^{1}

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Conceived and designed the experiments: ED SG. Analyzed the data: CB ED SG. Contributed reagents/materials/analysis tools: ED. Wrote the paper: CB ED SG. Performed the modeling: CB SG.

The authors have declared that no competing interests exist.

Chagas disease is the most important vector-borne disease in Latin America. Regional initiatives based on residual insecticide spraying have successfully controlled domiciliated vectors in many regions. Non-domiciliated vectors remain responsible for a significant transmission risk, and their control is now a key challenge for disease control.

A mathematical model was developed to predict the temporal variations in abundance of non-domiciliated vectors inside houses. Demographic parameters were estimated by fitting the model to two years of field data from the Yucatan peninsula, Mexico. The predictive value of the model was tested on an independent data set before simulations examined the efficacy of control strategies based on residual insecticide spraying, insect screens, and bednets. The model accurately fitted and predicted field data in the absence and presence of insecticide spraying. Pyrethroid spraying was found effective when 50 mg/m^{2} were applied yearly within a two-month period matching the immigration season. The >80% reduction in bug abundance was not improved by larger doses or more frequent interventions, and it decreased drastically for different timing and lower frequencies of intervention. Alternatively, the use of insect screens consistently reduced bug abundance proportionally to the reduction of the vector immigration rate.

Control of non-domiciliated vectors can hardly be achieved by insecticide spraying, because it would require yearly application and an accurate understanding of the temporal pattern of immigration. Insect screens appear to offer an effective and sustainable alternative, which may be part of multi-disease interventions for the integrated control of neglected vector-borne diseases.

Chagas disease is the most important vector-borne disease in Latin America. Residual insecticide spraying has been used successfully for the elimination of domestic vectors in many regions. However, some vectors of non-domestic origin are able to invade houses, and they are now a key challenge for further disease control. We developed a mathematical model to predict the temporal variations in abundance of non-domiciliated vectors inside houses, based on triatomine demographic parameters. The reliability of the predictions was demonstrated by comparing these with different sets of insect collection data from the Yucatan peninsula, Mexico. We then simulated vector control strategies based on insecticide spraying, insect, screens and bednets to evaluate their efficacy at reducing triatomine abundance in the houses. An optimum reduction in bug abundance by at least 80% could be obtained by insecticide application only when doses of at least 50 mg/m^{2} were applied every year within a two-month period matching the house invasion season by bugs. Alternatively, the use of insect screens consistently reduced bug abundance in the houses and offers a sustainable alternative. Such screens may be part of novel interventions for the integrated control of various vector-borne diseases.

Chagas disease is a major vector-borne parasitic disease in Latin America, with 9.8 to 11 million infected people, 60 million at risk of infection

However, it has become increasingly clear that several triatomine species do not establish permanent domestic colonies, but can occasionally infest domestic habitats by immigration from peridomestic and/or sylvatic habitats. These species include

Extensive field collections of

The control of house infestation by such non-domiciliated triatomine vectors is identified as a major problem and one of the new challenges for Chagas disease control since conventional spraying control strategies may be of limited efficacy in these conditions

In the present contribution, we use a combination of field and modelling studies to evaluate the efficacy of several strategies for the control of seasonal infestation by non-domiciliated triatomine populations. We took advantage of one of the best documented case of non-domiciliated triatomine vector; the populations of

We aimed to construct a model able 1) to reproduce and predict the temporal variations of vector abundance in the absence of control, and 2) to account for various control strategies. We expanded a previous population dynamics model

Data on the dynamics of house infestation by triatomines in the absence of vector control interventions were collected over 3 years of field studies, from October 1999 to December 2001 and from January to December 2003 ^{2} of cyfluthrin in November 2000, and monitored every 2 weeks for up to 9 months to detect re-infestation using a combination of manual searches, mouse traps and household collections

We modelled the dynamics of a non-domiciliated population of _{1}(n), n_{2}(n), n_{3}(n), n_{A}(n)) included the number of females in three immature age classes and the number of adult females at the n^{th} time step and M(n) = (m_{1}(n), m_{2}(n), m_{3}(n), m_{A}(n)) the number of immigrants of the same categories (Note that we use index n instead of t as in Gourbière et al. _{I} and S_{A} are survival of immature and adults (probabilities per trimester), and F is female fecundity (female immature offspring per female-trimester). Because only adults immigrate into houses and because this only occurs between April and June

Because the time unit desired to describe the control strategies in a flexible way is much shorter than the three-month time step previously selected, we adapted the above model to account for a daily description of the population dynamics, while keeping the three-month time step of the model. We divided each time step into T = 90 time units (t) and considered that immigrating individuals survive and reproduce proportionally to the time spent in the domestic habitat since their arrival at time τ. The population dynamics model is then divided into two parts, one describing the demography of individuals present in the domestic habitat since the beginning of the time step, and one accounting for the demography of individuals arriving at each time unit of the time step:_{I}(n,τ), S_{A}(n,τ) and fecundity F(n,τ) defined over the time T-τ spent in the domestic habitat within the n^{th} time step. Similarly, M(n,τ) includes the number of immigrants at time τ of the n^{th} time step. We then used Equation 3 to simulate the vector population dynamics with or without control by changing the definition of parameters S_{I}(n,τ), S_{A}(n,τ), F(n,τ) and M(n,τ) according to the control strategies to be considered and the assumptions about their effects on vector demography. Finally, bug collection over the time steps was incorporated by removing a percentage p of individuals at the end of each day. The removed insects were summed over the duration of the time step to obtain a number of collected bugs/house-trimester, which is the model outcome that we compared to field observations. The best fits were obtained for p values 1–10%, with very limited changes in the quality of predictions over this range. For consistency, we thus display all our results for p = 5%.

While subdividing the time step into smaller time units, we assumed that within the time step 1) immature and adult survival are constant over time 2) adults immigrate at a constant rate, and 3) adults lay eggs regularly within the time interval left after they immigrated into the house. All the demographic parameters, S_{I}(n,τ) S_{A}(n,τ), F(n,τ) and M(n,τ), were specified according to those assumptions (See

We estimated the demographic parameters by fitting the model with no control to field data from two villages during years 2000 and 2003. The data of both years and of all the immature stages were pooled to provide a reliable estimate of the yearly variations in vector abundance per house. The expected variations of the number of immature and adult individuals were calculated for a large range of values of each parameter. We calculated the sum of the square of the differences between observed and expected numbers of immatures and adults. The set of parameters providing the smallest sum of squares was retained and a Pearson correlation coefficient between observed and predicted bug abundance per house was used to measure the quality of the fit.

The ability of the model to predict bug abundance was measured by correlating the observed and predicted numbers of bugs. The test was performed on an independent data set coming from field studies carried out during years 2000, 2001 and 2003 in three villages different from those used to fit the model.

Insecticide spraying was considered to reduce immature and adults survival according to the dose of insecticide present in the house. This effect on survival probabilities was described by a classical sigmoid dose-response relationship. The insecticide dose present was evaluated daily according to an exponential decay of the active ingredient starting on the day of application. In absence of quantitative data on the potential interaction between these two sources of mortalities, we assumed that they act independently and thus combine them multiplicatively to define the overall survival probability. Fecundity of adults was also decreased as a result of the impact of insecticide on immature and adult survival. All the demographic parameters, S_{I}(n,τ), S_{A}(n,τ) and F(n,τ), were then modified to account for insecticide spraying (See

The model was fitted to field bug collections from a pilot insecticide trial performed in 2001 to estimate insecticide half-life (_{50} = 32.2 mg/m^{2} and a LD_{90} = 182.4 mg/m^{2} (

Parameter description | Estimate | Other tested values |

Immature survival probability over 3 months (S_{I}) |
1 |
0–0.01 |

Adult survival probability over 3 months (S_{A}) |
0.224 |
0.21 |

Fecundity of females over 3 months (F) | 0.434 |
0.29 |

Number of adult immigrating/year (M) | 21.1 |
1–25 |

Half-life of the insecticide in days (t_{1/2}) |
38 |
15 days to 6 months |

50% lethal dose in mg/m^{2} (LD_{50}) |
32.2 |
15–100 mg/m^{2} |

90% lethal dose in mg/m^{2} (LD_{90}) |
182.4 |
60–190 mg/m^{2} |

Dose sprayed in mg/m^{2} (Q) |
50 |
10–250 mg/m^{2} |

Trimester of first spraying (n_{fs}) |
4 |
1 to 4 (by 1) |

Day of first spraying (t_{ins}) |
45 |
0 to 60 (by 30) |

Number of trimesters between two interventions (P) | none |
2 to 12 (by 2) |

Reduction in immigration due to insect screens (r) | none | 0 to 1 (by 0.1) |

Reduction in survival and fecundity due to bednets (s) | none | 0 to 1 (by 0.1) |

Estimated from the fit to field data.

Values used to reproduce a unique spray on November 15^{th} as in the field trial

Values estimated in Gourbière et al.

For further simulation of interventions, we evaluated the effect of the spraying date of a single application by testing each month of the year (_{50} and LD_{90} within a range of possible values (

Since the immigration rate has been shown to be the overwhelming factor in explaining non-domiciliated vector population dynamics, we varied this parameter from 1 to 25 immigrants per year according to estimates obtained from various methods _{I} = 0.01/trimester, S_{A} = 0.21/trimester, F = 0.29 female offspring/female-trimester) correspond to a sink population, with a growth rate equal to λ = 0.20. As expected, this sensitivity analysis resulted in quantitative changes in the abundance of insects. However, it did not alter any of our conclusions about the relative efficacy of the various strategies of spraying. We then present only the results obtained for the demographic parameter values estimated in this contribution (

Door and window insect screens were considered as a physical barrier impeding the arrival of some of the immigrant vectors into the domestic habitat. Bednets were assumed to limit blood intake of the triatomines, leading to a decrease in survival and fecundity of the bugs. We thus modelled insect screens by multiplying the immigration M(n,τ) by a factor of bug exclusion r and bednets by weighting the fecundity F(n,τ) and survival S_{I}(n,τ), S_{A}(n,τ) by a factor of blood intake reduction s.

Because no field data are available to estimate the reduction in triatomine immigration which may be expected by insect screens or the magnitude of the reduction of survival and fecundity due to bednets, we tested a complete range of reduction by varying r and s from 0 to 100%. The efficacy of control is expressed as percent reduction in bug number/house for one year following installation of screens or bednets.

We also varied the demographic rates as described above. Again, because there were only quantitative changes in the abundance of vectors, we present only the results obtained for the demographic parameter values estimated in this contribution (

The model's demographic parameters were first fitted to two years of field data from two villages in the absence of vector control interventions. The optimal parameter values were M = 21.1 immigrants/year, S_{I} = 1/trimester, S_{A} = 0.434/trimester, F = 0.224 female offsprings/female-trimester, and these provided a very good fit of the model to field data for the total bug population (R^{2} = 0.953, _{I} was found to be 7 to 8 orders of magnitude lower than the effect of M (with Sobol standardized indices equal to 0.000005 and 0.89, respectively) ^{2} = 0.891, ^{2} = 0.985,

(A) Fit of the model with no control actions. (B) Test of the predictive power of the fitted model. (C) Fit of the model with insecticide spraying. Field data are given with a 95% confidence interval (shaded area).

Once we determined the model's parameters that best fitted field data, we predicted domestic bug abundance as a function of time after various control interventions. We first explored the effect of the timing of insecticide spraying during the year. The effects of a single insecticide spraying (50 mg/m^{2} at various dates) on bug abundance in the houses was only observed for a few months, and was followed by a rapid return to a normal cycle of infestation as soon as a new season of infestation occurred (

(A–D) Single spray. (A) Variations in bug abundance. (B) Efficacy as a function of the date of spraying. (C) Variations in bug abundance with application of various insecticide dose. (D) Efficacy as a function of insecticide dose. (E,F) Repeated spraying. (E) Variations in bug abundance with repeated spraying. (F) Efficacy as a function of time interval between spraying.

Although a standard cyfluthrin dose of 50 mg/m^{2} is commonly used for triatomine control ^{2}) only provided a limited improvement in the reduction of bug abundance compared with the standard dose, and was not enough to sustain triatomine control for more than one seasonal infestation cycle (^{2} thus provided a nearly optimal vector control. Nonetheless, an insecticide dose as low as 10 mg/m^{2} sprayed at the beginning of the infestation period was still able to reduce bug population by over 50% for a year (

Because a single insecticide spraying did not allow to achieve a sustainable vector control, we then evaluated the effects of repeated spraying and determined the optimal frequency of application. Our simulations clearly indicated that spraying once a year, just before the start of house invasion by adult triatomines, was the best strategy (

Although our insecticide half-life estimate was in good agreement with expected values, we evaluated the robustness of the results using various half-life values in simulations where 50 mg/m^{2} are applied with various frequency at the optimal timing (April 1^{st}). As expected, increasing insecticide half-life allowed for a more sustained vector control, leading to about 80% reduction in bug abundance by spraying every two years instead of one. However, a half-life of over 4 months was required for such a frequency of spraying to be effective (_{50} of the insecticide, provided the spraying dose is adjusted accordingly, regardless of the level of immigration considered (

(A) Efficacy of repeated insecticide spraying as a function of the spraying interval and the insecticide half-life (indicated on each curve). (B) Efficacy of a yearly insecticide spraying. (C) Efficacy of spraying every two years.

Given the importance of dispersal in triatomine population dynamics inside houses, we evaluated the effect of the presence of insect screens on doors and windows by reducing the immigration of bugs inside houses. Reduction in triatomine abundance in the houses was immediate following screens implementation, directly proportional to the reduction in bug immigration rate, and sustained for as long as the screens were maintained (

(A) Variations in bug abundance with insect screens (gray shaded area) reducing bug immigration by 10 (top), 50 (middle), and 90% (bottom). (B,C) Efficacy of insect screens and bednets as a function of the percent reduction in bug immigration and bug feeding, respectively.

The integrative studies performed in the Yucatan peninsula provide a rare opportunity to develop mathematical models rooted in several years of field data. It was used here for the first time in an attempt at optimizing control strategies for non-domiciliated vectors of Chagas disease. The quality of the fit and of the predictive value of our deterministic model allowed to produce reliable simulations of a variety of control interventions. Also, while stochastic variations in the number of immigrants, which ultimately determine the number of individuals present in a given house, were not considered in our model, these are unlikely to qualitatively affect our results as indicated by our sensitivity analysis of immigrant numbers.

Simulations aimed at optimizing insecticide spraying clearly indicated that efficacy depended dramatically on the timing and frequency of spraying, both of which had to match closely the immigration season. This implies that a good understanding of the temporal pattern of immigration, which may differ between non-domiciliated triatomine species

Our results clearly indicate that none of these insecticide spraying interventions would be sustainable, since as soon as they are interrupted, re-infestation by new immigrant bugs occurs during the next season, implying large costs associated with repeated spraying. Some authors even suggested that control of non-domiciliated triatomines should not be considered, and that resources should rather be devoted to patient detection and care

Our simulations of insect screen effects indicate that an effective and sustained control can be achieved when a significant reduction of bug immigration is obtained. While it is difficult to estimate the possible efficacy of such screens in the field, an exclusion of over 90% of other insects has been observed with some greenhouse screens

On the other hand, we found that bednets had little effect on bug abundance, possibly because triatomine reproductive output inside houses is already low in the absence of interventions

In conclusion, our study illustrates well the usefulness of coupling modelling and field studies to design and optimize effective control interventions and develop evidence-based public health policies, as previously done for the control of other diseases

Translation of the Abstract into Spanish by Eric Dumonteil

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Translation of the Author Summary into Spanish by Corentin Barbu

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Translation of the Author Summary into Portuguese by Sébastien Gourbière

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