# 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

**:**

## 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.

## References

- Jones, J.W.; Antle, J.M.; Basso, B.; Boote, K.J.; Conant, R.T.; Foster, I.; Godfray, H.C.J.; Herrero, M.; Howitt, R.E.; Janssen, S.; et al. Brief history of agricultural systems modeling. Agric. Syst.
**2017**, 155, 240–254. [Google Scholar] [CrossRef] [PubMed] - Lepenioti, K.; Bousdekis, A.; Apostolou, D.; Mentzas, G. Prescriptive analytics: Literature review and research challenges. Int. J. Inf. Manag.
**2020**, 50, 57–70. [Google Scholar] [CrossRef] - Archontoulis, S.V.; Miguez, F.E. Nonlinear Regression Models and Applications in Agricultural Research. Agron. J.
**2015**, 107, 786. [Google Scholar] [CrossRef] [Green Version] - Klosterman, R.E. Simple and Complex Models. Environ. Plan. B Plan. Des.
**2012**, 39, 1–6. [Google Scholar] [CrossRef] - Eva, Y.-H.; Wu, M.-C. Hung, Comparison of Spatial Interpolation Techniques Using Visualization and Quantitative Assessment. Appl. Spat. Stat.
**2016**, 11, 17–34. [Google Scholar] [CrossRef] [Green Version] - Patel, N.R.; Mandal, U.K.; Pande, L.M. Agro-ecological Zoning System-A Remote Sensing and GIS Perspective Upscaling of photosynthesis through Sun-induced fluorescence (SIF) View project 1. Regional Carbon Cycle Modeling for India and surrounding oceans View project. J. Agrometeorol.
**2000**, 2, 1–13. Available online: https://www.researchgate.net/publication/270683979 (accessed on 3 April 2020). - Gimond, M. Intro to GIS and Spatial Analysis. 2019. Available online: https://mgimond.github.io/Spatial/index.html (accessed on 9 October 2019).
- FAO. Agro-Ecological Zoning: Guidelines, Rome. 1996. Available online: https://books.google.com/books?hl=fr&lr=&id=IWFD2zGLyrYC&oi=fnd&pg=PA1&dq=AGRO-ECOLOGICAL+ZONING+Guidelines&ots=bAH-Or-Nn0&sig=XdhPDm3WBbN8ckFjP2d5Zj5qaIc (accessed on 3 April 2020).
- Mardani, A.; Jusoh, A.; Nor, K.M.D.; Khalifah, Z.; Zakwan, N.; Valipour, A. Multiple criteria decision-making techniques and their applications—A review of the literature from 2000 to 2014. Econ. Res. Istraž.
**2015**, 28, 516–571. [Google Scholar] [CrossRef] - Belton, V.; Stewart, T.J. Multiple Criteria Decision Analysis: An Integrated Approach; Springer: New York, NY, USA, 2002. [Google Scholar]
- Stojčić, M.; Zavadskas, E.; Pamučar, D.; Stević, Ž.; Mardani, A. Application of MCDM Methods in Sustainability Engineering: A Literature Review 2008–2018. Symmetry (Basel)
**2019**, 11, 350. [Google Scholar] [CrossRef] [Green Version] - Sprague, R.H.; Carlson, E. Building Effective Decision Support Systems; Prentice Hall College Div: Englewood Cliffs, NJ, USA, 1982. [Google Scholar]
- Dan, P. Ask Dan about DSS—What Are the Components of A Decision Support System? 2005. Available online: http://dssresources.com/faq/index.php?action=artikel&id=101 (accessed on 5 April 2017).
- Karacapilidis, N. An Overview of Future Challenges of Decision Support Technologies; Springer: London, UK, 2006; pp. 385–399. [Google Scholar] [CrossRef] [Green Version]
- Huber, G.P. Organizational science contributions to the design of decision support systems. 1980. Available online: http://pure.iiasa.ac.at/id/eprint/1221/1/XB-80-512.pdf#page=55 (accessed on 17 January 2020).
- Fick, G.; Sprague, R.H. Decision Support Systems: Issues and Challenges: Proceedings of the an International Task Force Meeting June 23–25, 1980; Elsevier: Oxford, UK, 1980. Available online: http://www.sciencedirect.com/science/book/9780080273211 (accessed on 5 April 2017).
- Wierzbicki, A.P.; Makowski, M.; Wessels, J. Model-Based Decision Support Methodology with Environmental Applications. Interfaces
**2000**, 32(2), 84. [Google Scholar] [CrossRef] - Power, D.J. Understanding Data-Driven Decision Support Systems. Inf. Syst. Manag.
**2008**, 25, 149–154. [Google Scholar] [CrossRef] - Hijmans, R.J.; Cameron, S.E.; Parra, J.L.; Jones, P.G.; Jarvis, A. Very high resolution interpolated climate surfaces for global land areas. Int. J. Climatol.
**2005**, 25, 1965–1978. [Google Scholar] [CrossRef] - Jaqaman, K.; Danuser, G. Linking data to models: Data regression. Nat. Rev. Mol. Cell Biol.
**2006**, 7, 813–819. [Google Scholar] [CrossRef] [PubMed] - Marquardt, D. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Gavin, H. The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems. 2011. Available online: http://people.duke.edu/~hpgavin/ce281/lm.pdf (accessed on 5 December 2019).
- Lourakis, M.I.A. A Brief Description of the Levenberg-Marquardt Algorithm Implemened by levmar. Found. Res. Technol.
**2005**, 4, 1–6. [Google Scholar] [CrossRef] - Silva, V. Practical Eclipse Rich Client Platform Projects; Apress: New York, NY, USA, 2009. [Google Scholar] [CrossRef]
- Gaujoux, R. doRNG: Generic Reproducible Parallel Backend for “Foreach” Loops. 2017. Available online: https://cran.r-project.org/web/packages/doRNG/index.html (accessed on 9 October 2017).
- Pebesma, E.; Bivand, R.; Rowlingson, B.; Gomez-Rubio, V.; Hijmans, R.; Sumner, M.; MacQueen, D.; Lemon, J.; O’Brien, J. sp: Classes and Methods for Spatial Data. 2017. Available online: https://cran.r-project.org/web/packages/sp/index.html (accessed on 9 October 2017).
- Ripley, B.; Venables, B.; Bates, D.M.; Hornik, K.; Gebhardt, A.; Firth, D. MASS: Support Functions and Datasets for Venables and Ripley’s MASS. 2017. Available online: https://cran.r-project.org/web/packages/MASS/index.html (accessed on 9 October 2017).
- Elzhov, T.V.; Mullen, K.M.; Spiess, A.-N.; Bolker, B. Minpack.lm: R Interface to the Levenberg-Marquardt Nonlinear Least-Squares Algorithm Found in MINPACK, Plus Support for Bounds. 2016. Available online: https://cran.r-project.org/web/packages/minpack.lm/index.html (accessed on 9 October 2017).
- Bivand, R.; Lewin-Koh, N.; Pebesma, E.; Archer, E.; Baddeley, A.; Bearman, N.; Bibiko, H.-J.; Brey, S.; Callahan, J.; Carrillo, G.; et al. Maptools: Tools for Reading and Handling Spatial Objects. 2017. Available online: https://cran.r-project.org/web/packages/maptools/index.html (accessed on 9 October 2017).
- Becker, R.A.; Wilks, A.R.; Brownrigg, R.; Minka, T.P.; Deckmyn, A. Maps: Draw Geographical Maps. 2017. Available online: https://cran.r-project.org/web/packages/maps/index.html (accessed on 9 October 2017).
- Urbanek, S. Rserve—A Fast Way to Provide R Functionality to Applications. 2003. Available online: https://www.r-project.org/conferences/DSC-2003/Proceedings/Urbanek.pdf (accessed on 14 April 2017).
- Augustyniuk-Kram, A.; Kram, K.J. Entomopathogenic Fungi as an Important Natural Regulator of Insect Outbreaks in Forests (Review). 2012. Available online: https://www.intechopen.com/books/forest-ecosystems-more-than-just-trees/entomopathogenic-fungi-as-an-important-natural-regulator-of-insect-outbreaks-in-forests-review- (accessed on 21 September 2016).
- Lacey, L.A.; Grzywacz, D.; Shapiro-Ilan, D.I.; Frutos, R.; Brownbridge, M.; Goettel, M.S. Insect pathogens as biological control agents: Back to the future. J. Invertebr. Pathol.
**2015**, 132, 1–41. [Google Scholar] [CrossRef] [Green Version] - Roberts, D.W.; Humber, R.A. Entomogenous Fungi. Biol. Conidial Fungi
**1981**, 2(201), 201–236. Available online: http://linkinghub.elsevier.com/retrieve/pii/B9780121795023500145 (accessed on 21 September 2016). - Shahid, A.A.; Rao, A.Q.; Bakhsh, A.; Husnain, T. Entomopathogenic fungi as biological controllers: New insights into their virulence and pathogenicity. 2012. Available online: http://agris.fao.org/agris-search/search.do?recordID=RS2012000992 (accessed on 21 September 2016).
- Bayissa, W.; Ekesi, S.; Mohamed, S.A.; Kaaya, G.P.; Wagacha, J.M.; Hanna, R.; Maniania, N.K. Selection of fungal isolates for virulence against three aphid pest species of crucifers and okra. J. Pest Sci.
**2004**, 90, 355–368. [Google Scholar] [CrossRef] - Migiro, L.N.; Maniania, N.K.; Chabi-Olaye, A.; Vandenberg, J. Pathogenicity of Entomopathogenic Fungi Metarhizium anisopliae and Beauveria bassiana (Hypocreales: Clavicipitaceae) Isolates to the Adult Pea Leafminer (Diptera: Agromyzidae) and Prospects of an Autoinoculation Device for Infection in the Field. Environ. Entomol.
**2010**, 39, 468–475. [Google Scholar] [CrossRef] [Green Version] - Niassy, S.; Maniania, N.K.; Subramanian, S.; Gitonga, M.L.; Maranga, R.; Obonyo, A.B.; Ekesi, S. Compatibility of Metarhizium anisopliae isolate ICIPE 69 with agrochemicals used in French bean production. Int. J. Pest Manag.
**2012**, 58, 131–137. [Google Scholar] [CrossRef] - Akhtar, K.U.S.; Dey, D. Spatial Distribution of Mustard Aphid Lipaphis erysimi (Kaltenbach) Vis-à-vis its Parasitoid, Diaeretiella rapae (M’intosh). World Appl. Sci.
**2010**, 11, 284–288. [Google Scholar] - CABI. Mustard Aphid (Lipaphis Erysimi) Plantwise Technical Factsheet, Plantwise Knowledge Bank. 2020. Available online: http://www.plantwise.org/KnowledgeBank/Datasheet.aspx?dsid=30913 (accessed on 20 June 2017).
- Awaneesh, Mustard Aphid agropedia. 2009. Available online: http://agropedia.iitk.ac.in/node/4578 (accessed on 29 June 2017).
- Scott, D. 6 Reasons for Component-based UI Development. 2016. Available online: https://www.tandemseven.com/technology/6-reasons-component-based-ui-development/ (accessed on 14 July 2017).
- Jones, V.P.; Brunner, J.F.; Grove, G.G.; Petit, B.; Tangren, G.V.; Jones, W.E. A web-based decision support system to enhance IPM programs in Washington tree fruit. Pest Manag. Sci.
**2010**, 66(6), 587–595. [Google Scholar] [CrossRef] [PubMed] - Damos, P. Modular structure of web-based decision support systems for integrated pest management: A review. Agron. Sustain. Dev.
**2015**, 35, 1347–1372. [Google Scholar] [CrossRef] [Green Version] - Klass, J.I.; Blanford, S.; Thomas, M.B. Use of a geographic information system to explore spatial variation in pathogen virulence and the implications for biological control of locusts and grasshoppers. Agric. For. Entomol.
**2007**, 9, 201–208. [Google Scholar] [CrossRef] - Klass, J.I.; Blanford, S.; Thomas, M.B. Development of a model for evaluating the effects of environmental temperature and thermal behaviour on biological control of locusts and grasshoppers using pathogens. Agric. For. Entomol.
**2007**, 9, 189–199. [Google Scholar] [CrossRef] - Mishra, S.; Kumar, P.; Malik, A. Effect of temperature and humidity on pathogenicity of native Beauveria bassiana isolate against Musca domestica L. J. Parasit. Dis.
**2015**, 39, 697–704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hsiao, W.-F.; Bidochka, M.J.; Khachatourians, G.G. Effect of temperature and relative humidity on the virulence of the entomopathogenic fungus, Verticillium lecanii, toward the oat-bird berry aphid, Rhopalosiphum padi (Hom., Aphididae). J. Appl. Entomol.
**1992**, 114, 484–490. [Google Scholar] [CrossRef] - Athanassiou, C.G.; Kavallieratos, N.G.; Rumbos, C.I.; Kontodimas, D.C. Influence of Temperature and Relative Humidity on the Insecticidal Efficacy of Metarhizium anisopliae against Larvae of Ephestia kuehniella (Lepidoptera: Pyralidae) on Wheat. J. Insect Sci.
**2017**, 17, 22. [Google Scholar] [CrossRef]

**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