As the distribution of survival times may undergo complex changes across a range of temperatures we also chose to construct a non-parametric model that would not share the same restrictions as the aforementioned parametric models. For this we chose regression spline generalised additive models (GAMs) which apply a smoothing variable to the explanatory variables in order to model the response variable [58 ]. This method has the advantage of being able to model unknown and non-linear effects of covariates and thus elucidate the potentially complex effect of temperature on adult mosquito survival.
To evaluate the improvement of using GAMs over the parametric alternatives, we fitted parametric and GAM models to each laboratory experiment and calculated the difference in AIC between parametric and non-parametric models across all experiments.
A second GAM was then formulated to use the data from all experiments in one model to recreate the relationship between survival, time and temperature. The GAM was formulated as follows:
Sij = number of mosquitoes surviving at observation i in experiment jNij = number of mosquitoes at start of time step at i, jPij = survival probability for a mosquito at i, jf() = smooth term
Di = day of observation iTi = temperature of observation iϵj = random error term for experiment jϵd = random error term for mosquito diet dθj2 = variance across experiments
θd2 = variance across mosquito diets
Smoothing parameters were selected by restricted maximum likelihood with a data-driven basis dimension choice of kD = 8 and 5 and kT = 5 and 5 for Ae. aegypti and Ae. albopictus respectively [59 ]. Confidence intervals for the interquartile range of predictions were obtained by bootstrapping with 200 repeats, each the size of the original dataset. This model was fit using D ≥ 1 to be consistent with the experimental observations that record mortality. Extrapolated model predictions for 0 ≤ D < 1 were scaled proportionally to ensure 100% survival at D = 0. All GAMs were implemented using the “mgcv” package in R [60 (link)].
For the model to fit biologically appropriate responses, additional data defining the limits of prediction were required. Observations from Christophers [18 ], suggest 4°C and 42-43°C as suitable minimum and maximum critical temperatures at which survival of Ae. aegypti is minimal (<24 hours). Similar observations for Ae. albopictus suggest values between -5°C and 40–40.6°C [61 ,62 (link)]. To constrain mortality in the model, all non right censored experimental observations were extended to 120 days at 0% survival and a maximum lifetime of 120 days was imposed at all temperatures, the maximum longevity observed in our dataset. Furthermore, to produce meaningful estimates of longevity, survival of less than 0.1% of the initial mosquito population was considered sufficient to indicate complete mortality.
To evaluate the improvement of using GAMs over the parametric alternatives, we fitted parametric and GAM models to each laboratory experiment and calculated the difference in AIC between parametric and non-parametric models across all experiments.
A second GAM was then formulated to use the data from all experiments in one model to recreate the relationship between survival, time and temperature. The GAM was formulated as follows:
Sij = number of mosquitoes surviving at observation i in experiment jNij = number of mosquitoes at start of time step at i, jPij = survival probability for a mosquito at i, jf() = smooth term
Di = day of observation iTi = temperature of observation iϵj = random error term for experiment jϵd = random error term for mosquito diet dθj2 = variance across experiments
θd2 = variance across mosquito diets
Smoothing parameters were selected by restricted maximum likelihood with a data-driven basis dimension choice of kD = 8 and 5 and kT = 5 and 5 for Ae. aegypti and Ae. albopictus respectively [59 ]. Confidence intervals for the interquartile range of predictions were obtained by bootstrapping with 200 repeats, each the size of the original dataset. This model was fit using D ≥ 1 to be consistent with the experimental observations that record mortality. Extrapolated model predictions for 0 ≤ D < 1 were scaled proportionally to ensure 100% survival at D = 0. All GAMs were implemented using the “mgcv” package in R [60 (link)].
For the model to fit biologically appropriate responses, additional data defining the limits of prediction were required. Observations from Christophers [18 ], suggest 4°C and 42-43°C as suitable minimum and maximum critical temperatures at which survival of Ae. aegypti is minimal (<24 hours). Similar observations for Ae. albopictus suggest values between -5°C and 40–40.6°C [61 ,62 (link)]. To constrain mortality in the model, all non right censored experimental observations were extended to 120 days at 0% survival and a maximum lifetime of 120 days was imposed at all temperatures, the maximum longevity observed in our dataset. Furthermore, to produce meaningful estimates of longevity, survival of less than 0.1% of the initial mosquito population was considered sufficient to indicate complete mortality.
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