Optimal control approach to dengue reduction and prevention in Cali, Colombia
The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible...
- Autores:
-
Sepúlveda Salcedo, Lilian Sofía
Vasillieva, Olga
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 2016
- Institución:
- Universidad Autónoma de Occidente
- Repositorio:
- RED: Repositorio Educativo Digital UAO
- Idioma:
- eng
- OAI Identifier:
- oai:red.uao.edu.co:10614/11048
- Acceso en línea:
- http://hdl.handle.net/10614/11048
https://doi.org/10.1002/mma.3932
- Palabra clave:
- Ross–Macdonald model
SIR-SI model; optimal control
cost-effectiveness analysis
Dengue
- Rights
- closedAccess
- License
- Derechos Reservados - John Wiley and Sons Ltd
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Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| title |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| spellingShingle |
Optimal control approach to dengue reduction and prevention in Cali, Colombia Ross–Macdonald model SIR-SI model; optimal control cost-effectiveness analysis Dengue |
| title_short |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| title_full |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| title_fullStr |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| title_full_unstemmed |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| title_sort |
Optimal control approach to dengue reduction and prevention in Cali, Colombia |
| dc.creator.fl_str_mv |
Sepúlveda Salcedo, Lilian Sofía Vasillieva, Olga |
| dc.contributor.author.none.fl_str_mv |
Sepúlveda Salcedo, Lilian Sofía Vasillieva, Olga |
| dc.subject.eng.fl_str_mv |
Ross–Macdonald model SIR-SI model; optimal control cost-effectiveness analysis |
| topic |
Ross–Macdonald model SIR-SI model; optimal control cost-effectiveness analysis Dengue |
| dc.subject.spa.fl_str_mv |
Dengue |
| description |
The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd |
| publishDate |
2016 |
| dc.date.issued.none.fl_str_mv |
2016-05-27 |
| dc.date.accessioned.none.fl_str_mv |
2019-09-04T21:47:27Z |
| dc.date.available.none.fl_str_mv |
2019-09-04T21:47:27Z |
| dc.type.spa.fl_str_mv |
Artículo de revista |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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http://purl.org/coar/resource_type/c_6501 |
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Text |
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publishedVersion |
| dc.identifier.issn.spa.fl_str_mv |
1099-1476 (en línea) 0170-4214 (impresa) |
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http://hdl.handle.net/10614/11048 |
| dc.identifier.doi.spa.fl_str_mv |
https://doi.org/10.1002/mma.3932 |
| dc.identifier.instname.spa.fl_str_mv |
Universidad Autónoma de Occidente |
| dc.identifier.reponame.eng.fl_str_mv |
Repositorio Educativo Digital UAO |
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1099-1476 (en línea) 0170-4214 (impresa) Universidad Autónoma de Occidente Repositorio Educativo Digital UAO |
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http://hdl.handle.net/10614/11048 https://doi.org/10.1002/mma.3932 |
| dc.language.iso.eng.fl_str_mv |
eng |
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eng |
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18 |
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39 |
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Mathematical methods in the applied sciences |
| dc.relation.references.spa.fl_str_mv |
BrownJ,McBrideC,JohnsonP,RitchieS,PaupyC,BossinH,LutomiahJ,Fernandez-SalasI,PonlawatA,CornelA,BlackW,Gorrochotegui-EscalanteN, Urdaneta-MarquezL,SyllaM,SlotmanM,MurrayK,WalkerC,PowellJ.Worldwidepatternsofgeneticdifferentiationimplymultiple’domestications Tapia-ConyerR,Mendez-GalvanJF,Gallardo-RinconH.ThegrowingburdenofdengueinLatinAmerica.JournalofClinicalVirology 2009;46:S3–S6 Lopez-Gatell H, Hernandez-Avila M, Avila JEH, Alpuche-Aranda CM. Dengue in Latin America: a persistent and growing public health challenge. InNeglectedTropicalDiseases-LatinAmericaandtheCaribbean.Springer:Vienna,2015;203–224 Dick OB, San Martín JL, Montoya RH, del Diego J, Zambrano B, Dayan GH. The history of dengue outbreaks in the Americas. TheAmericanjournalof tropicalmedicineandhygiene2012;87(4):584–593 GuzmánMG,KouríG.Denguediagnosis,advancesandchallenges.InternationalJournalofInfectiousDiseases2004;8(2):69–80 Messer W, de Alwis R, Yount B, Royal S, Huynh J, Smith S, Crowe J, Doranz B.J, Kahle K, Pfaff J, Whitej L, Sariolk C, de Silvac A, Baric R. Dengue virus envelopeproteindomainI/IIhingedetermineslong-livedserotype-specificdengueimmunity.ProceedingsoftheNationalAcademyofSciences2014; 111(5):1939–1944 StateoftheArtinthePreventionandControlofDengueintheAmericas.PAHO–WHO:Washington,DC,2014 MendezF,BarretoM,AriasJ,RengifoG,MunozJ,BurbanoM,ParraB.Humanandmosquitoinfectionsbydenguevirusesduringandafterepidemics inadengue–endemicregionofColombia.TheAmericanJournalofTropicalMedicineandHygiene2006;74(4):678–683 Ocazionez RE, Cortés FM, Villar LA, Gómez SY. Temporal distribution of dengue virus serotypes in Colombian endemic area and dengue incidence: re-introductionofdengue-3associatedtomildfebrileillnessandprimaryinfection. MemóriasdoInstitutoOswaldoCruz 2006;101(7):725–731 Ocampo CB, Mina NJ, Carabalí M, Alexander N, Osorio L. Reduction in dengue cases observed during mass control of Aedes (Stegomyia) in street catchbasinsinanendemicurbanareainColombia.ActaTropica 2014;132:15–22 Rial MJ, Alarcón N, Ferrario C, Szefner M, Califano G. Corredores endémicos: Una herramienta útil para la vigilancia epidemiológica de la influenza [Endemiccorridors:ausefultoolfortheepidemiologicalsurveillanceofinfluenza].RevistaArgentinademicrobiologia2007;40(1):37–40 CaetanoM,YoneyamaT.Optimalandsub-optimalcontrolindengueepidemics.OptimalControlApplicationsandMethods2001;22(2):63–73 BlaynehK,CaoY,KwonHD.Optimalcontrolofvector-bornediseases:treatmentandprevention.DiscreteandContinuousDynamicalSystemsB2009; 11(3):587–611 Aldila D, Götz T, Soewono E. An optimal control problem arising from a dengue disease transmission model. Mathematical Biosciences 2013; 242(1):9–16 AlpheyN,AlpheyL,BonsallMB.Amodelframeworktoestimateimpactandcostofgenetics-basedsterileinsectmethodsfordenguevectorcontrol. PLoSOne2011;6(10):e25384–e25384 Okosun KO, Ouifki R, Marcus N. Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waningimmunity.Biosystems2011;106(2):136–145 OkosunKO,RachidO,MarcusN.Optimalcontrolstrategiesandcost-effectivenessanalysisofamalariamodel. BioSystems2013;111(2):83–101 Parham PE, Hughes DA. Climate influences on the cost-effectiveness of vector-based interventions against malaria in elimination scenarios. PhilosophicalTransactionsoftheRoyalSocietyofLondonB:BiologicalSciences2015;370(1665):1–15.PaperID:20130557 RossR.ThePreventionofMalaria.JohnMurray:London,1911 MacdonaldG.TheEpidemiologyandControlofMalaria.OxfordUniversityPress:Oxford,1957 Aron JL, May RM. The population dynamics of malaria. In Population Dynamics and Infectious Disease: Theory and Applications, Anderson RM (ed). ChapmanandHall:London,1982;139–179 AndersonRM,MayRM.InfectiousDiseasesofHumans:DynamicsandControl,OxfordSciencePublications.OUPOxford,1992 SmithDL,BattleKE,HaySI,BarkerCM,ScottTW,McKenzieFE.Ross,Macdonald,andatheoryforthedynamicsandcontrolofmosquito-transmitted pathogens.PLoSPathog2012;8(4):1–13.PaperID:e1002588 Villar LA, Rojas DP, Besada-LS, Sarti E. Epidemiological trends of dengue disease in Colombia (2000–2011): a systematic review. PLoS neglected tropicaldiseases2015;9(3):e0003499–e0003499 SepúlvedaLS.Manejo óptimoyviableenmodelosepidemiológicosdeldengue[Optimalandviablemanagementindengueepidemiologicalmodels], Ph.D.Thesis,UniversitéParis-Est&UniverdidadNacionaldeColombia,2015 Sepúlveda-SalcedoLS,VasilievaO,Martínez-RomeroHJ,Arias-CastroJH.Ross-Macdonald:UnmodeloparaladinámicadeldengueenCali,Colombia [Ross-Macdonald:AmodelforthedenguedynamicinCali,Colombia].RevistadeSaludPública2015;17(5):749–761 JoshiHR.OptimalcontrolofanHIVimmunologymodel.OptimalControlApplicationsandMethods2002;23(4):199–213 JungE,LenhartS,FengZ.Optimalcontroloftreatmentsinatwo-straintuberculosismodel.DiscreteandContinuousDynamicalSystemsSeriesB2002; 2(4):473–482 Gaff H, Schaefer E. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences andEngineering2009;6(3):469–492 Neilan RM, Lenhart S. An introduction to optimal control with an application in disease modeling. DIMACSSeriesinDiscreteMathematics 2010; 75: 67–81 BiswasMHA,PaivaLT,dePinhoMdR.ASEIRmodelforcontrolofinfectiousdiseaseswithconstraints.MathematicalBiosciencesandEngineering2014; 11(4):761–784 MoulayD,Aziz-AlaouiMA,KwonHD.Optimalcontrolofchikungunyadisease:larvaereduction,treatmentandprevention.MathematicalBiosciences andEngineering2012;9(2):369–392 CoddingtonE,LevinsonN.TheoryofOrdinaryDifferentialEquations 9thed.McGraw-Hill:NewYork,1987 WalterW.OrdinaryDifferentialEquations,GraduateTextsinMathematics.Springer:NewYork,1998 FlemingW,RishelR.DeterministicandStochasticOptimalControl.Springer:NewYork,1975 Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. TheMathematicalTheoryofOptimalProcesses. Interscience Publishers, John Wiley & Sons:NewYork,1962 LenhartS,WorkmanJT.OptimalControlAppliedtoBiologicalModels.Chapman&Hall/CRC:BocaRaton,FL,2007 BrauerF,Castillo-ChávezC.MathematicalModelsinPopulationBiologyandEpidemiology,TextsinAppliedMathematics,vol.40.Springer-Verlag:New York,2001 Castillo-Chávez C, Thieme HR. Asymptotically autonomous epidemic models. Mathematical Population Dynamics: Analysis of Heterogeneity, Winnipeg,Canada:Wuerz,1995,33–50 JansenCC,BeebeNW.ThedenguevectorAedesaegypti:whatcomesnext.MicrobesandInfection2010;12(4):272–279 Santacoloma L, Chaves B, Brochero HL. Susceptibility of natural populations of dengue vector to insecticides in Colombia. Biomédica 2012; 32(3):333–343 Menger DJ, Van Loon JJA, Takken W. Assessing the efficacy of candidate mosquito repellents against the background of an attractive source that mimicsahumanhost.MedicalandVeterinaryEntomology 2014;28(4):407–413 Bello S, Díaz E, Rojas J, Romero M, Salazar V. Medición del impacto económico del dengue en Colombia: una aproximación a los costos médicos directos en el periodo 2000-2010 [MeasuringtheeconomicimpactofdengueinColombia:anapproximationtothedirectmedicalcostsin2000-2010]. Biomédica2011;31(sup3):110–113 SuayaD,ShepardJ,SiqueiraJ,MartelliC,LumL,TanL,KongsinS,JiamtonS,GarridoF,MontoyaR,ArmienB, R.,CastilloL,CaramM,SahB,Sughayyar R,TyoK,HalsteadS.CostofdenguecasesineightcountriesintheAmericasandAsia:aprospectivestudy.TheAmericanJournalofTropicalMedicine andHygiene2009;80(5):846–855 BangH,ZhaoH.Averagecost-effectivenessratiowithcensoreddata.Journalofbiopharmaceuticalstatistics2012;22(2):401–415 HersheyJC,AschDA,JepsonC,BaronJ,UbelPA.Incrementalandaveragecost-effectivenessratios:willphysiciansmakeadistinction?.RiskAnalysis 2003;23(1):81–89 HyewonHL,LevineM.DeterminingthethresholdforacceptabilityofanICERwhennaturalhealthunitsareused.JournalofPopulationTherapeutics andClinicalPharmacology 2012;19(2):e234–e238 Escobar-Morales G (ed). Cali en cifras 2010 [Cali in numbers 2010], Departamento Administrativo de Planeación. Alcaldia de Santiago de Cali, 2011. Availablefrom:http://planeacion.cali.gov.co/Publicaciones/Cali_en_Cifras/Caliencifras2010.pdf[Accessedon29March2016] TomashekKM,SharpTM,MargolisHS.Dengue.CDC?sHealthInformationforInternationalTravel:2014(YellowBook)2014 Costero A, Edman J, Clark G, Scott T. Life table study of Aedes aegypti (Diptera: Culicidae) in Puerto Rico fed only human blood versus blood plus sugar.JournalofMedicalEntomology 1998;35(5):809–813 ScottT,AmerasingheP,MorrisonA,LorenzL,ClarkG,StrickmanD,KittayapongP,EdmanJ.LongitudinalstudiesofAedesaegypti(Diptera:Culicidae) inThailandandPuertoRico:bloodfeedingfrequency.JournalofMedicalEntomology 2000;37(1):89–101 Scott TW, Morrison A, Lorenz L, Clark G, Strickman D, Kittayapong P, Zhou H, Edman J. Longitudinal studies of Aedesaegypti (Diptera: Culicidae) in ThailandandPuertoRico:populationdynamics.JournalofMedicalEntomology 2000;37(1):77–88 SorensenDC.Newton’smethodwithamodeltrustregionmodification.SIAMJournalonNumericalAnalysis1982;19(2):409–426 MoréJJ.TheLevenberg–Marquardtalgorithm:implementationandtheory.InNumericalAnalysis.Springer:BerlinHeidelberg,1978;105–116 |
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Sepúlveda Salcedo, Lilian Sofíavirtual::4685-1Vasillieva, Olga8192a3cb04cd7d52c219424345b35dc3Cali, Colombia2019-09-04T21:47:27Z2019-09-04T21:47:27Z2016-05-271099-1476 (en línea)0170-4214 (impresa)http://hdl.handle.net/10614/11048https://doi.org/10.1002/mma.3932Universidad Autónoma de OccidenteRepositorio Educativo Digital UAOThe aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd22 páginasapplication/pdfengJohn Wiley and Sons LtdDerechos Reservados - John Wiley and Sons Ltdhttps://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/closedAccessAtribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)http://purl.org/coar/access_right/c_14cbRoss–Macdonald modelSIR-SI model; optimal controlcost-effectiveness analysisDengueOptimal control approach to dengue reduction and prevention in Cali, ColombiaArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTREFinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a851839Mathematical methods in the applied sciencesBrownJ,McBrideC,JohnsonP,RitchieS,PaupyC,BossinH,LutomiahJ,Fernandez-SalasI,PonlawatA,CornelA,BlackW,Gorrochotegui-EscalanteN, Urdaneta-MarquezL,SyllaM,SlotmanM,MurrayK,WalkerC,PowellJ.Worldwidepatternsofgeneticdifferentiationimplymultiple’domesticationsTapia-ConyerR,Mendez-GalvanJF,Gallardo-RinconH.ThegrowingburdenofdengueinLatinAmerica.JournalofClinicalVirology 2009;46:S3–S6Lopez-Gatell H, Hernandez-Avila M, Avila JEH, Alpuche-Aranda CM. Dengue in Latin America: a persistent and growing public health challenge. InNeglectedTropicalDiseases-LatinAmericaandtheCaribbean.Springer:Vienna,2015;203–224Dick OB, San Martín JL, Montoya RH, del Diego J, Zambrano B, Dayan GH. The history of dengue outbreaks in the Americas. TheAmericanjournalof tropicalmedicineandhygiene2012;87(4):584–593GuzmánMG,KouríG.Denguediagnosis,advancesandchallenges.InternationalJournalofInfectiousDiseases2004;8(2):69–80Messer W, de Alwis R, Yount B, Royal S, Huynh J, Smith S, Crowe J, Doranz B.J, Kahle K, Pfaff J, Whitej L, Sariolk C, de Silvac A, Baric R. Dengue virus envelopeproteindomainI/IIhingedetermineslong-livedserotype-specificdengueimmunity.ProceedingsoftheNationalAcademyofSciences2014; 111(5):1939–1944StateoftheArtinthePreventionandControlofDengueintheAmericas.PAHO–WHO:Washington,DC,2014MendezF,BarretoM,AriasJ,RengifoG,MunozJ,BurbanoM,ParraB.Humanandmosquitoinfectionsbydenguevirusesduringandafterepidemics inadengue–endemicregionofColombia.TheAmericanJournalofTropicalMedicineandHygiene2006;74(4):678–683Ocazionez RE, Cortés FM, Villar LA, Gómez SY. Temporal distribution of dengue virus serotypes in Colombian endemic area and dengue incidence: re-introductionofdengue-3associatedtomildfebrileillnessandprimaryinfection. MemóriasdoInstitutoOswaldoCruz 2006;101(7):725–731Ocampo CB, Mina NJ, Carabalí M, Alexander N, Osorio L. Reduction in dengue cases observed during mass control of Aedes (Stegomyia) in street catchbasinsinanendemicurbanareainColombia.ActaTropica 2014;132:15–22Rial MJ, Alarcón N, Ferrario C, Szefner M, Califano G. Corredores endémicos: Una herramienta útil para la vigilancia epidemiológica de la influenza [Endemiccorridors:ausefultoolfortheepidemiologicalsurveillanceofinfluenza].RevistaArgentinademicrobiologia2007;40(1):37–40CaetanoM,YoneyamaT.Optimalandsub-optimalcontrolindengueepidemics.OptimalControlApplicationsandMethods2001;22(2):63–73BlaynehK,CaoY,KwonHD.Optimalcontrolofvector-bornediseases:treatmentandprevention.DiscreteandContinuousDynamicalSystemsB2009; 11(3):587–611Aldila D, Götz T, Soewono E. An optimal control problem arising from a dengue disease transmission model. Mathematical Biosciences 2013; 242(1):9–16AlpheyN,AlpheyL,BonsallMB.Amodelframeworktoestimateimpactandcostofgenetics-basedsterileinsectmethodsfordenguevectorcontrol. PLoSOne2011;6(10):e25384–e25384Okosun KO, Ouifki R, Marcus N. Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waningimmunity.Biosystems2011;106(2):136–145OkosunKO,RachidO,MarcusN.Optimalcontrolstrategiesandcost-effectivenessanalysisofamalariamodel. BioSystems2013;111(2):83–101Parham PE, Hughes DA. Climate influences on the cost-effectiveness of vector-based interventions against malaria in elimination scenarios. PhilosophicalTransactionsoftheRoyalSocietyofLondonB:BiologicalSciences2015;370(1665):1–15.PaperID:20130557RossR.ThePreventionofMalaria.JohnMurray:London,1911MacdonaldG.TheEpidemiologyandControlofMalaria.OxfordUniversityPress:Oxford,1957Aron JL, May RM. The population dynamics of malaria. In Population Dynamics and Infectious Disease: Theory and Applications, Anderson RM (ed). ChapmanandHall:London,1982;139–179AndersonRM,MayRM.InfectiousDiseasesofHumans:DynamicsandControl,OxfordSciencePublications.OUPOxford,1992SmithDL,BattleKE,HaySI,BarkerCM,ScottTW,McKenzieFE.Ross,Macdonald,andatheoryforthedynamicsandcontrolofmosquito-transmitted pathogens.PLoSPathog2012;8(4):1–13.PaperID:e1002588Villar LA, Rojas DP, Besada-LS, Sarti E. Epidemiological trends of dengue disease in Colombia (2000–2011): a systematic review. PLoS neglected tropicaldiseases2015;9(3):e0003499–e0003499SepúlvedaLS.Manejo óptimoyviableenmodelosepidemiológicosdeldengue[Optimalandviablemanagementindengueepidemiologicalmodels], Ph.D.Thesis,UniversitéParis-Est&UniverdidadNacionaldeColombia,2015Sepúlveda-SalcedoLS,VasilievaO,Martínez-RomeroHJ,Arias-CastroJH.Ross-Macdonald:UnmodeloparaladinámicadeldengueenCali,Colombia [Ross-Macdonald:AmodelforthedenguedynamicinCali,Colombia].RevistadeSaludPública2015;17(5):749–761JoshiHR.OptimalcontrolofanHIVimmunologymodel.OptimalControlApplicationsandMethods2002;23(4):199–213JungE,LenhartS,FengZ.Optimalcontroloftreatmentsinatwo-straintuberculosismodel.DiscreteandContinuousDynamicalSystemsSeriesB2002; 2(4):473–482Gaff H, Schaefer E. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences andEngineering2009;6(3):469–492Neilan RM, Lenhart S. An introduction to optimal control with an application in disease modeling. DIMACSSeriesinDiscreteMathematics 2010; 75: 67–81BiswasMHA,PaivaLT,dePinhoMdR.ASEIRmodelforcontrolofinfectiousdiseaseswithconstraints.MathematicalBiosciencesandEngineering2014; 11(4):761–784MoulayD,Aziz-AlaouiMA,KwonHD.Optimalcontrolofchikungunyadisease:larvaereduction,treatmentandprevention.MathematicalBiosciences andEngineering2012;9(2):369–392CoddingtonE,LevinsonN.TheoryofOrdinaryDifferentialEquations 9thed.McGraw-Hill:NewYork,1987WalterW.OrdinaryDifferentialEquations,GraduateTextsinMathematics.Springer:NewYork,1998FlemingW,RishelR.DeterministicandStochasticOptimalControl.Springer:NewYork,1975Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. TheMathematicalTheoryofOptimalProcesses. 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