# Decision Support System for Fitting and Mapping Nonlinear Functions with Application to Insect Pest Management in the Biological Control Context

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Input Dataset

#### 2.1.1. Experimental Data

#### 2.1.2. Climate Data

#### 2.2. Model Fitting Description

_{0}; then, x is adjusted by ∆ only for downhill steps verifying (J

^{T}J + λI) ∆ = J

^{T}r, where J is the Jacobian matrix of derivatives of the residuals for the parameters, λ is the damping parameter between the two steps, and r is the residual vector. A set of equations used in the study that can be selected by the user are presented in the Supplementary Materials.

#### 2.3. Features of the DSS

#### 2.4. Software Design and Architecture

#### 2.4.1. RComponent

#### 2.4.2. Eclipse RCP

#### 2.4.3. Rserve

#### 2.4.4. Udig-SDK

#### 2.4.5. Graphical User Interface (GUI)

#### 2.4.6. DSS Output Evaluation

## 3. Case Study: Using the DSS to Fit Time–Dose–Mortality Data to Mathematical Expressions and Mapping the Potential Zone of Efficacy of Fungal-Based Biopesticides in the Killing of Insect Pests

#### 3.1. Data Input, Visualization, and Model Fitting Features

#### 3.2. Mapping Features

_{opt}= 21.98 °C ± 0.29). Based on the Lorentzian four-parameter model results, the maps of the potential zones of the efficacy of EPF isolate ICIPE 62 when applied against mustard aphid were produced for Kenya and Cameroon (Figure 8 and Figure 9, respectively). A level of efficacy that varies between 0 and 1 characterizes the virulence level of ICIPE 62 against the targeted pest.

## 4. Discussion

**Application:**The Lorentzian 4-parameter model obtained in the application section estimates an optimum temperature for the higher virulence of ICIPE62 in killing the aphid at about 21 °C. When comparing the current distribution map of the Lipaphis pseudobrassicae [40] with the map of the potential zone of the efficacy of ICIPE 62 isolate in Kenya and Cameroon, we observed that many invaded locations fit well with a potential zone of efficacy with virulence level greater than or equal to 0.5. Although the estimates were made without full inclusion of other environmental variables that have impacts on the fungi efficacy in killing insects, the outputs are promising. However, it will be useful to explore the association of temperature with other factors such as relative humidity to improve the accuracy of the prediction, especially for mapping the virulence level of the EPF. Indeed, many studies highlighted the key role played by both temperature and relative humidity in enhancing the virulence of EPF on insect pests [47,48,49]. On this note, a good perspective to consider for improving the current tool will be to consider the association of at least two factors (temperature and relative humidity, for example) as independent variables in the fitting process.

## 5. Conclusions and Future Works

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Experimental data: Var1 represents the range of the independent variable (temperature); Var2 is the number of replicates in the experiment, Var3 is the time duration of each experiment; Var4 and Var5 are records of the dependent variable (mortality) with the variation of the independent variable.

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**Figure 1.**Use case diagram presenting the interactions among the platform components and their functionalities. Each circle represents a feature implemented within the decision support system (DSS). The links from the user to a feature represent the direct interactions of the user with the DSS, while the arrows with labels “include” characterize the relationships and level of dependency between features.

**Figure 2.**Detailed flowchart diagram of the platform; the figure displays two processes: the fitting process in which experimental data are fitted to nonlinear mathematical expressions using the Marquardt optimization algorithm and the mapping process that begins by linking the obtained best-fitted model to the climate database for map creation. The GUI represents the graphical user interface, and the KMS is the knowledge management system that processes all the simulations.

**Figure 3.**Unified modeling language (UML) component diagram. The users interact with the inbuilt software environment through the perspectives (model designer and mapping perspective) of the GUI. The GUI is based on the Eclipse-RCP (Rich Client Platform) and Udig-RCP components. Rserve allows the communication between R and Java by creating objects with port 6311.

**Figure 5.**Wizard for the selection list for the fitting process. This frame assists the user in the selection of equations to be fitted with experimental data.

**Figure 7.**Evaluation criteria and goodness of fit for each fitted equation. Cells in red suggest the best-performing function for each evaluation criterion.

**Figure 8.**Kenyan map of the potential efficacy of ICIPE 62 isolate when used against mustard aphid, modeled with the software. The level of efficacy varies between 0 and 1. Locations with 0% level of efficacy are displayed in white, values between 0 and 0.5 are displayed in blue, values between 0.5 and 0.75 are displayed in green, and values between 0.75 and 1 are displayed in red. Red indicates the highest efficacy levels. The yellow circles surround areas in Kenya where the EPF isolate ICIPE 62 is successfully used, and results were used for validation of the developed model.

**Figure 9.**Cameroon map of the potential efficacy of ICIPE 62 isolate when used against mustard aphid, modeled with the software. The level of efficacy varies between 0 and 1. Locations with 0% level of efficacy are displayed in white, values between 0 and 0.5 are displayed in blue, values between 0.5 and 0.75 are displayed in green, and values between 0.75 and 1 are displayed in red. Red indicates the highest efficacy levels.

**Table 1.**Summary of key functions used for fitting. The “Name” column gives the name of the model, the “Equation” column gives the mathematical expression of the model, the “comment” column gives the number of derived sub-models from the original main model, and the last column gives the reference for the model. T is the independent variable, and m(T) represents the virulence model. ID—identifier.

ID | Model Name | Model Main Mathematical Expression | Comment | Reference |
---|---|---|---|---|

1 | Sharpe and DeMichele | $m\left(T\right)=\frac{p.\frac{T}{{T}_{0}}.{e}^{\left[\frac{\Delta {H}_{A}}{R}\left(\frac{1}{{T}_{0}}-\frac{1}{T}\right)\right]}}{1+{e}^{\left[\frac{\Delta {H}_{L}}{R}\left(\frac{1}{{T}_{L}}-\frac{1}{T}\right)\right]}+{e}^{\left[\frac{\Delta {H}_{H}}{R}\left(\frac{1}{{T}_{H}}-\frac{1}{T}\right)\right]}}$ | 12 sub-models | Sharpe and DeMichele 1977 |

Sharpe and DeMichele 1–13 | ||||

2 | Deva | $m(T)=b\left(T-{T}_{\mathrm{min}}\right)$ T ≥ Tmax $m(T)=0$ T < Tmax | 1 sub-model | Dallwits and Higgins 1992 |

Deva 1 and 2 | ||||

3 | Logan | $m(T)=Y*({e}^{p*T}-{e}^{\left(p*{T}_{\mathrm{max}}-\frac{\left({T}_{\mathrm{max}}-T\right)}{v}\right)})$ | 4 sub-models | Longan 1976 |

Logan 1–5 | ||||

4 | Briere | $m(T)=a*T\left(T-{T}_{o}\right)\left(\sqrt{{T}_{\mathrm{max}}-T}\right)$ | 1 sub-model | Briere et al. 1999 |

Briere 1 and 2 | ||||

5 | Stinner | $m(T)=\frac{{R}_{\mathrm{max}}\left(1+{e}^{{k}_{1}+{k}_{2}\left({T}_{opc}\right)}\right)}{1+{e}^{{k}_{1}+{k}_{2}\left(T\right)}}$ | 3 sub-models | Stinner et al. 1974 |

Stinner 1–4 | ||||

6 | Hilber and Logan | $m(T)=Y\left(\frac{{T}^{2}}{{T}^{2}+{d}^{2}}-{e}^{-\frac{\left({T}_{\mathrm{max}}-T\right)}{v}}\right)$ | 2 sub-models | Hilber and logan 1983 |

Logan 1–3 | ||||

7 | Lactin 1 | $m(T)={e}^{p*T}-{e}^{-\frac{\left(p*{T}_{l}-\left(T-{T}_{l}\right)\right)}{dt}}+\lambda $ | 2 sub-models | Lactin et al. 1995 |

Logan 1–3 |

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## Share and Cite

**MDPI and ACS Style**

Guimapi, R.A.; Mohamed, S.A.; Biber-Freudenberger, L.; Mwangi, W.; Ekesi, S.; Borgemeister, C.; Tonnang, H.E.Z. Decision Support System for Fitting and Mapping Nonlinear Functions with Application to Insect Pest Management in the Biological Control Context. *Algorithms* **2020**, *13*, 104.
https://doi.org/10.3390/a13040104

**AMA Style**

Guimapi RA, Mohamed SA, Biber-Freudenberger L, Mwangi W, Ekesi S, Borgemeister C, Tonnang HEZ. Decision Support System for Fitting and Mapping Nonlinear Functions with Application to Insect Pest Management in the Biological Control Context. *Algorithms*. 2020; 13(4):104.
https://doi.org/10.3390/a13040104

**Chicago/Turabian Style**

Guimapi, Ritter A., Samira A. Mohamed, Lisa Biber-Freudenberger, Waweru Mwangi, Sunday Ekesi, Christian Borgemeister, and Henri E. Z. Tonnang. 2020. "Decision Support System for Fitting and Mapping Nonlinear Functions with Application to Insect Pest Management in the Biological Control Context" *Algorithms* 13, no. 4: 104.
https://doi.org/10.3390/a13040104