a modified version of the Besag-York-Mollié model Riebler et al. (2016): \(u_j\,\,\) is split into (i) a spatially-structured component with a conditional auto-regressive (CAR) distribution, and (ii) a “white noise” component. A mixing parameter \(\Phi\) - ranging from 0 to 1, gives the CAR proportion in \(u_j\). Interpretable parameters and meaningful priors for their precision are provided by the modified version

• Deviance information criterion (DIC) used to compare four assumptions about boosted SIT effect on SAD

  • a constant-slope model (presented above)

  • a constant-effect model: initial drop, then constant SAD

  • a three-slope model: one for each period

  • With Mixed-effect models, the estimation of population means must account for both fixed and random effects. Thus, for the one-slope model

\[ E(y_i) = m_i = \frac{1}{J} \left[ \sum_{j=1}^J \exp(b \times x_i + u_j) \right]\]

• \(m_1\) is the survival at the end of period i. An estimate of duration n of boosted SIT to reach a target residual population is

\[n \geq \frac{log(\alpha)}{\log(m_1)}\]

• Finally, we mapped the bym2 random effect and its two components to identify traps with outstanding effects