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Interdisciplinary Applied Mathematics 52 Xue-Zhi Li Junyuan Yang Maia Martcheva Age Structured Epidemic Modeling Interdisciplinary Applied Mathematics Volume 52 Editors AnthonyBloch,UniversityofMichigan,AnnArbor,MI,USA CharlesL.Epstein,UniversityofPennsylvania,Philadelphia,PA,USA AlainGoriely,UniversityofOxford,Oxford,UK LeslieGreengard,NewYorkUniversity,NewYork,USA Advisors L.Glass,McGillUniversity,Montreal,QC,Canada R.Kohn,NewYorkUniversity,NewYork,NY,USA P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA AndrewFowler,UniversityofOxford,Oxford,UK C.Peskin,NewYorkUniversity,NewYork,NY,USA S.S.Sastry,UniversityofCaliforniaBerkeley,CA,USA J.Sneyd,UniversityofAuckland,Auckland,NewZealand RickDurrett,DukeUniversity,Durham,NC,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/1390 Xue-Zhi Li • Junyuan Yang (cid:129) Maia Martcheva Age Structured Epidemic Modeling Xue-ZhiLi JunyuanYang CollegeofMathematics ComplexSystemsResearchCenter andInformationSciences ShanxiUniversity HenanNormalUniversity Taiyuan,Shanxi,China Xinxiang,Xinxiang,China MaiaMartcheva DeptofMath,LittleHall358 UniversityofFlorida Gainesville,FL,USA ISSN0939-6047 ISSN2196-9973 (electronic) InterdisciplinaryAppliedMathematics ISBN978-3-030-42495-4 ISBN978-3-030-42496-1 (eBook) https://doi.org/10.1007/978-3-030-42496-1 Mathematics Subject Classification (2010): 34D05, 34D20, 34D23, 34D45, 34F10, 35A09, 35B09, 35B10,35B30,35D30,35B32,35B40,35D35,35E15,37C10,37C75,37G15,45A05,45D05,45E10, 45J05,45P05,46E15,46E30,47A10,47A11,47A75,47B07,47B33,47B34,47B38,47H05,47H09, 47H10,49K40,49G50,62P10,62P25,65F50,65L11,65R10,74G30,74G35,91A05,91A22,92B05, 92D25,92D30,93C20,93D20,93C15 ©SpringerNatureSwitzerlandAG2020 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,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Age-structuredepidemicmodelsdatebacktotheearlytwentiethcenturywhenKer- mackandMcKendrickcreatedtheirwell-knownSIRmodeltoexplaindiseaseout- breakssuchastheplagueorcholera.EvenoneoftheoriginalKermack-McKendrick modelsis,infact,discreteage-since-infectionstructuredtocapturebetterthepoten- tial changes in infectivity and recovery, depending on time-since-infection. Since KermackandMcKendrick,age-structuredepidemicmodelshavehadasteadypres- ence among the infectious disease literature. Their major contribution to mathe- matics was a novel type of partial differential equations with nonlocal boundary conditions,whoseanalysishaschallengedmathematiciansinthelast70years.The mathematicalsetupforthesefirstorderPDEsiscompletelydifferentthanthe“tradi- tionalsecondorderPDEs”—itrequiresworkinginL1 spaceswhicharenotHilbert spacesandarenotreflexive. Interestinage-structuredmodelswasrenewedatthebeginningofthetwenty-first centurywiththeappearanceofseveraltypesofmulti-scaleimmuno-epidemiological models.Allthesenewlydevelopedmodels,someofwhichmaybecomeastandard of modeling in infectious diseases, will continue challenging mathematicians for yearstocome. Therearealotofveryniceintroductionbookstoage-structuredmodeling.Some oftheminvolvetheeditedbook“MathematicalEpidemiology”(Springer),thetext- book “Introduction to Mathematical Epidemiology” by M. Martcheva (Springer, 2015), the “Mathematical Models in Population Biology and Epidemiology” by F. Brauer and C. Castillo-Chavez (Springer, 2002), “The Basic Approach to Age- Structured Population Dynamics” by M. Iannelli and F. Milner (Springer, 2017), and others. The present book is not meant as an introduction to age-structured modeling—it is meant as a tool to develop the skills of graduate students and re- searcherswhowishtomakeage-structuredepidemicmodelingtheirareaofexper- tise.Themathematicaltoolsthatcurrentlyserveage-structuredepidemicmodeling arepresentedinthebookthroughexamples.Manyofthesetoolsalreadyexistinthe literature—publishedinpapers.However,thebookdemonstratesthemondifferent examplesandataslowerpace,withmoreexplanationscomparedtoregulararticles. The presentation of the tools in the book is also graded in difficulty, from regu- v vi Preface lar functional-analytic tools to C -semigroup tools to integrated semigroup tools. 0 The only other book that the authors know of that comes this close in developing the mathematical tools is a recent book by H. Inaba “Age-Structured Population DynamicsinDemographyandEpidemiology”(Springer,2017).Ourgoalwiththe bookisthatthisbookservestotrainawholenewgenerationofmathematicalepi- demiologists, ready to tackle the analytical and numerical challenges of the novel age-structuredpopulationandepidemicmodeling. The book starts with an introduction to chronological age-structured popula- tionmodelswhichisonlymeanttopreparethereadertotacklechronologicalage- structuredepidemicmodels,whicharediscussedinChap.2.Alongwiththemodels, we introduce here a general theorem for proving well posedness of age-structured models,wediscuss,basicanalysis,someearlytoolsfordetecting backward bifur- cationinage-structuredmodelsaswellassettingupandsimulatingoptimalcontrol problems. All these techniques are routinely used with ODE epidemic models but are rarely in the arsenal of mathematical epidemiologists when it comes to age- structuredmodeling.Chapter3focusesonimmuno-epidemiologicalmodeling.We believe that age-structured population modeling, analysis, and simulations will be drivenintheyearstocomeprimarilybydevelopmentsinimmuno-epidemiological or multi-scale modeling. Chapter 3 describes some well-known and some novel (bidirectionallylinked)immuno-epidemiologicalmodelsaswellassomeoftheex- isting analysis tools but much of the tools necessary for the analysis and simula- tionofthesetypesofmodelsarestillyettocome.Someoftheopenquestionshere are:Howdowefindoscillationsinunidirectionallylinkedimmuno-epidemiological models,ifthetoolsdevelopedforregularage-since-infectionmodelsdonotapply here? How do we analyze bidirectionally linked immuno-epidemiological models giventhattheircomplexityishigherandpresentschallengestoexistingtools?Chap- ter4introducesage-since-infectionstructuredepidemicmodelscoupledwithgame theory.Gametheoryhasbeenusedtoanalyzebehavioralimplicationsoninfectious diseaseepidemiology.Itisoftenusedtostudythedecisionspeoplemaketocomply (or not to comply) with some control strategies, such as vaccination. Most of the modelsinthisdirectionareODEmodels.Inthischapter,weintroduceanumberof age-since-infectionstructuredPDEmodelstogetherwiththeappropriatefunctional- analytictoolsfortheiranalysis.Chapter5discussesnetworkage-structuredmodels. Complexnetworksarethestandardofmodelingofcomplexsystemsbutthemod- elsaretypicallysimulationalanddonotlendthemselvestoclosed-formequations thatcanbeanalyzed.Herewepresent,undersomeassumptions,closed-formage- structured epidemic models built on complex, typically scale-free, networks. We developfunctional-analytictoolsfortheiranalysis.Chapter6introducesanumber ofchronologicalageandage-since-infectionstructuredmodelsofvector-bornedis- eases.Althoughmodelslikethesecanbefoundreadilyintheliterature,weusethem asexamplestoillustrateanumberoffunctional-analytictechniques,aswellastech- niquesbasedonC -semigrouptheory.C -semigroupsareanalyticaltoolsthathave 0 0 had long existence but have been applied to age-structured epidemic models only sincethelatetwentiethcentury.OneniceintroductorytextbooktoC -semigroupsis 0 theonebyA.Pazy“SemigroupsofLinearOperatorsandApplicationstoPartialDif- Preface vii ferentialEquations”(Springer,1983)andamorerecentreferencebyK.Engeland R. Nagel “One-Parameter Semigroups for Linear Evolution Equations” (Springer, 2000).C -semigroups arefurtherusedasthemaintoolofanalysis alsoinChap.7 0 whichintroducesandtacklesmetapopulationepidemicmodelsandmulti-groupepi- demic models with chronological age.C -semigroups are a powerful tool to study 0 age-structuredmodelsbuttheyoftenfallshortintheanalysisofage-since-infection or other class-age structured models. The main reason is that class-age structured models often have as a boundary condition of the PDE a term that depends in a nonlinear way on the dependent variables. That, in general, precludes the use of C semigroups,unlessanappropriatechangeofvariablesismadetotransformthe 0 problemintoonewithazeroboundarycondition.Sincethischangeofvariablesis nonobviousorimpossibletomake,thistypeofproblemsarebesttreatedbytheso- called integrated semigroups or semigroups with non-densely defined generators. ThebestintroductionheremaybeanarticlebyH.Thieme“IntegratedSemigroups and Integrated Solutions to Abstract Cauchy Problems” in JMAA, 1990. We use integratedsemigroupstoanalyzeanumberofexampleclass-agemodelsinChap.8. Weincludesomefrequentlyusedmathematicaltoolsintheappendix. Thisbookemergedfromthelong-termfriendshipofitsauthorsanditisaresult ofcollaborationofscientistsfromacrosstheworld,oftensupportedandencouraged by the National Funding Agencies. The authors would like to thank the National ScienceFoundationoftheUSAandtheNationalScienceFoundationofChinafor bringingthemtogether.Further,theauthors thankthewarmhospitalityofUniver- sityofFloridathatmadetheirfacecollaborationonthebookpossible.Theauthors would like to thank their Springer editors Donna Chernyak and Danielle Walker who believed in the book and encouraged them to work. The authors also thank theirfamiliesfortheirpatiencewhiletheauthorswereworkingonthebook. Webelievethatthisbookisuniqueinthesensethatitisperhapsthefirstattempt tomergethebestofthemathematicseducationandtraininginAsia,NorthAmerica, and Europe. The differences in these educational systems made the writing of the booksomewhatchallenging.Wehopewehaverisentothechallengeandthebook that has emerged will contribute seriously to the training of the next generation expertsinage-structuredpopulationandepidemicmodeling. Xinxiang,China Xue-ZhiLi Taiyuan,China JunyuanYang Gainesville,FL,USA MaiaMartcheva Contents 1 Linear Age-Structured Population Models as a Base of Age-StructuredEpidemicModels ................................ 1 1.1 TheLotka-McKendrick-vonFoersterModel .................... 1 1.1.1 TheMcKendrickAge-StructuredPDEModel ............ 1 1.1.2 TheLotkaIntegralEquationModel ..................... 3 1.2 PropertiesoftheSolutionsofLotka-McKendrickModel.......... 4 1.2.1 Existence and Uniqueness of Solutions oftheLotka-McKendrickModel ....................... 5 1.2.2 Long-TermBehavior ................................. 6 1.2.3 PersistentSolutionsoftheLotka-McKendrickModel ...... 8 1.2.4 TheSharpe-LotkaTheorem............................ 9 1.3 CombiningtheAge-StructuredPopulationwithEpidemicModels.. 10 1.3.1 HomogeneousAge-StructuredEpidemicModels.......... 12 1.3.2 Age-StructuredEpidemicModelswithStationaryPopulation 15 1.3.3 Age-StructuredModelswithDisease-InducedMortality.... 16 1.3.4 Age-StructuredEpidemicModelswithConstantTotal BirthRate .......................................... 18 1.4 EarlyAge-StructuredEpidemicModels........................ 19 2 Age-StructuredEpidemicModels ................................ 23 2.1 AnalysisofAge-StructuredEpidemicModels................... 23 2.1.1 WellPosednessofAge-StructuredEpidemicModels ...... 23 2.1.2 ComputingR forAge-StructuredEpidemicModels ...... 27 0 2.1.3 BackwardBifurcationinAge-StructuredModels.......... 30 2.2 AnAge-StructuredSIRModelwithReinfection................. 32 2.2.1 BasicAnalysisoftheSIRModelwithReinfection ........ 33 2.2.2 StabilityoftheDisease-FreeEquilibrium ................ 38 2.3 NumericalMethodsforAge-StructuredEpidemicModels......... 40 2.4 OptimalControlofAge-StructuredModels..................... 42 2.4.1 TheOptimalControlProblem.......................... 43 ix x Contents 2.4.2 DerivingtheAdjointSystemandtheCharacterization oftheControl ....................................... 44 2.4.3 NumericalMethods .................................. 48 2.4.4 NumericalSimulations ............................... 50 2.5 Two-StrainAge-StructuredEpidemicModels ................... 53 2.5.1 Disease-FreeEquilibriumandReproductionNumbers ..... 54 2.5.2 StrainOneandStrainTwoEquilibriaandInvasionNumbers 58 Appendix ...................................................... 60 3 NestedImmuno-EpidemiologicalModels ......................... 69 3.1 NestedImmuno-EpidemiologicalModeling .................... 69 3.1.1 Within-HostModels.................................. 69 3.1.2 ComposingImmuno-EpidemiologicalModels ............ 72 3.1.3 An Immuno-Epidemiological Model of Disease withRecovery....................................... 74 3.2 AnalysisofImmuno-EpidemiologicalModels................... 76 3.2.1 AnalysisoftheSIRImmuno-EpidemiologicalModel ...... 77 3.2.2 StabilityoftheSIImmuno-EpidemiologicalModelofHIV . 80 3.2.3 ImpactofWithin-HostParametersontheBetween-Host Dynamics .......................................... 83 3.3 BidirectionallyLinkedImmuno-EpidemiologicalModels ......... 84 3.3.1 ABidirectionallyLinkedModelofHIV ................. 85 3.3.2 BidirectionallyLinkedImmuno-EpidemiologicalModel ofCholera .......................................... 89 3.4 Immuno-EpidemiologicalMulti-StrainModels.................. 94 3.4.1 An n-Strain Immuno-Epidemiological Competitive ExclusionModel..................................... 95 3.4.2 ATwoStrainModelStructuredbyInoculumFraction ..... 96 3.4.3 Multi-Strain Models with Trade-Off Mechanisms ontheBetween-HostScale ............................ 99 4 Age-Since-InfectionStructuredModelsBasedonGameTheory .....105 4.1 Introduction ...............................................105 4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors BasedontheGameTheory ..................................109 4.2.1 ExistenceofEquilibriaandTheirLocalStability..........115 4.2.2 The Attractivity of Boundary Equilibria and Disease Persistence..........................................124 4.2.3 NumericalSimulations ...............................131 4.2.4 Discussion..........................................133 4.3 ImitationDynamicsintheCaseofVaccinatingSusceptibles.......134 4.3.1 StabilityoftheDisease-FreeEquilibrium ................137 4.3.2 BoundaryEquilibriumandtheEndemicEquilibrium ......140 4.3.3 Discussion..........................................148 4.4 Comparison ...............................................151

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