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Quantitative Methods for Investigating Infectious Disease Outbreaks PDF

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Texts in Applied Mathematics 70 Ping Yan Gerardo Chowell Quantitative Methods for Investigating Infectious Disease Outbreaks Texts in Applied Mathematics Volume 70 Editors-in-chief A.Bloch,UniversityofMichigan,PublicUniversity,AnnArbor,USA C.L.Epstein,UniversityofPennsylvania,Philadelphia,USA A.Goriely,UniversityofOxford,Oxford,UK L.Greengard,NewYorkUniversity,NewYork,USA SeriesEditors J.Bell,LawrenceBerkeleyNationalLab,Berkeley,USA R.Kohn,NewYorkUniversity,NewYork,USA P.Newton,UniversityofSouthernCalifornia,LosAngeles,USA C.Peskin,NewYorkUniversity,NewYork,USA R.Pego,CarnegieMellonUniversity,Pittsburgh,USA L.Ryzhik,StanfordUniversity,Stanford,USA A.Singer,PrincetonUniversity,Princeton,USA A.Stevens,UniversitätMünster,Münster,Germany A.Stuart,UniversityofWarwick,Coventry,UK T.Witelski,DukeUniversity,Durham,USA S.Wright,UniversityofWisconsin,Madison,USA Themathematizationofallsciences,thefadingoftraditionalscientificboundaries, theimpactofcomputertechnology,thegrowingimportanceofcomputermodeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The aim of this series is to provide such textbooks in applied mathematics for the student scientist. Books should be well illustrated and have clear exposition and sound pedagogy. Large number of examples and exercises at varying levels are recommended. TAM publishes textbooks suitable for advanced undergraduate and beginning graduate courses, and complements the Applied Mathematical Sciences (AMS) series, which focuses on advanced textbooks and research-level monographs. Moreinformationaboutthisseriesathttp://www.springer.com/series/1214 Ping Yan (cid:129) Gerardo Chowell Quantitative Methods for Investigating Infectious Disease Outbreaks 123 PingYan GerardoChowell InfectiousDiseasesPrevention SchoolofPublicHealth andControlBranch GeorgiaStateUniversity PublicHealthAgencyofCanada Atlanta,GA,USA Ottawa,ON,Canada DepartmentofStatistics andActuarialScience FacultyofMathematics UniversityofWaterloo Waterloo,ON,Canada ISSN0939-2475 ISSN2196-9949 (electronic) TextsinAppliedMathematics ISBN978-3-030-21922-2 ISBN978-3-030-21923-9 (eBook) https://doi.org/10.1007/978-3-030-21923-9 MathematicsSubjectClassification:Primary:92D30;92C60;60J28;60K20;60K37;62P10.Secondary: 97M6;37N2 ©Crown2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ToLouise,Genevieve,andVeronique ToPiaandCatalina Preface Mathematicalandstatisticalmodelsandmethodscanplayacentralroleinoutbreak investigationsandinpublichealthdecision-making.Thepurposeofthisbookisto providereaderswithbalancedperspectivesbetweentheoryandpractice.Toprovide insight between models driven by scientific hypotheses intended to characterize the agent-host-environment interface in complex disease transmission dynamics, and models driven by observational data intended to capture the data-generating process; and between the unobservable variables predicted by most disease trans- mission dynamic models and data collected based on observed outcomes. As for prerequisites,beforeembarkingintoChaps.2–4ofthisbook,thereaderswillneed an essential understanding of random variables, distribution theory, and stochastic processes(see,forinstance,thetextbookbyRoss(2019)). The modeling process in this book is illustrated in the following flowchart. Unlike most other scientific investigations, in which questions are formulated and dataarisefromexperimentstoaddressthosequestions,dataarisingfromoutbreak investigations are mostly observational and collected by different agencies for a variety of purposes. In this book, we put equal emphasis on answering the right questionsandunderstandingthedata-generatingprocesses. We started our collaboration in 2003 when we met at a modeling workshop focused on social responses to bioterrorism involving infectious agents, orga- nized by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University. In subsequent years, we actively participated in and co-organized some of the workshops and summer schools on disease modeling supported by the Mathematics of Information Technology and Complex Systems (Canada); the Pacific Institute for the Mathematical Sciences; the Fields Institute; the Banff International Research Station for Mathematical Innovation and Discovery; the Simon A. Levin Mathematical, Computational, and Modeling Science Center at Arizona State University; the Centro Internacional de Ciencias, Cuernavaca, Mexico; and the Centre for Disease Modelling at York University, GeorgiaStateUniversity,UniversityofBritishColumbia,andUniversityofAlberta, amongothers. vii viii Preface Much of our research on integrating mathematical and statistical models in infectiousdiseaseoutbreakinvestigationshasbeenmotivatedbydiscussionsamong appliedmathematiciansandstatisticalscientistsduringtheseworkshops.Thisbook containsmaterialsfromourownpresentationsandlecturenotes,ourideasandviews basedonpersonalcommunicationsthroughouttheseyears,and,moreimportantly, inspirationsfromquestionsduringtheworkshopsandsummerschoolsfromworld- renownedscientistsaswellasyoungresearchersandgraduatestudents. We would like to thank the following mentors and long-term collaborators: Carlos Castillo-Chavez, Jerald F. Lawless, Fred Brauer, Mac Hyman, Nicholas Hengartner, Paul W. Fenimore, Hiroshi Nishiura, Cecile Viboud, Lone Simonsen, Mark Miller, Selcuk Candan, Charles Perrings, Alexandra Smirnova, Jianhong Wu; colleagues at the Public Health Agency of Canada, Donald Sutherland, Chris Archibald,DenaSchanzer,FanZhang,andPascalMichel;colleaguesattheGeorgia StateUniversitySchoolofPublicHealth,MichaelEriksenandRichardRothenberg; andcolleaguesattheUniversityofWaterloo,MaryThompsonandCharmaineDean. We are thankful to Donna Chernyk, our Springer Editor, for providing detailed guidancethroughoutthepublicationprocess. SpecialthanksgotograduatestudentsKimberlynRoosa,AmnaTariq,andYiseul Lee in the Department of Population Health Sciences, Georgia State University SchoolofPublicHealth,fortheirhelpinproofreadingandediting. Finally,wewouldliketothankourfamiliesfortheirsupportandunderstanding. Ottawa,ON,Canada PingYan Atlanta,GA,USA GerardoChowell Contents 1 Introduction .................................................................. 1 1.1 TheMotivation.......................................................... 4 1.2 StructureoftheBookwithBriefSummary............................ 6 2 ShapesofHazardFunctionsandLifetimeDistributions................ 11 2.1 DefinitionsandtheScaleParameter ................................... 12 2.1.1 TheHazardFunction,theDistributionFunctions,and SomeCommonlyUsedSummaryMeasures .................. 12 2.1.2 TheScaleParameter............................................ 14 2.2 TheShapesofHazardFunctions....................................... 14 2.2.1 TheConstantHazardFunctionandtheExponential Distribution ..................................................... 15 2.2.2 MonotonicHazardFunctionsWithoutUpperLimit .......... 16 2.2.3 HazardFunctionsthatConvergetoaPositiveConstant asx →∞....................................................... 18 2.2.4 TwoEmpiricalDistributionsforDiseaseProgression CharacterizedbyNon-monotoneHazardFunctions .......... 22 2.2.5 ParametricLifetimeDistributionswithMorethanTwo Parameters....................................................... 27 2.3 TheResidualLifeDistributionandtheTailProperty................. 27 2.3.1 The Residual Life Distribution as Uniquely DeterminedbytheHazardFunction........................... 27 2.3.2 SomeHighlySkewed,HeavyTailedDistributions............ 28 2.4 TheLaplaceTransformforLifeDistributions......................... 29 2.4.1 LaplaceTransformoftheSumofTwoIndependent RandomVariables .............................................. 30 2.4.2 MomentGeneratingProperty .................................. 31 2.4.3 As a Probability Comparing X Against an ExponentiallyDistributedLifetimeY ......................... 31 2.4.4 LaplaceTransformasaSurvivalFunction .................... 31 ix x Contents 2.5 ComparingTwoLifetimesX andX ................................. 32 1 2 2.5.1 ComparingMagnitudes......................................... 32 2.5.2 ComparingVariabilities ........................................ 34 2.6 MixtureofDistributionsandFrailtyModels .......................... 38 2.6.1 FrailtyandDampenedHazardFunctions...................... 39 2.7 ProblemsandSupplements............................................. 42 3 RandomCountsandCountingProcesses ................................. 47 3.1 SomeImportantDistributionsForRandomCounts................... 48 3.1.1 TheProbabilityFunctionsandRelatedQuantities............ 48 3.1.2 TwoClassesofDistributions................................... 49 3.2 Random Count Distributions as Generated by Stochastic DiseaseTransmissionModels.......................................... 55 3.2.1 MixtureofPoissonDistributionsandProcesses .............. 56 3.2.2 HighlySkewedData:Proneness,Contagion,orSpells?...... 61 3.3 GeneralFormulationofaCountingProcess........................... 66 3.3.1 ReviewofSomeoftheCountingProcessesthatHave BeenMentionedEarlier ........................................ 68 3.3.2 MartingalesandTheirRelationswithCountingProcesses... 72 3.4 ProblemsandSupplements............................................. 73 4 Behaviors of a Disease Outbreak During the Initial Phase andtheBranchingProcessApproximation............................... 79 4.1 TheBranchingProcessApproximation................................ 79 4.1.1 TheGalton-WatsonBranchingProcess........................ 80 4.1.2 EmbeddingtheGalton-WatsonBranchingProcess intoaContinuousTimeFramework ........................... 82 4.2 ExtinctionandtheInvasionProbability................................ 84 4.2.1 The Effects of Variability of N on the Invasion Probability1−δandGenerationsTowardExtinction........ 86 4.2.2 WhenN FollowsthePowerSeriesDistributions ............. 89 4.2.3 FinalSizeDistributionsforSmallOutbreaks ................. 92 4.2.4 Examples........................................................ 98 4.2.5 Estimation for R Based on the Galton-Watson 0 BranchingProcess.............................................. 100 4.3 TheInitialGrowthGivenNon-extinction.............................. 105 4.3.1 TheExponentialGrowthbyGeneration....................... 105 4.3.2 GrowthinReal(Continuous)Time............................ 106 4.3.3 TheEuler-LotkaEquationsUnderModelswithSEI Structure ........................................................ 109 4.4 OnAssumptionsandConditions....................................... 118 4.4.1 TheInitialPhase................................................ 118

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