Table 1.
Definition and description of terms and acronyms.
Fig 1.
Diagrammatic Representation of Smooth and Truncation Selection. Left panel: Probabilistic Selection (implemented in the “polysmooth” model).
Probabilistic Selection [14] defines the survival probability of an individual mosquito to be a function of its own polygenic resistance score (PRS). Note, while it looks like only a small proportion of more resistant individuals survives, this only appears to be the case because there were initially only a smaller proportion of higher resistance individuals. Right panel: Truncation Selection (implemented in the “polytruncate” model). Truncation selection occurs when only mosquitoes with a resistance level above a certain threshold survive insecticide contact. In our model this means that if the mean survival probability of the population was calculated as 10%, the surviving individuals constitute the top 10% of the population. This threshold value is dependent on insecticide efficacy and the level of resistance in the population. The black dashed line indicates the threshold for selection (calculated in Equation 2b(i)) and is therefore the proportion of individuals expected to survive based on the mean PRS of the population. The individuals in red section of the distribution therefore all die, and the individuals in the blue section of the distribution all survive.
Table 2.
Summary of model methodology and key assumptions. This table summarises and highlights the key points and assumptions for each model development methodology section. Technical details of their implementation and evaluation is given in Methods section 4.
Table 3.
Summary of the capabilities of the dynamic model “polysmooth” and “polytruncate” compared to the previously published “polyres” model.
Fig 2.
The Polygenic Resistance Score (PRS) relationship with Bioassay Survival and Field Survival.
Panel A shows the Polygenic Resistance Score (PRS) on a scale of 0 to 10000 (Equation 2a), the zoomed inset shows the same data on a scale from 0 and 1000, values especially important in the evaluation of novel insecticides. The orange line indicates 10% bioassay survival (PRS = 100) (our default for insecticidal withdrawal), when = 900 (green line). Panel B is the relationship between bioassay survival and field survival (Equation 2b), and Panel C is the relationship between the polygenic resistance score and field survival. These are identical to the relationships used in [11].
Fig 3.
An example insecticide decay profile for an ITN.
In this example the insecticide is deployed at its manufacturers recommended concentration with an initial deployed insecticide efficacy of =1. It is a two-step function. Step 1 describes the first two years (~20 mosquito generations) as the insecticide decays slowly at its base decay rate (blue line). Step 2 occurs after the decay threshold (
) has been reached (vertical yellow line) and the insecticide efficacy decays rapidly (red line). In this example the base decay rate
was 0.015, the rapid decay rate
was 0.08 and the decay threshold
was 20 generations (~2 years). These values are user-defined inputs and can be varied allowing insecticides to decay faster/slower or sooner/later. Setting
and
to 0 prevents insecticide decay from occurring (a key assumption in previous models). Insecticides in mixture can have different decay properties. Setting decay rates to be identical before and after the threshold eliminates the second phase and converts the decay profile to a single dynamic if required.
Fig 4.
Impact of insecticide efficacy on field survival at given polygenic resistance scores.
This plot displays the field survival of mosquitoes which encounter an insecticide of a particular efficacy, calculated using Equation 2b(i). The colour of the line corresponds to the polygenic resistance score. Blue = 0, Green = 10, Red = 100, Orange = 900, Purple = 8100; corresponding to 0, 2, 10, 50 and 80% bioassay survivals. The black vertical line indicates the insecticide is at its manufacturers recommended concentration for 100% efficacy, which is 1. The red background indicates the insecticide is above the manufactures recommended dosing/concentration as may occur, for example, with over-spraying when performing indoor residual spraying.
Fig 5.
Compartmental diagram of truncation selection process for a single generation.
The “polytruncate” model implements truncation selection, and determines which equations are used to calculate each stage of the selection process. Note this process is calculated separately for males and females. Panel 1. The mosquito population emerges, with a PRS that is Normally distributed with mean (black line) and standard deviation
. At emergence there are a total of
individuals. Panel 2. A proportion (males:
and females:
) avoids the insecticide and insecticide selection; the mean is therefore unchanged at
, there will be
of these individuals (Equation 4d). Panel 3. A proportion do encounter the insecticide (males:
and females:
) (Equation 5d). Panel 4. These individuals are selected by truncation selection (Equation 5c). Individuals with a PRS less than the defined threshold (red line) are killed (red area). Individuals with a PRS above the threshold survive (blue area). Panel 5. Only the most resistant individuals in the population will have survived the insecticide encounter, these individuals have a mean of
(purple line) (Equation 5c), and there are
of these individuals (Equation 5d). Panel 6. The unexposed group (Panel 2) and the exposed survivors (Panel 5) form the final breeding parental population. They have a mean of
(orange line) (Equation 4b), which would be expected to be higher than the original population mean (black line). The insecticide selection differential is the difference between the orange and black lines (Equation 3b).
Fig 6.
Compartmental diagram of a probabilistic selection process for a single generation.
This is implemented in “polysmooth”. Panels 1-3 and 6 have same description as Fig 5. Where the two models diverge is the calculation of who survives/dies insecticide exposure. Panel 4: The probability of surviving insecticide encounter depends on an individual’s polygenic resistance score i.e., individuals with a higher polygenic resistance score have a higher probability Panel 5. The mosquitoes surviving exposure will have a higher mean polygenic resistance score (purple line (Equation 6c) and there are individuals (Equation 6b).
Fig 7.
Conceptualisation of insecticide selection over multiple gonotrophic cycles.
1): The number of females in each subsequent cycle decreases due to insecticide killing (e.g., ) and natural survival (
). This results in fewer eggs being laid in subsequent cycles. 2): As a result of insecticide selection, surviving females have a higher mean PRS than the previous cycle as only resistant individuals would be able to survive multiple insecticide encounters. 3): An increasing mean PRS of females gives an increased female insecticide selection differential (height of blue bars). The male selection differential (height of red bars) remains constant as female mosquitoes mate only once. 4): As the female selection differential increases, the response increases. Eggs laid by females in subsequent cycles produce more resistant individuals. The updated mean PRS for the next generation is the mean PRS of the eggs laid over all the cycles.
Fig 8.
Compartmental diagram of the selection process over multiple gonotrophic cycles for IRM strategies.
The framework of the multiple cycle model allows for multiple different strategies to be evaluated: Red indicates only a single insecticide is deployed, orange is a mixture, blue is a micro-mosaic, and green is a combination of ITN and IRS. These different strategies apply differing levels of selection producing different mean PRS values post insecticide selection and different number of females surviving each cycle.
Fig 9.
Demonstration of the “polysmooth” model: assessing IRM strategies in the presence/absence of cross resistance, insecticide and multiple gonotrophic cycles.
Plots are stratified by insecticide resistance management strategy (columns), and model features (rows). The green line is insecticide and the purple line insecticide
, with their ranking (shown in the numbered boxes: 1 is best and 5 is worst) based on mean bioassay survival across the five IRMs in each row. The yellow boxes contain rankings based on the mean of both insecticides.
Fig 10.
Comparison of Sequences, Rotations and Mixtures for Polysmooth.
Left column: Sequences versus Rotations; Centre column: Mixtures vs Rotations; Right column: Mixtures vs Sequences. Colours indicate which strategy had the longer operational lifespan in the direct comparison. Red = rotations longer. Blue = Sequences longer. Purple = Full-Dose Mixtures longer. Draws were excluded from the histogram, but reported in the grey boxes. The number of draws is that obtained from the 5000 parameter sets summarised in the panel. The rows indicate the amount of cross resistance between the two insecticides. Simulations were terminated when no insecticides were available for deployment (because resistance had evolved to them all), or the 500-generation (~50 years) cap was reached. Simulation duration (in years) therefore measures operational lifespan of the IRM strategy. Alternating high and low bars are because simulations with two insecticides are more likely to terminate in even years, thus a difference as a multiple of two years is more likely than as a multiple of 1 year.
Fig 11.
Comparison of Sequences, Rotations and Mixtures for Polytruncate.
As Fig 10, but with results obtained from “polytruncate”.
Fig 12.
Comparing Outcomes between Polysmooth and Polytruncate.
The data shows the extent of agreement for the strategy outcome between the “polysmooth” and “polytruncate” models for the same parameter inputs. Colours indicate how the strategies performed in both models. Dark blue = both models indicate the same strategy performed best (“full agreement”). Light blue = both strategies draw in both models (“agreement”). Light red = both strategies drew in one model, but one strategy performed better than the other in the other model (“partial disagreement”). Dark red = Models diverge as to which strategy performed best (“full disagreement”).
Fig 13.
The impact of including multiple gonotrophic cycles and natural survival on the performance of deploying insecticides
and
in rotation versus micro-mosaics. Values above zero indicate the rotation strategy performed best, and values below zero indicates the micro-mosaic strategy performed best. Left Plot: The model is run only for a single cycle. Centre Plot: The model is run allow for multiple gonotrophic cycles (5 cycles) but natural survival is not included. Right plot: The model is run with multiple cycles and allowing for natural mortality (gonotrophic length = 3 days, natural daily survival = 0.8).
Fig 14.
How mosquito age distributions change given insecticide efficacy and resistance.
The grey bars indicate the age profile if no intervention had been deployed. The colored bars represent age distribution at different levels of IR quantified as bioassay survival. The value at the top of each panel is insecticide efficacy.
Fig 15.
Changes in mosquito age distributions given resistance and insecticide efficacy for mixtures.
The grey shaded area is the age profile if the intervention was not deployed. The blue bars are the age profile given the values of insecticide efficacies and resistance specified at the top of each plot. A red vertical dashed line is plotted at 10 days of age which is the minimum age mosquitoes can become infectious.
Fig 16.
Age profile comparisons depending on whether IRM is deployed as micro-mosaics, half-dose mixtures and full-dose mixtures.
The colored bars are the age distributions that occur if there is no intervention (grey), if full-dose mixtures are deployed (purple), if micro-mosaics are deployed (green) or if half-dose mixtures (HD50%_HD50%) are deployed (orange).
Fig 17.
Impact of the standard deviation of the PRS on time to resistance.
Each grey line represents a single parameter set, with a total 30 parameter sets generated from sampling uniform distributions of heritability, female insecticide exposure and male insecticide exposure), with the time taken to reach 10% bioassay survival. The black line is the mean value for these simulations.