Quantitative Methods for Investigating Infectious Disease Outbreaks

361 Pages·2019·7.75 MB·English

by  Ping YanGerardo Chowell

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
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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