Stochastic Population and Epidemic Models: Persistence and Extinction PDF
Preview Stochastic Population and Epidemic Models: Persistence and Extinction
Mathematical Biosciences Institute Lecture Series 1.3 Stochastics in Biological Systems Linda J. S. Allen Stochastic Population and Epidemic Models Persistence and Extinction MathematicalBiosciencesInstituteLectureSeries The Mathematical Biosciences Institute (MBI) fosters innovation in the application of mathematical, statistical and computational methods in the resolution of significant problems in the biosciences, and encourages the development of new areas in the mathematical sciences motivated by important questions in the biosciences. To accomplish this mission, MBI holds many week-long research workshops each year, trains postdoctoral fellows,andsponsorsavarietyofeducationalprograms. The MBI lecture series are readable up to date introductions into exciting research areas that are inspired by annual programs at the MBI. The pur- pose is to provide curricular materials that illustrate the applications of the mathematical sciences to the life sciences. The collections are organized as independent volumes, each one suitable for use as a module in standard graduatecoursesinthemathematicalsciencesandwritteninastyleaccessi- bletoresearchers,professionals,andgraduatestudentsinthemathematical andbiologicalsciences.TheMBIlecturescanalsoserveasanintroduction for researchers to recent and emerging subject areas in the mathematical biosciences. MartyGolubitsky,MichaelReed MathematicalBiosciencesinstitute Moreinformationaboutthisseriesathttp://www.springer.com/series/13083 MathematicalBiosciencesInstituteLectureSeries Volume1.3:StochasticsinBiological Systems Stochasticity is fundamental to biological systems. In some situations the system can be treated as a large number of similar agents interacting in a ho- mogeneously mixing environment, and so the dynamics are well-captured by deterministic ordinary differential equations. However, in many situations, the systemcanbedrivenbyasmallnumberofagentsorstronglyinfluencedbyan environmentfluctuatinginspaceandtime.Forexample,fluctuationsarecritical in the early stages of an epidemic; a small number of molecules may deter- minethedirectionofcellularprocesses;changingclimatemayalterthebalance among competing populations. Spatial models may be required when agents aredistributedinspaceandinteractionsbetweenagentsarelocal.Systemscan evolvetobecomemorerobustorco-evolveinresponsetocompetitiveorhost- pathogen interactions. Consequently, models must allow agents to change and interact in complex ways. Stochasticity increases the complexity of models in someways,butmayalsosimplifyandsmoothresultsinotherways. Volume1providesaseriesoflecturesbyinternationallywell-knownauthors basedontheyearonStochasticsinBiologicalSystemswhichtookplaceat theMBIin2011–2012. MichaelReed,RichardDurrett Editors MathematicalBiosciencesInstituteLectureSeries Volume1:StochasticsinBiological Systems StochasticPopulationandEpidemicModels LindaJ.S.Allen StochasticAnalysisofBiochemicalSystems DavidAnderson;ThomasG.Kurtz StochasticModelsforStructuredPopulations VincentBansaye;SylvieMe´le´ard BranchingProcessModelsofCancer RichardDurrett StochasticNeuronModeling PricillaGreenwood;LawrenceWard TheMathematicsofIntracellularTransport ScottMcKinley;PeterKramer PopulationModelswithInteraction EtiennePardoux CorrelationsfromCoupledEnzymaticProcessing RuthWilliams Linda J.S. Allen Stochastic Population and Epidemic Models Persistence and Extinction 123 LindaJ.S.Allen DepartmentofMathematicsandStatistics TexasTechUniversity Lubbock,TX,USA ISSN2364-2297 ISSN2364-2300 (electronic) MathematicalBiosciencesInstituteLectureseries ISBN978-3-319-21553-2 ISBN978-3-319-21554-9 (eBook) DOI10.1007/978-3-319-21554-9 LibraryofCongressControlNumber:2015946862 MathematicsSubjectClassification(2010):60J85,92D25,92D30,92D40,34D20 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 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,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Preface Theintentofthismonographistointroducegraduatestudentstobranchingprocess applicationsofpopulationsandepidemics.Deterministicmodelsofpopulationsand epidemicsarewellknowninthescientificliteratureandtheyprovideusefulinfor- mation on the dynamics when population and epidemic sizes are large. However, when sizes are not large, stochastic models and theory are required, for example, to estimate the probability of extinction. The stochastic theory of branching pro- cesses has a long history and can be used as a tool in understanding extinction in manysituations.Inthemid19thcentury,GaltonandWatsonintroducedbranching processes to explain the extinction of family names. Whittle applied the theory in 1955 to an SIR epidemic to estimate the probability of a major outbreak. In this brief monograph, a summary is presented of single-type and multi-type branching process theory. This theory is used to estimate the probability of ultimate extinc- tion in some classic population and epidemic models such as SEIR epidemic and logisticgrowth,andsomenewapplicationsofspeciesinvasionsandspatialspread of disease. Some MatLaB programs of stochastic simulations are provided in the Appendix, and some references are given to additional applications of branching processestopopulationsandepidemics. I thank Rick Durrett and Mike Reed for the invitation to develop these lecture notes as part of the Mathematical Biosciences Institute Graduate Lecture Series, Volume 1: Stochastics in Biological Systems. In addition, I thank Edward Allen, TexasTechUniversity,whoprovidedvaluablefeedbackonChapter4,andananony- mousreviewerwhoprovidedsuggestionsandcorrectionsonChapters1–4.Iespe- ciallythanktheeditorialandproductionstaffatSpringer. Lubbock,TX,USA LindaJ.S.Allen vii Contents 1 Continuous-TimeandDiscrete-StateBranchingProcesses .......... 1 1.1 Introduction ............................................... 1 1.2 Single-TypeBranchingProcesses ............................. 1 1.2.1 Birth-Death ......................................... 5 1.3 Multi-TypeBranchingProcesses.............................. 7 1.3.1 Birth-Death-Dispersal ................................ 9 1.4 Summary ................................................. 12 2 ApplicationsofSingle-TypeBranchingProcesses .................. 13 2.1 Introduction ............................................... 13 2.2 SIRSEpidemic ............................................ 13 2.3 SpeciesInvasion ........................................... 17 2.4 Summary ................................................. 20 3 ApplicationsofMulti-TypeBranchingProcesses................... 21 3.1 Introduction ............................................... 21 3.2 SEIREpidemic ............................................ 22 3.3 EpidemicDispersal......................................... 24 3.4 Summary ................................................. 27 4 Continuous-TimeandContinuous-StateBranchingProcesses ....... 29 4.1 Introduction ............................................... 29 4.2 Single-TypeBranchingProcesses ............................. 30 4.3 Applications............................................... 32 4.3.1 LogisticGrowth ..................................... 32 4.3.2 SIREpidemic ....................................... 34 4.4 Summary ................................................. 34 ix
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