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Dynamics of Low and High Pathogenic Avian Influenza in Birds

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Title:
Dynamics of Low and High Pathogenic Avian Influenza in Birds
Creator:
Torres, Juan Carlos
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (100 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
MARTCHEVA,MAIA
Committee Co-Chair:
MCKINLEY,SCOTT
Committee Members:
KEESLING,JAMES E
HAGER,WILLIAM WARD
PONCIANO CASTELLANOS,JOSE MIGUEL
Graduation Date:
8/8/2015

Subjects

Subjects / Keywords:
Birds ( jstor )
Cross immunity ( jstor )
Eigenvalues ( jstor )
Infections ( jstor )
Influenza ( jstor )
Pathogens ( jstor )
Poultry ( jstor )
Simulations ( jstor )
Viruses ( jstor )
Wild birds ( jstor )
Mathematics -- Dissertations, Academic -- UF
age-structured-differential-equations -- avian-influenza -- h5n1 -- hpai -- invasion-number -- lpai -- mathematical-models -- reproduction-number
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, Ph.D.

Notes

Abstract:
Influenza A viruses infect many species of wild and domestic birds and are classified into two groups based on their ability to cause disease: low pathogenic avian influenza (LPAI) and high pathogenic avian influenza (HPAI). Prior infection with LPAI provides partial immunity towards HPAI. The first two models (chapters 2 and 3) introduced in this dissertation examine the dynamics of LPAI and HPAI in birds taking into account the cross-immunity provided by prior infection with LPAI. In the first model (chapter 2), cross immunity is treated as a constant parameter. The model is a compartmental ODE model, and a special case of the structured model of the next chapter. The second model (chapter 3) structures the LPAI-recovered individuals by time-since-recovery and involves the cross-immunity that LPAI infection generates toward the HPAI. Reproduction numbers ( RLw0 ,RHw0 ) and invasion reproduction numbers ( ^ RHw , ^ RLw ) of LPAI and HPAI are computed. It is shown that the system has a unique disease-free equilibrium that is locally and globally stable if RLw0 1 a unique LPAI dominance equilibrium exists. Similarly, if RHw0 > 1 a unique HPAI dominance equilibrium exists. The equilibria are locally stable if ^ RHw 1 (RH > 1) and it is locally asymptotically stable if HPAI (LPAI) cannot invade the equilibrium, that is, if the invasion number ^ RHL < 1 ( ^ RLH < 1 ). The pathogens LPAI and HPAI can coexist with sustained oscillations in both populations. It is well known that age structured models involving cross-immunity are capable of oscillations. However, we show that the ODE version of the model, which is obtained by discarding the time-since-recovery structures (making cross immunity constant), also can exhibit oscillations. Through simulations, we show that even if both populations (wild and domestic) are sinks, LPAI and HPAI can persist in both populations combined. Thus, reducing the reproduction numbers of LPAI and HPAI in each populations to below unity is not enough to eradicate the disease. The pathogens can continue to coexists in both populations unless transmission between the populations is reduced. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: MARTCHEVA,MAIA.
Local:
Co-adviser: MCKINLEY,SCOTT.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-02-29
Statement of Responsibility:
by Juan Carlos Torres.

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UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
2/29/2016
Classification:
LD1780 2015 ( lcc )

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DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINBIRDSByJUANCARLOSTORRESADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015JuanCarlosTorres

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Tomyfamilyandfriends

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ACKNOWLEDGMENTS Ithankeveryoneinvolvedinmyeducation.Frommyprofessors,tomyfamily,tomyfriends.IamespeciallygratefultomyadvisorDr.Martcheva.Ithankherforallherguidance,patience,andsupport.IwouldliketothankmycommitteemembersDr.Keesling,andDr.McKinleyforintroducingmetothebeautyofmathematicalbiology.IwouldalsoliketothankmycommitteemembersDr.Hager,Dr.Ponciano,andthemathematicsdepartmentattheUniversityofFloridaforhelpingmealongtheway.Finally,IthankmyparentsandespeciallymybrotherRaul,andmybestfriendAlicia. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1BACKGROUND,PREVIOUSMODELS,MATHPRELIMINARIES ........ 11 1.1AvianInuenzaBackgroundAndModels ................... 11 1.1.1LowPathogenicAvianInuenza .................... 12 1.1.2HighPathogenicAvianInuenza ................... 13 1.1.3CrossImmunity ............................. 13 1.1.4MathematicalModelingofAvianInuenza .............. 14 1.2MathematicalPreliminaries .......................... 16 1.2.1Matrices ................................. 16 1.2.2TheNextGenerationMatrix ...................... 17 2DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINBIRDPOPULATIONS .................................... 19 2.1Introduction ................................... 19 2.2MathematicalModel .............................. 20 2.3TheCompetitionbetweenLPAIandHPAI .................. 21 2.4Discussion ................................... 25 3DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINWILDBIRDPOPULATION ................................. 26 3.1Introduction ................................... 26 3.2TheModel ................................... 26 3.3LPAI-HPAIdynamicsinwildbirds ....................... 27 3.3.1Disease-FreeEquilibrium ....................... 28 3.3.2GlobalStabilityoftheDisease-FreeEquilibrium ........... 31 3.3.3LPAI-onlyandHPAI-onlyEquilibria .................. 33 3.3.4CoexistenceEquilibrium ........................ 37 3.3.5StabilityoftheCoexistenceEquilibrium: ............... 39 3.4NumericalSimulations ............................. 42 3.5Discussion ................................... 48 5

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4DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINWILDANDDOMESTICBIRDPOPULATIONS ...................... 50 4.1Introduction ................................... 50 4.2TheModel .................................... 53 4.3LPAI-HPAIdynamicsinwildanddomesticbirdpopulations ......... 58 4.3.1Disease-FreeEquilibrium ....................... 58 4.3.2LPAI-onlyandHPAI-onlyEquilibria .................. 63 4.3.3CoexistenceEquilibrium ........................ 72 4.4Simulations ................................... 80 4.4.1EstimatingParameterValues ..................... 81 4.4.2Mainquestions ............................. 82 4.4.3SimulationswiththefullODEsystem ................. 83 4.4.4LPAIandHPAIdynamicsinthewildbirdsystemonly ........ 88 4.5Discussion ................................... 91 REFERENCES ....................................... 95 BIOGRAPHICALSKETCH ................................ 100 6

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LISTOFTABLES Table page 3-1Parametervaluesofthewildbirdsmodel( 3 )whichexhibitsoscillations. ... 44 4-1Denitionofthevariables .............................. 56 4-2Denitionoftheparameters ............................. 58 4-3Parameterranges .................................. 82 4-4Source-sinkstatusofbirdstoAIviruses. ...................... 83 4-5Sink-SinkTable .................................... 87 7

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LISTOFFIGURES Figure page 2-1Flow-ChartoftheModel ............................... 20 2-2CoexistenceofLPAIandHPAIinwildbirds .................... 21 2-3RegionsofdominanceandcoexistenceofLPAIandHPAI ............ 23 2-4CoexistenceofLPAIandHPAIintheformofsustainedoscillations ....... 24 3-1OscillationsinWildSystem ............................. 43 3-2LPAIandHPAIprevalenceforthreedifferentvalues ................ 45 3-3Effectofcross-immunityonHPAIprevalence ................... 45 3-4HPAIcoexistenceequilibrium ............................ 46 3-5Theregionsofcoexistenceanddominance .................... 47 4-1FlowChart ...................................... 57 4-2CoexistencewithRealisticParameterValues ................... 85 4-3WildandDomesticOscillations ........................... 86 4-4Sink-Sink ....................................... 87 4-5WildBirdsOnlyOscillations1 ............................ 89 4-6WildBirdsOnlyOscillations2 ............................ 90 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINBIRDSByJuanCarlosTorresAugust2015Chair:MaiaMartchevaMajor:MathematicsInuenzaAvirusesinfectmanyspeciesofwildanddomesticbirdsandareclassiedintotwogroupsbasedontheirabilitytocausedisease:lowpathogenicavianinuenza(LPAI)andhighpathogenicavianinuenza(HPAI).PriorinfectionwithLPAIprovidespartialimmunitytowardsHPAI.Thersttwomodels(chapters2and3)introducedinthisdissertationexaminethedynamicsofLPAIandHPAIinbirdstakingintoaccountthecross-immunityprovidedbypriorinfectionwithLPAI.Intherstmodel(chapter2),crossimmunityistreatedasaconstantparameter.ThemodelisacompartmentalODEmodel,andaspecialcaseofthestructuredmodelofthenextchapter.Thesecondmodel(chapter3)structurestheLPAI-recoveredindividualsbytime-since-recoveryandinvolvesthecross-immunitythatLPAIinfectiongeneratestowardtheHPAI.Reproductionnumbers(RLw0,RHw0)andinvasionreproductionnumbers(^RHw,^RLw)ofLPAIandHPAIarecomputed.Itisshownthatthesystemhasauniquedisease-freeequilibriumthatislocallyandgloballystableifRLw0<1andRHw0<1.IfRLw0>1auniqueLPAIdominanceequilibriumexists.Similarly,ifRHw0>1auniqueHPAIdominanceequilibriumexists.Theequilibriaarelocallystableif^RHw<1(^RLw<1correspondingly).Auniquecoexistenceequilibriumispresentifbothinvasionnumbersarelargerthanone.Simulationsshowthatthiscoexistenceequilibriumcanlosestabilityandcoexistenceintheformofsustainedoscillationsispossible.Cross-immunityanddurationofprotectionincreasetheprobabilityof 9

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coexistence.SimulationsalsoshowthatincreasingLPAItransmissionincreasesLPAIprevalenceanddecreasesHPAIprevalence.ThisobservationinpartmayexplainwhywildbirdswhichhavemuchhighertransmissionofLPAIcomparedtodomesticbirdsalsohavemuchlowerprevalenceofHPAI.Thethirdmodel(chapter4)introducesatime-since-recoverystructured,multi-strain,multi-populationmodelofavianinuenza.ThismodelstructuresLPAI-recoveredbirds(wildanddomestic)withtime-since-recoveryandincludescross-immunitytowardHPAIthatcanfadewithtime.Themodelhasauniquediseasefreeequilibrium(DFE),uniqueLPAI-onlyandHPAI-onlyequilibriaandatleastonecoexistenceequilibrium.WecomputethereproductionnumbersofLPAI(RL),andHPAI(RH)andshowthattheDFEislocallyasymptoticallystablewhenRL<1andRH<1.AuniqueLPAI-only(HPAI-only)equilibriumexistswhenRL>1(RH>1)anditislocallyasymptoticallystableifHPAI(LPAI)cannotinvadetheequilibrium,thatis,iftheinvasionnumber^RHL<1(^RLH<1).ThepathogensLPAIandHPAIcancoexistwithsustainedoscillationsinbothpopulations.Itiswellknownthatagestructuredmodelsinvolvingcross-immunityarecapableofoscillations.However,weshowthattheODEversionofthemodel,whichisobtainedbydiscardingthetime-since-recoverystructures(makingcrossimmunityconstant),alsocanexhibitoscillations.Throughsimulations,weshowthatevenifbothpopulations(wildanddomestic)aresinks,LPAIandHPAIcanpersistinbothpopulationscombined.Thus,reducingthereproductionnumbersofLPAIandHPAIineachpopulationstobelowunityisnotenoughtoeradicatethedisease.Thepathogenscancontinuetocoexistsinbothpopulationsunlesstransmissionbetweenthepopulationsisreduced. 10

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CHAPTER1BACKGROUND,PREVIOUSMODELS,MATHPRELIMINARIES 1.1AvianInuenzaBackgroundAndModelsAvian(bird)Inuenza(u)virusesbelongtoagroupofvirusescalledInuenzaA.Therearethreetypesofinuenzaviruses:InuenzaA,BandC.InuenzaAvirusesinfectmanydifferentavianandmammalianspeciesincludinghumans( 18 ).HumanscanbeinfectedwithallthreetypesofinuenzaviruseswhilebirdscanonlybeinfectedwithInuenzaAvirus( 65 ).InuenzaAstrainsareclassiedbytheirsurfaceproteins:haemagglutinin(HA)andneuraminidase(NA).ThemajorityofallHA/NAcombinationshavebeenisolatedinwildbirdsespeciallywaterfowlsandshorebirds( 1 ; 65 ).ThesubtypeHAhas16distinctmolecules(H1-H16)andNAhas9distinctmolecules(N1-N9).TheHandNcombinationnamesthesubtype,forinstancethevirusthatcausedoneofthedeadliestpandemicsinhistory,whichisknownasthe“SpanishFlu,”wasH1N1subtype.Twoothermajorinuenzapandemicshaveoccurredinthe20thcentury:“AsianInuenza”causedbyH2N2subtypein1957and“HongKongInuenza”causedbyH3N2subtypein1968.Aninuenzapandemichappenswhenanewinuenzasubtypespreadsinthehumanpopulation.Inuenzaviruseshavethecapabilityofevolvingrapidlyandjumpingbetweenspecies.Theyevolvethroughtwoevolutionarymechanisms:driftandshift.Driftissmallandgradualchangeonthesurfaceproteins(antigens)whichoccurbothinInuenzaAandInuenzaBviruses.Shiftevolutionoccursthroughreassortmentwhichisthemixingoftwoinuenzastrainsintoanewstrainwithcapabilitiesofbothstrains.SinceInuenzaAvirusesinfectmanydifferentspecies,shiftoccursonlyinInuenzaAtypeviruses.TheresearchoninuenzasubtypesH2N2andH3N2whichcausedAsianandHongKonginuenzarespectively,recoveredthatthesesubtypeshadsurfaceproteingenesalmostcertainlyfrombothinuenzavirusofavianandhumanorigins( 4 ; 50 ).Thisevidenceindicatesthatantigenicshiftoccurredthroughreassortmentof 11

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bothavianandhumaninuenzaviruses.Ontheotherhand,inuenzasubtypeH1N1thatcausedthe“SpanishFlu”wasmorerelatedtoavianinuenzathantoinuenzafromanyotherspecieswhichsuggestedthatanavianinuenzaviruswasadaptedtohuman-to-humantransmissiblepathogenandcausedapandemic( 2 ).Itishighlypossiblethatthenextpandemicinuenzacanbecausedbyavianinuenza.ThepandemicpotentialofhighlypathogenicavianinuenzaofsubtypeH5N1leadstocontinuedconcerns.ThersthumanavianinuenzacaseofsubtypeH5N1appearedinHongKongin1997,whichcausedthedeathofaboy.Sincethen,asofMarch2015,therearetotalof784reportedH5N1humancasesworldwide,outofwhich429resultedindeath.Eventhough,thenumberofH5N1infectedhumancasesissmall,thecasefatalityrate,whichisapproximately60%isveryhigh.HighlypathogenicavianinuenzaofsubtypeH5N1iscurrentlyatthetopofthelistforapandemicthreat.Themainreasonforthehighpandemicpotentialisthatthevirusiscapableofrapidevolutionandatsomepointmightemergeasaneffectivehuman-to-humantransmissiblepathogenandcauseapandemic.Avianinuenzavirusesarefurtherclassiedintotwogroupsbasedontheirabilitytocausedisease:lowpathogenicavianinuenza(LPAI)andhighpathogenicavianinuenza(HPAI).Virusesofmostsubtypespersistinlowpathogenicform,typicallyproducingasymptomaticormildillnessinwildordomesticbirds.StudiessuggestthatLPAIstrainsfromtheH5andH7subtypescirculatinginwildbirdscanevolveintostrainsofHighlyPathogenicAvianInuenza(HPAI),afterspilloverinfectiontodomesticbirds( 1 ; 10 ; 51 ; 58 ).TheWorldOrganizationforAnimalHealth(OIE)denesavirusoftheH5orH7subtypeasahighlypathogenicavianinuenzavirusifitcancauseatleast75%mortalityin4-weekto8-weekoldchickensinfectedintravenously( 68 ; 58 ). 1.1.1LowPathogenicAvianInuenzaThehostrangeofAIinwildbirdsisnotknownbutsomespeciesofaquaticbirds(suchasducks,geese,andshorebirds),serveasnaturalreservoirsofAIviruses( 10 ; 12

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56 ).AllsubtypesofAIvirusesisolatedthisfarpersistinwildbirdsaslowpathogenicforms.LPAIvirusesoftheH5subtype,whichisofprimaryinteresthere,aregenerallyreportedatverylowprevalenceratesinducks( 31 ; 44 ),buttheprevalencemayvarybylocationoryear.LPAIviruseshavebeenreportedindomesticbirds,mostfrequentlyturkeys,ducksandchickens.AlthoughLPAIvirusesofmanysubtypescanbefoundinpoultry,virusesfromtheH5(andH7)subtypesaremostfrequentlyreportedthere( 59 ). 1.1.2HighPathogenicAvianInuenzaHPAIwasrstdescribedinpoultryinItalyin1878( 39 ).Unlikethelowpathogenicinuenza,whichmostlyaffectstherespiratorytract,thehighlypathogenicforminfectsmultipleorgansandsystemsofinfectedbirds.Inpoultry,HPAIischaracterizedbyhighmortalityrate,oftenover80%within48hours.Before2002,HPAIwasrarelyfoundinwildbirds.Forthisreason,itisstillanopenquestionwhetherHPAIvirusesareendemicinwildbirds.Since2002,HPAIhasbeenisolatedfrommultiplespeciesofwildbirds( 56 ).TheHPH5N1,whichisnowthemaincauseofconcern,isbelievedtohaveemergedfromanLPAIviruscirculatinginchickenssometimein1996( 64 ).Afterpotentiallyundergoingadditionalmutation,theHPAIH5N1virusinfected18peopleinHongKongin1997,sixofwhomdied( 11 ).Since2003HPAIH5N1hasbeenregularlyinfectinghumansprimarilythoughbird-to-humaninfection.Theemergenceofavirusthatcanpassdirectlyfrombirdstohumansshowedthatpigsarenotanecessarylinkinthechain.Sincethen,virusesofothersubtypeshavealsomadethetransitionLPAI!HPAIandcanposeasignicantthreat( 50 ). 1.1.3CrossImmunityInfectionwithonestrainofInuenzaAtypevirusesprovidescrossprotectionagainstinfectionwithantigenicallysimilarstrains.Thepartialprotectionofferedbythecrossimmunityishighonlyifthenewstraininfectingthehostandthestrainresponsibleforimmunityarecloselyrelated.Theprotectionisineffectiveagainstviruseswithmajorantigenicdivergence( 6 ; 7 ).However,theuniqueevolutionarycapabilitiesofInuenza 13

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typeAviruses(driftandshift)stillcausesaconcern,sincepriorinfectionmayonlyreducethevirusspreadbutcannotpreventtheinfection( 15 ).Studiesin( 54 )and( 15 )suggestthatpriorinfectionwithLPAIprovidescrossprotectionagainstinfectionwithHPAI.JustbeforetherstH5N1humancasein1997,therewasH5N1poultryoutbreakinchickenfarmsinHongKong.Eventhough,evidenceshowedthepresenceofH5N1virus,mostofthechickensinthepoultrymarketsdidnotshowanysymptomsandappearedhealthy( 53 ).SeoandWebster,motivatedbythispuzzlingsituation,setupanexperimenttotestthehypothesisthatpriorinfectionwithH9N1(whichisLPAI)providespartialprotectionagainsthighlypathogenicavianH5N1infections( 54 ).Therearemainlytwotypesofimmunitycausedbypriorinfection:cellmediatedimmunityandantibodymediatedimmunity.ThepriorinfectionwithLPAIprovidescellmediatedimmunitytowardHPAI( 54 ; 15 ).TheprotectionmediatedbycellularimmuneresponseisestablishedbyCD8+Tcells.Thesendingsoftheresearchdonebytheauthorsin( 54 ; 15 )indicatethatitispossibletoinducecellmediatedimmunitytowardhighpathogenicavianinuenzavirusesbylowpathogenicavianinuenza.Intermsofcontrolmeasures,itispromisingtodevelopvaccinesthatemphasizecellmediatedcellularimmunity. 1.1.4MathematicalModelingofAvianInuenzaAvianinuenzaisperhapsthemostdangerousdiseaselinkinghumansandanimalsatpresent.Becauseofitsdeadlypandemicpotential,eveninitscurrentpre-pandemicstageithascausedsignicanthardshipandeconomicloss( 46 ).Inthelast5-10yearsavianinuenzahasenjoyedsignicantattentionfrommathematicalmodelers,largelyduetoitskeypositionamonginfectiousdiseases.Earlymodelsofavianinuenzafocusedonhumans,investigatingthepotentialimpactofahypotheticalpandemicandexploringstrategiesforitspossiblemitigation( 17 ; 20 ; 19 ; 37 ; 43 ).Othermodelsfocusedonthepresentstatusquocenteredoninfectionofdomesticbirds,andcurrentcontrolstrategieswhichprimarilytargetpoultry. 14

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Forexample,aspatialfarm-basedmodeltreatingpoultry-farmsasunits( 34 )andSIRmodelsforwithin-ocktransmissionofH5N1( 61 )weredeveloped.Despitetheimportanceofanumberofemergentdiseases,manyofwhicharisefromspilloverinfectionsfromanimals,fewmodelstodateinvolvebothanimalsandhumanslinkedbyapathogen.Thissituationhasbeenchangingrecently,particularlyinrelationtoAI.Thesimplestmodelthatcapturesabird-to-humantransmissionpathwayofHPAIinvolvesdomesticbirdsandhumans( 24 ).ThedynamicsofHPAIinpoultryaregivenbyasimpleSImodel,asinfecteddomesticbirdseitherdiefromtheinfectionorareculledtopreventfurtherspread.Tointroducethemodel,wedenotethenumberofsusceptibledomesticbirdsbySd(t),andthenumberofdomesticbirdsinfectedbyHPAIbyIHd(t).Furthermore,wedenotesusceptiblehumansbyS(t),andhumansinfectedwithHPAIbyI(t).TheAImodelconsistsofthefollowingtwosystems: Domesticbirds:8>>><>>>:dSd dt=d)]TJ /F7 11.955 Tf 11.95 0 Td[(HdIHdSd)]TJ /F7 11.955 Tf 11.95 0 Td[(dSd,dIHd dt=HdIHdSd)]TJ /F6 11.955 Tf 11.95 0 Td[((d+Hd)IHd(1)wheretheparametersforthedomesticbirdpopulationare:d–therecruitmentrateofdomesticbirds,d–thenaturaldeathrateofdomesticbirds,Hd–thetransmissioncoefcientofHPAIamongdomesticbirds,andHd–thedeathrateofdomesticbirdsduetoHPAI. Humans:8>>><>>>:dS dt=)]TJ /F7 11.955 Tf 11.96 0 Td[(IHdS)]TJ /F7 11.955 Tf 11.96 0 Td[(SdI dt=IHdS)]TJ /F6 11.955 Tf 11.96 0 Td[((++)I(1)Theparametersrelatedtohumandynamicsare:–thebirth/recruitmentrateofhumans,–thenaturaldeathrateofhumans,–thetransmissioncoefcientofHPAIfrombirdstohumans,–thedeathrateofhumansduetoHPAI,and–therecoveryrate.Model( 1 1 )hasverysimpledynamics.However,itfailstocapturemostofthecomplexityandcharacteristicfeaturesofAItransmissionandevolution.Recognizingthe 15

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importanceofbothbirdsandhumansinthetransmissionandevolutionofAI,anumberofmodelsinvolvingdomesticbirdsandhumans( 24 ; 26 ; 25 ; 30 )weredeveloped.Somemoreelaboratemodelseveninvolvewildbirds,aswellasdomesticbirdsandhumans( 38 ; 36 ; 21 ; 3 ).ThesemodelstypicallyincorporatethehypotheticalscenarioinwhichHPH5N1avianinuenzabecomesadaptedtohumansandstartstransmittingefcientlyfromhuman-to-human.Thismodelstheshiftevolutionarymechanism.ThedriftevolutionarymechanismwasoriginallymodeledbyPease( 48 ).Pease'sdriftmodelhasrecentlybeenextensivelystudied( 23 ; 40 ).AnovelmodelcombiningeffectsofbothdriftandshiftevolutionofinuenzaAwasdiscussedbyMartcheva( 41 ),whereitwasfoundthatdriftevolutionmayberesponsibleforthe365-dayoscillationofhumanu,aswellas365-dayoscillationsofthenumberofhumansinfectedwithHPH5N1,asobservedindata.InthisdissertationweinvestigatethedynamicsofLPAIandHPAIinbirds.TheinteractionbetweenLPAIandHPAIinbirdshasbeenpriorlystudiedbyseveralauthors( 38 ; 5 )buttheeffectofcross-immunitythatLPAIgivestoHPAIhasnotbeeninvestigated. 1.2MathematicalPreliminaries 1.2.1MatricesInthefollowingdefenitions,thematrixAhaseigenvalues1,2,....,n.Denition.ThespectralradiusofAis(A):=maxijij.Denition.ThespectralboundofAism(A):=supfReigDenition.Amatrixiscalledpositiveifallofit'sentriesarepositive.Denition.AisprimitiveifthereexistsapositiveintegerksuchAkispositive. Theorem1.1(Perron-FrobeniusTheorem). IfAisaprimitivennmatrix,thenAhasapositiveeigenvaluewiththefollowingproperties: (i) isasimplerootofthecharacteristicpolynomialofA, (ii) hasapositiveeigenvectorv, 16

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(iii) anyothereigenvalueofAhasmodulusstrictlylessthan, (iv) anynon-negativeeigenvectorofAisapositivemultipleofv. 1.2.2TheNextGenerationMatrixThissubsectiongivesaquickviewatonewayofdeningthereproductionnumberforacompartmentalmodelofdiseasetransmission.Inthefollowing,letx2Rnrepresenttheinfectedcompartmentsandy2Rmrepresentthenon-infectedcompartments.x0i=fi(x,y))]TJ /F10 11.955 Tf 11.96 0 Td[(vi(x,y)i=1,...,n (1)y0j=gj(x,y)j=1,...,m (1)wheretheratesoftheinfectedcompartmentshavebeenseparatedinto,fi(x,y),therateofnewlyinfectedincompartmenti,andvi(x,y),theremainingtransitionaltermssuchasdiseaseprogression,death,orrecovery.LetthematricesFandVbedenedasfollows:F=@fi(0,y0) @xjandV=@vi(0,y0) @xjwhere(0,yo)isthediseasefreeequilibrium.Fromthese,thenextgenerationmatrixKisdenedas:K=FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1andthereproductionnumberR0=(K),isthespectralradiusofK.Notethatthelinearizationoftheinfectedcompartmentsatthediseasefreeequilibriumcanbewrittenasx0=(F)]TJ /F10 11.955 Tf 11.95 0 Td[(V)x.Thenwehavethefollowingstatements: 17

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Theorem1.2. m(F)]TJ /F10 11.955 Tf 12.64 0 Td[(V)<0iff(FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1)<1,andm(F)]TJ /F10 11.955 Tf 12.64 0 Td[(V)>0iff(FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1)>1wherem(A),thespectralboundand(A),thespectralradiusaredenedintheprevioussubsection.ThisallowsustouseR0asacriterionforlocalstability. Theorem1.3. IfR0<1thenthediseasefreeequilibriumislocallyasymptoticallystable,andifR0>1thenthediseasefreeequilibriumisunstable. 18

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CHAPTER2DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINBIRDPOPULATIONS 2.1IntroductionTheHPAIH5N1isdangerousbecauseitcanmutatetobecomeeffectivelyhuman-to-humantransmissible.TheemergenceofapandemicHPAIH5N1strainmayhappeninoneoftwoways:(1)reassortmentand(2)mutation.Evenifthemortalityofthepandemicstrainismuchsmallerthantheoneofthecurrentbird-to-humantransmittablestrains,itwillkillmillionsofpeople.Becauseofitsimportancetopublichealth,AIhasbeenextensivelymodeled.TherstfewarticleswerepublishedbyIwamietal.( 24 ; 25 ; 26 ).Theauthorsintroducedasimpledomesticbirds-humansmodel,inwhichthecurrentcirculatingHPAIH5N1strainmutatesintoahuman-to-humantransmittablestrain.TheyconcludethatcontinuingcullingofH5N1-infecteddomesticbirdsinthefaceoftheemergenceofapandemicstrainwillincreasethechanceofthatpandemicstrainandwillresultofhigherprevalenceofpandemicstraininfectedhumans.UsingH5N1humancasedata,Martcheva( 42 )showsthattheinterferencethatthebird-to-humanstrainexercisesoverthepandemicstrainisverylowandshouldnotinuencethedecisiontocontinueculling.HPAIusuallyevolvesfromLPAIindomesticbirds.However,itistheHPAIthatisdangerousanddeadlybothforhumansandforpoultry.PriorexposuretoLPAIbothfordomesticandforwildbirdscanmakeasubsequentinfectionwithHPAImuchmoremildandevenasymptomatic.Thispropertyiscalledcross-immunity,thatisLPAIprovidespartialprotectionandcross-immunitytoHPAI.SeveralarticleshavestudiedtheinteractionbetweenLPAIandHPAI.Lucchettietal( 38 )showthatHPAIpersistsinpoultrybutLPAIcanbefoundinpoultryonlybecauseofspill-overinfectionfromwildbirds.Bourouibaetal( 5 )investigatestheimpactofcross-immunityonthetwotypesofAI. 19

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2.2MathematicalModelTounderstandtheinterplaybetweenLPAIandHPAIweintroduceasimplemathematicalmodelofcross-immunitybetweenLPAIandHPAI.Tobuildthemodel,weassume: InfectedbirdswithHPAIcanrecover.Thatassumptionisvalidforwildbirdsandmayholdforvaccinatedpoultry InfectedbirdswithLPAIcanrecover RecoveredfromLPAIbirdscangetinfectedwithHPAIbutnotviceversa,thatis,weassumethatonceinfectedwithHPAIthebirdexitsthesystem. RecoveredbirdsfromLPAIcangetinfectedwithHPAIatalowerprobability.WedonotkeeptrackoftherecoveryrateofbirdsinfectedwithHPAIwhichhavebeenpriorlyinfectedwithLPAI.Theow-chartofthemodelisshowninFigure 2-1 .Intheowchart,Sisthenumberof Figure2-1. Flow-ChartoftheModel susceptiblebirds,IListhenumberofbirdsinfectedwithLPAI,IHisthenumberofbirdsinfectedwithHPAI,RListhenumberofbirdsrecoveredfromLPAI,RHisthenumberof 20

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birdsrecoveredfromHPAI.Themodeltakestheform. 8>>>>>>>>>><>>>>>>>>>>:S0=)]TJ /F7 11.955 Tf 11.96 0 Td[(LILS)]TJ /F7 11.955 Tf 11.95 0 Td[(HIHS)]TJ /F7 11.955 Tf 11.96 0 Td[(S,I0L=LILS)]TJ /F6 11.955 Tf 11.96 0 Td[((+L)IL,R0L=LIL)]TJ /F10 11.955 Tf 11.96 0 Td[(qHIHRL)]TJ /F7 11.955 Tf 11.95 0 Td[(RL,I0H=HIHS+qHIHRL)]TJ /F6 11.955 Tf 11.96 0 Td[((+H+H)IH,R0H=HIH)]TJ /F7 11.955 Tf 11.96 0 Td[(RH(2)HereLandHarethetransmissionratesofLPAIandHPAIrespectively,LandHaretherecoveryratesofLPAIandHPAI,Histhedisease-inducedmortalityofHPAIandqisthereducedsusceptibilitywhenpriorlyinfectedwithLPAIbirdsgetinfectedwithHPAI.istherecruitmentrateandisthedeathrateofbirds. 2.3TheCompetitionbetweenLPAIandHPAIDynamically,long-term,therearefouroptionsfortheLPAIandHPAI:(1)BothLPAIandHPAIdeclinetozero;(2)LPAIpersists,HPAIdeclinestozero;(3)HPAIpersists,LPAIdeclinestozero;(4)BothLPAIandHPAIpersist.Thislastscenario,calledcoexistence,isillustratedinFigure 2-2 .TheprevalenceofLPAIislargerthantheprevalenceofLPAI,whichisthecaseinwildbirdsinnature. Figure2-2. CoexistenceofLPAIandHPAIinwildbirds.RedlineisHPAI,bluelineisLPAI Theseoutcomescorrespondtofourtypesofequilibriaofmodel( 2 ).ThersttypeistheDisease-FreeEquilibrium(DFE),E0=(=,0,0,0,0).Thisequilibrium 21

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correspondstothecasewhenbothLPAIandHPAIdieoutlongterm.ThestabilityofthisequilibriumdependsonthereproductionnumbersofLPAIandHPAI: RL=L (+L)RH=H (+H)(2)ThereproductionnumberofLPAI(HPAI)givesthenumberofsecondaryinfectionsoneinfectedwithLPAI(HPAI)birdwillproduceinanentirelysusceptiblebirdpopulation.Itcanbeshown( 62 )thatifRL<1andRH<1thentheDFEislocallyasymptoticallystable.IfRL>1orRH>1,thenitisunstable.Moreover,itcanbeshownthatifRL<1andRH<1thentheDFEisgloballyasymptoticallystable.Ideally,throughcontrolmeasureswewouldliketopushthereproductionnumbersofbothLPAIandHPAIbelowone.However,thatmaybedifcultinpractice.Furthermore,wendthatthesystemhasLPAI-onlyequilibriumandHPAI-onlyequilibrium.TheLPAI-onlyequilibriumisgivenbyEL=(S,IL,RL,0,0)whereS=+L L,IL= L(RL)]TJ /F6 11.955 Tf 11.95 0 Td[(1),RL=L L(RL)]TJ /F6 11.955 Tf 11.96 0 Td[(1).ClearlytheLPAI-onlyequilibriumexistsifandonlyifRL>1.TheHPAI-onlyequilibriumisgivenbyEH=(S,0,0,IH,RH)whereS=+H+H H,IH= H(RH)]TJ /F6 11.955 Tf 11.96 0 Td[(1),RH=H H(RH)]TJ /F6 11.955 Tf 11.96 0 Td[(1).ClearlytheHPAI-onlyequilibriumexistsifandonlyifRH>1.Thestabilityofthesemi-trivialequilibriaand,therefore,theoutcomeofthecompetitionbetweenLPAIandHPAIisgivenbytheinvasionreproductionnumbers.TheinvasionreproductionnumberofLPAI(HPAI)attheequilibriumofHPAI(LPAI),denotedbyRL(RH),givesthenumberofsecondarycasesoneLPAI-infectedbirdwillproduceinapopulationinwhichtheHPAIisatequilibrium.Thevaluesoftheinvasionnumbersaregivenbelow: 22

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RL=RL RHRH=RH RL1+qL (L+)(RL)]TJ /F6 11.955 Tf 11.95 0 Td[(1)Furthermore,theLPAIdominanceequilibriumELislocallyasymptoticallystableifRH<1,thatisiftheHPAIcannotinvadetheequilibriumoftheLPAI.TheLPAIequilibriumisunstableifRH>1.Similarly,theHPAIdominanceequilibriumEHislocallyasymptoticallystableifRL<1,thatisiftheLPAIcannotinvadetheequilibriumoftheHPAI.TheHPAIequilibriumisunstableifRL>1.Itcanbeshownthatifbothinvasionnumbersaregreaterthanone,RH>1andRL>1,thenauniquecoexistenceequilibriumexists,E=(S,IL,RL,IH,RH).TogetabetterideaoftheparameterspacewhereLPAI-onlyexists,HPAI-onlyexistsorthetwocoexist,inthe(RL,RH)-plane,weplottheregionsofcoexistenceanddominance.TheseregionsaregivenbytheequalitiesRL=1andRH=1.TheresultinggureisgivenasFigure 2-3 . Figure2-3. RegionsofdominanceandcoexistenceofLPAIandHPAI ExaminingFigure 2-3 ,wecanmakeseveralinterestingconclusions.First,wemaynoticethatregionofparameterspaceforwhichHPAIpersistismuchlarger. 23

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Thus,HPAIispresentformorevaluesofRLandRH.Second,thepresenceofLPAIactuallyincreasestheareawhereHPAIpersists.Thatisvisiblefromthecoexistencearea.Third,simulationssuggeststhatthelargertheq,thelargerthecoexistenceareaandthesmallertheareawhereLPAIpersistsalone.Ifq=0thentherewillnotbecoexistencebutcompetitiveexclusionbetweentheLPAIandHPAI.Inthiscasewecallcross-immunityacoexistencemechanismbecauseitspresenceleadstocoexistence.Whencompetitiveexclusionistheonlypossibleoutcome,thatisinthecaseq=0,thenwhetherLPAIpersistsorHPAIpersistsisdeterminedbytheirreproductionnumbers.TheAIwiththelargerreproductionnumberwillpersistsaslongasthisreproductionnumberisaboveone.Inthelate1980'sCarlos-Castillo-Chavezetal( 45 )investigatedthequestionwhethercross-immunitymayberesponsiblefortheoscillationsobservedininuenzadynamics.TheyconcludedthatasimpleODEmodelwithcross-immunitycannotproducesustainedoscillations.Surprisingly,thisisnotthecaseformodel( 2 ).WendthatthecoexistenceequilibriumcanbedestabilizedthroughHopfbifurcation.InthiscaseoscillationsinwhichbothLPAIandHPAIpersistoccur.Thisdestabilizationseemstoneedq1,thatisitoccursinthecasewhenLPAIprovidesnearlynoprotectionagainstHPAI.TheoscillationsareshowninFigure 2-4 . Figure2-4. CoexistenceofLPAIandHPAIintheformofsustainedoscillations 24

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2.4DiscussionWeintroduceanLPAIandHPAImodelwhichdescribeswellthetransmissionofthetwopathogenvariantsbothinwildanddomesticbirds.ThemodelhasauniqueDFEwhichisgloballystableifthetworeproductionnumbersarebelowone.ThemodelhasLPAIandHPAIdominanceequilibriawhichexistwhentheLPAI(HPAI)reproductionnumberisaboveone.TheLPAI(HPAI)dominanceequilibriumislocallyasymptoticallystableiftheinvasionreproductionnumberoftheHPAI(LPAI)isbelowone.Themodelhasauniquecoexistenceequilibriumifbothinvasionnumbersareaboveone.ThecoexistenceequilibriumcanbecomeunstableandpersistenceofLPAIandHPAIispossibleintheformofsustainedoscillations.LPAIandHPAIcompeteforsusceptibleindividuals:IncreasingprevalenceofLPAIdecreasesprevalenceofHPAI.ThismayexplainwhyhigherprevalenceofLPAIleadstolowerprevalenceofHPAIinwildbirdsandviceversaindomesticbirds. 25

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CHAPTER3DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINWILDBIRDPOPULATION 3.1IntroductionInuenzastrainscancompeteintwoways:depletionofsusceptiblehosts,andcross-immunity( 6 ; 7 ; 45 ).Cross-immunityisamechanismbywhichinfectionwithonestrainprovidespartialprotectionagainstinfectionwithanother.AnumberofreferencesinthebiologicalliteraturesuggestthatinfectionwithLPAIcanprovidepartialprotectionagainstHPH5N1bothinpoultryandinwildbirds.SeoandWebster( 54 )infectedagroupofchickenswithLPAIH9N2inuenzavirus.TheyfoundthatthegroupinfectedbyH5N1within30daysofinoculationbyLPAIhada100%survivalrateandreducedclinicalsigns.Astimebetweenthetwoinfectionsgrew,theprotectionstartedtofadeandthemorbidityofinfectionwithHPAIgrew.ThisstudysuggeststhattheimmunityprovidedbypriorinfectionwithLPAIistemporary,decliningwithtime-since-recoveryfromLPAI. 3.2TheModelToincorporatethevariabilityofcross-immunityinwildbirds,weletbethetime-since-recoveryfromlowpathogenicinuenzaandqw()bethevariablecross-immunity(highcross-immunityismodeledbylowqw)impartedfromLPAItoinfectionwithHPAI.Tointroducethemodel,letthebirth/recruitmentrateofwildbirdsbedenotedbywandthenaturalmortalityrateforwildbirdsbyw.ThenumberofsusceptiblewildbirdsisSw(t),thenumberofLPAIinfectedwildbirdsisILw(t),andthenumberofHPAIinfectedwildbirdsisIHw(t).ThedensityofindividualsthathaverecoveredfromLPAIisrLw(,t)andthenumberofindividualsthathaverecoveredfromHPAIisRHw(t).Thedynamicsof 26

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LPAIandHPAIinwildbirdsarecapturedbythefollowingmodel 8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:dSw dt=w)]TJ /F7 11.955 Tf 11.95 0 Td[(L11ILwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,dILw dt=L11ILwSw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)ILw@rLw @t+@rLw @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()H11IHwrLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wrLwrLw(0,t)=LwILwdIHw dt=H11IHwSw+H11IHwR10qw()rLw(,t)d)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)IHwdRHw dt=HwIHw)]TJ /F7 11.955 Tf 11.95 0 Td[(wRHw.(3)whereLwandHwaretherecoveryratesforwildbirds,andHwistheHPAI-inducedmortality.AsimplifyingassumptionisthatmostwildbirdsthatrecoverfromHPAIdosobecausetheymayhavehadpriorexposuretoLPAI.ThisassumptionisjustiedbasedontheexperimentalstudiesthatsuggestthatLPAIstrainsofthesamesubtypeordifferentsubtypeinducepartialcross-immunitytowardinfectionwithHPAIstrains( 16 ; 28 ).Wemakethesimplifyingassumptionthatrecruitmentdoesnotdependuponfocalbirdnumbers.Thiscouldberelaxedinfuture,forinstancebyintroducingalogisticgrowthterm.Inthefollowingsectionweanalyzethedynamicsoflowpathogenicandhighpathogenicavianinuenzawithinthewildbirdpopulation.ThemodelthatdescribestheinteractionofLPAIandHPAIinpoultryisquitesimilar.Therefore,symmetricalresultsaretrueforthedomesticbirdpopulation. 3.3LPAI-HPAIdynamicsinwildbirdsWestudytheexistenceandstabilityofequilibriaofthewildbirdsystem( 3 ).Todeterminetheequilibria,wesolvethefollowingsystemwhichisobtainedbysettingthe 27

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timederivativesin( 3 )equaltozero. 8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:0=w)]TJ /F7 11.955 Tf 11.95 0 Td[(L11ILwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,0=L11ILwSw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)ILw@rLw @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()H11IHwrLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wrLwrLw(0)=LwILw0=H11IHwSw+H11IHwR10qw()rLw(,t)d)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)IHw0=HwIHw)]TJ /F7 11.955 Tf 11.95 0 Td[(wRHw.(3)Thewildbirdsystemhas4equilibria.Therstisthedisease-freeequilibrium.ThesecondandthethirdaretheLPAI-onlyandHPAI-onlyequilibria(i.e.boundaryequilibria).Thefourthoneisthecoexistenceequilibrium(i.e.interiorequilibrium)whichrepresentsthestateinwhichbothLPAIandHPAIarepresentinthewildbirdpopulation. 3.3.1Disease-FreeEquilibriumThewild-bird-model( 3 )hasadisease-freeequilibrium(DFE)"0givenby"0=(Sw,0,0,0,0),whereSw=w w.ThebasicreproductionnumberforLPAIinwildbirds,denotedbyRLw0,isgivenbyRLw0=L11w w(w+Lw),andthebasicreproductionnumberforHPAIinwildbirds,denotedbyRHw0,isgivenbyRHw0=H11w w(w+Hw+Hw). 28

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ThesebasicreproductionnumbersarethresholdvalueswhichdeterminewhetherLPAIorHPAIcaninvadethedisease-freeequilibrium. Theorem3.1. IfRLw0<1andRHw0<1thentheDFE,"0,islocallyasymptoticallystable. Proof. Let(u,v,x,y,z)betheperturbationsaroundthesteadystate.ExpressingtheperturbationsasSw(t)=Sw+u(t),ILw(t)=v(t),rLw(,t)=x(,t),IHw(t)=y(t),RHw(t)=z(t),wesubstituteintothePDEsystem( 3 ).Usingtheequationforthedisease-freeequilibriumanddroppingthequadratictermsintheperturbations,weobtainthefollowinglinearsysteminvolvingonlyperturbations.du dt=L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F7 11.955 Tf 11.95 0 Td[(wu,dv dt=L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)v@x @t+@x @=)]TJ /F7 11.955 Tf 9.3 0 Td[(wx (3)x(0,t)=Lwvdy dt=H11Swy)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)ydz dt=Hwy)]TJ /F7 11.955 Tf 11.96 0 Td[(wzToinvestigatethelocalstabilityoftheDFE,westudythesolutionsofthesystem( 3 ).Supposethatthelinearsystem( 3 )hasexponentialsolutions,thatiswelookforsolutionsofthefollowingform: u=uet,v=vet,x=x()et,y=yet,z=zet.(3) 29

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Substitutingtheabovesolutionsintolinearizedsystem( 3 ),wegetthefollowingeigenvalueproblem;u=)]TJ /F7 11.955 Tf 9.29 0 Td[(L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F7 11.955 Tf 11.95 0 Td[(wu,v=L11Swv)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)vy=H11Swy)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)y (3)z=Hwy)]TJ /F7 11.955 Tf 11.96 0 Td[(wzwhichiscombinedwiththefollowingrstorderODE;x+dx d=)]TJ /F7 11.955 Tf 9.3 0 Td[(wx,x(0)=Lwv.Solvingtheabovedifferentialequation,weobtain;x=Lwve)]TJ /F5 7.97 Tf 6.59 0 Td[((+w).Solutionsof( 3 )givetheeigenvaluesfig4i=1ofthelinearizeddifferentialoperatorin( 3 ).Theeigenvalueproblem( 3 )islinear.Thesecondequationinvolvesonlyvandisindependentfromu,yandz.Similarstatementistrueforthethirdequationinvolvingonlyy.Solvingthesecondequationweget2=L11Sw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw),whichisnegativesinceRLw0<1.Solvingthethirdequationweget3=H11Sw)]TJ /F6 11.955 Tf -424.9 -23.9 Td[((w+Hw+Hw).Clearly,3<0,sinceRHw0<1.Theothertwoeigenvaluesare1=4=)]TJ /F7 11.955 Tf 9.3 0 Td[(w<0. ThebasicreproductionnumberRLw0(RHw0)measurestheaveragenumberofnewlow(high)pathogenicinfectionsgeneratedbyasinglewildbirdinfectedwithlow(high)pathogenicavianinuenzainacompletelysusceptiblewildbirdpopulation. 30

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Thus,Theorem 3.1 impliesthattheLPAIandHPAIcanbeeliminatedfromthewildbirdpopulationifRLw0<1andRHw0<1andifinitiallythenumberofwildbirdsinfectedwithLPAIandHPAIareinthethebasinofattractionoftheDFE. 3.3.2GlobalStabilityoftheDisease-FreeEquilibriumNow,weprovetheglobalasymptoticstabilityofthedisease-freeequilibrium"0. Theorem3.2. IfRLw0<1andRHw0<1thentheDFE,"0,isgloballyasymptoticallystable. Proof. Fromtherstequationinthesystem( 3 )weobtainthefollowinginequalityS0ww)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw.Fromthisinequalitywehave limsuptSw(t)w w.(3)Integratingthesecondequalityinsystem( 3 )wehaveILw(t)=e)]TJ /F5 7.97 Tf 6.59 0 Td[((w+Lw)tILw(0)+L11Zt0e)]TJ /F5 7.97 Tf 6.58 0 Td[((w+Lw)(t)]TJ /F14 7.97 Tf 6.59 0 Td[()Sw()d.Changingthevariableofintegrationintheintegral,wehaveILw(t)=e)]TJ /F5 7.97 Tf 6.59 0 Td[((w+Lw)tILw(0)+L11Zt0e)]TJ /F5 7.97 Tf 6.58 0 Td[((w+Lw)Sw(t)]TJ /F7 11.955 Tf 11.96 0 Td[()d.Takingalimsupofbothsidesofthisequality,weobtainthefollowinginequality:limsuptILw(t)L11w w(w+Lw)limsuptILw(t)wherethecoefcientinfrontthelimsupontherighthandsideisexactlyRLw0.SinceRLw0<1,thisinequalityisonlypossibleiflimsuptILw(t)=0.Hence,ILw(t)!0ast!1.Next,weintegratethepartialdifferentialequationalongthecharacteristiclines. 31

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Weobtain, rLw(,t)=8><>:LwILw(t)]TJ /F7 11.955 Tf 11.95 0 Td[()e)]TJ /F15 7.97 Tf 8 6.42 Td[(R0qw()IHw(t)]TJ /F14 7.97 Tf 6.59 0 Td[(+)d)]TJ /F14 7.97 Tf 6.59 0 Td[(wt(3)wherer0()=rLw(,0).ConsiderthetermR10qw()rLw(,t)d.WeclaimlimsuptZ10qw()rLw(,t)d=0.Indeed,using( 3 )wehave limsuptR10qw()rLw(,t)dLwRt0qw()ILw(t)]TJ /F7 11.955 Tf 11.95 0 Td[()e)]TJ /F14 7.97 Tf 6.59 0 Td[(wd+R1tqw()r0()]TJ /F10 11.955 Tf 11.96 0 Td[(t)e)]TJ /F14 7.97 Tf 6.58 0 Td[(wtdLwRt0qw()ILw(t)]TJ /F7 11.955 Tf 11.95 0 Td[()e)]TJ /F14 7.97 Tf 6.59 0 Td[(wd+e)]TJ /F14 7.97 Tf 6.58 0 Td[(wtR10r0()d.(3)Takingthelimsupfrombothsidesoftheaboveinequalitygivestheclaim.WedenotebyQ(t)=Z10qw()rLw(,t)d.Toconcludetheproof,weintegratetheequationforhighpathogenicinuenzaandchangethevariableofintegration. IHw(t)=e)]TJ /F5 7.97 Tf 6.59 0 Td[((w+Hw+Hw)tIHw(0)+H11Rt0e)]TJ /F5 7.97 Tf 6.59 0 Td[((w+Hw+Hw)Sw(t)]TJ /F7 11.955 Tf 11.96 0 Td[()IHw(t)]TJ /F7 11.955 Tf 11.96 0 Td[()d+H11Rt0e)]TJ /F5 7.97 Tf 6.59 0 Td[((w+Hw+Hw)Q(t)]TJ /F7 11.955 Tf 11.95 0 Td[()IHw(t)]TJ /F7 11.955 Tf 11.95 0 Td[()d.(3)Takinglimsupofbothsidesoftheaboveequality,wehavelimsuptIHw(t)H11w w(w+Hw+Hw)limsuptIHw(t).Thecoefcientontheright-handsideinfrontlimsuptIHw(t)isexactlyRHw0.SinceRHw0<1,theonlywaytheaboveinequalitycanholdisiflimsuptIHw(t)=0,thatisifIHw(t)!0ast!1.Thatcompletestheproof. 32

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3.3.3LPAI-onlyandHPAI-onlyEquilibriaWestudythecompetitionofthelowandhighpathogenicavianinuenzainwildbirdpopulation.Likethebasicreproductionnumber,theinvasionnumberisathresholdquantitythatdeterminesifonepathogencaninvadetheotherpathogen'sequilibrium.Theinvasionnumbersareveryusefulinunderstandingthedynamicsbetweenlowandhighpathogensinwildbirdpopulation.Wedenoteby^RLwtheinvasionnumberofLPAIwhenthesystemisatHPAI-onlyequilibrium.TheinvasionnumberofLPAIis^RLw=L11(w+Hw+Hw) H11(w+Lw)=RLw0 RHw0.Theinvasionnumber^RLwgivestheabilityofLPAItoinvadetheHPAI-onlyequilibriumwhichismeasuredasthesecondaryinfectionsoneLPAI-infectedwildbirdcanproduceinawildbirdpopulationwhereHPAIisatequilibrium.Wedenoteby^RHwtheinvasionnumberofHPAI.SimilardenitionistruefortheinvasionnumberofHPAI.TheinvasionnumberofHPAIis ^RHw=H11^Sw+H11R10qw()^rwd w+Hw+Hw=RHw0 RLw01+Lww Lw+w(RLw0)]TJ /F6 11.955 Tf 11.95 0 Td[(1)R10qw()e)]TJ /F14 7.97 Tf 6.58 0 Td[(wd.(3)Thewildbirdsystem( 3 )hastwoboundaryequilibria:LPAI-onlyandHPAI-onlyequilibria.WedenotetheLPAI-onlyequilibriumby(^Sw,^ILw^rLw,0,0)andtheHPAI-onlyequilibriumby(~Sw,0,0,~IHw~RHw).Inthefollowingtwotheoremsweprovetheexistenceofboundaryequilibria. Theorem3.3. IfRLw0>1,thenthereexistsauniqueLPAI-onlyequilibrium(^Sw,^ILw^rLw,0,0)inwhich^Sw=w+Lw L11,^ILw=w L11)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(RLw0)]TJ /F6 11.955 Tf 11.95 0 Td[(1and^rLw=Lw^ILwe)]TJ /F14 7.97 Tf 6.59 0 Td[(w. Proof. TondtheLPAI-onlyequilibrium,welookfortime-independentsolutionsoftheform(^Sw,^ILw^rLw(),0,0).Wesetthetimederivativesinwildbirdsystem( 3 )equaltozeroandobtainthefollowingsystemfortheLPAI-onlyequilibrium. 33

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0=w)]TJ /F7 11.955 Tf 11.96 0 Td[(L11^ILw^Sw)]TJ /F7 11.955 Tf 11.95 0 Td[(w^Sw0=L11^ILw^Sw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+w)^ILw (3)d^rLw d=w^rLw^rLw(0)=Lw^ILwThesystemconsistsofonerstorderODEwhoseinitialconditiondependsonthesolution^ILwandtwoalgebraicequations.Thesecondequationcanberewrittenas0=L11^Sw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+w)^ILw.Solvingfor^Swweget,^Sw=w+w L11.Substitutingtheexpressionfor^Swintotherstequationandsolvingitfor^ILwweget,^ILw=w w+Lw)]TJ /F7 11.955 Tf 13.98 8.09 Td[(w L11.SincethebasicreproductionnumberforLPAIisRLw0=L11w w(w+Lw),werearrangethetermsin^ILwbyfactoringoutw L11toobtain^ILw=w L11)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(RLw0)]TJ /F6 11.955 Tf 11.96 0 Td[(1.Finallywesolvethedifferentialequationd^rLw d=)]TJ /F7 11.955 Tf 9.29 0 Td[(w^rLwwith^rLw(0)=Lw^ILwwhosesolutionis^rLw()=Lw^ILwe)]TJ /F14 7.97 Tf 6.59 0 Td[(w Theorem3.4. IfRHw0>1,thenthereexistsauniqueHPAI-onlyequilibrium(~Sw,0,0,~IHw~RHw)where~Sw=w+Lw+Hw H11,^IHw=w H11)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(RHw0)]TJ /F6 11.955 Tf 11.95 0 Td[(1and~RHw=Hw H11)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(RHw0)]TJ /F6 11.955 Tf 11.96 0 Td[(1. 34

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Proof. ProofisverysimilartotheproofofTheorem 3.3 ,andwillnotberepeated. Theorem3.5. TheLPAI-onlyequilibriumislocallyasymptoticallystableif^RHw<1andunstableif^RHw>1. Proof. Asbefore,westartbylinearizingthesystem.Wedenoteby(u,v,x,y,z)theperturbationsaroundthesteadystateandsetSw(t)=^Sw+u(t),ILw(t)=^ILw+v(t),rLw(,t)=^rLw()+x(,t),IHw(t)=y(t),RHw(t)=z(t).Substitutingtheaboveexpressionsonto( 3 ),weobtainthefollowinglinearsystemforperturbations.du dt=)]TJ /F7 11.955 Tf 11.96 0 Td[(L11^ILwu)]TJ /F7 11.955 Tf 11.95 0 Td[(L11^Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(H11^Swy)]TJ /F7 11.955 Tf 11.96 0 Td[(wu,dv dt=L11^ILwu+L11^Swv)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)v@x @t+@x @=)]TJ /F7 11.955 Tf 11.96 0 Td[(H11^rLwqw()y)]TJ /F7 11.955 Tf 11.96 0 Td[(wxx(0,t)=Lwv (3)dy dt=H11^Swy+H11yZ10qw()^rLwd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)ydz dt=Hwy)]TJ /F7 11.955 Tf 11.96 0 Td[(wzWeexpectthatthesolutionsareexponentialandseekforsolutionsoftheform( 3 ).Substitutingthesesolutions( 3 )intolinearizedsystem( 3 ),wegetthefollowinglineareigenvalueproblemwhichconsistsofalgebraicequationsandadifferentialequation.Thealgebraicequationscanberepresentedinthefollowingmatrixform A!=!(3) 35

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where!=(u,v,y,z)TandthematrixAis;A=0BBBBBBB@)]TJ /F7 11.955 Tf 9.3 0 Td[(L11^ILw)]TJ /F7 11.955 Tf 11.95 0 Td[(w)]TJ /F7 11.955 Tf 9.3 0 Td[(L11^Sw)]TJ /F7 11.955 Tf 9.3 0 Td[(H11^Sw0L11^ILwL11^Sw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)0000D000Hw)]TJ /F7 11.955 Tf 9.3 0 Td[(w1CCCCCCCAwhereD=H11^Sw+H11R10qw()^rLwd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw).Thedifferentialequationis:x+dx d=)]TJ /F7 11.955 Tf 9.3 0 Td[(H11^rLwqw()y)]TJ /F7 11.955 Tf 11.95 0 Td[(wx,x(0)=Lwv.Wesolvethenon-homogeneousdifferentialequationbyrstmultiplyingwiththeintegralfactore(+w).Wethenintegrateandobtainthefollowingsolution;x()=Lwve)]TJ /F5 7.97 Tf 6.59 0 Td[((+w))]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z0H11^rLwqw(s)ye)]TJ /F5 7.97 Tf 6.58 0 Td[(()]TJ /F14 7.97 Tf 6.59 0 Td[(w)()]TJ /F3 7.97 Tf 6.59 0 Td[(s)ds.TheLPAI-onlyequilibriumisstable,ifandonlyiftheeigenvaluesfig4i=1ofthealgebraiceigenvalueproblem( 3 )areallnegative.Thethirdequationin( 3 )involvesonlyy.Wesolvethethirdequationandobtain3=H11^Sw+H11Z10qw()^rLwd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw).Theeigenvalue3<0if^RHw<1.Weseethateigenvalue4=)]TJ /F7 11.955 Tf 9.3 0 Td[(w.NotethatL11^Sw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)=0Thus,thecharacteristicequationforeigenvalues1and2is;2+H11^ILw+w+L11^SwL11^ILw=0Since1+2<0and12>0,thereareeither1<0and2<0ortwocomplexconjugateeigenvaluesthatsatisfy<1<0,<2<0. Theorem3.6. TheHPAI-onlyequilibriumislocallyasymptoticallystableif^RLw<1andunstableif^RLw>1. 36

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Proof. ProofofTheorem 3.6 isverysimilartotheproofofTheorem 3.5 ,andwillbeomitted. 3.3.4CoexistenceEquilibriumInthissubsection,weinvestigatetheexistence,uniquenessandthestabilityofthecoexistenceequilibrium(i.e.interiorequilibrium).Wedenotethecoexistenceequilibriumby(Sw,ILw,rLw,IHw,RHw).Coexistenceequilibriumrepresentsthestateforwhichbothlowpathogenicandhighpathogenicavianinuenzaareendemicinthewildbirdpopulation.Werstshowtheexistenceanduniquenessofthecoexistenceequilibriumbythefollowingtheorem. Theorem3.7. Thereexistsauniquecoexistenceequilibrium(Sw,ILw,rLw,IHw,RHw)iff^RLw>1,and^RHw>1. Proof. Thecoexistenceequilibriumsatisesthefollowingsteadystateequation 8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:0=w)]TJ /F7 11.955 Tf 11.96 0 Td[(L11ILwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,0=L11ILwSw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)ILwdrLw d=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()H11IHwrLw)]TJ /F7 11.955 Tf 11.96 0 Td[(wrLwrLw(0)=LwILw0=H11IHwSw+H11IHwR10qw()rLw()d)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)IHw0=HwIHw)]TJ /F7 11.955 Tf 11.95 0 Td[(wRHw(3)SolvingthesecondequationforSw,weget;Sw=w+Lw L11.WethensubstitutetheexpressionforSwtotherstequationandsolveforILwandobtain:ILw=w)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(RLw0)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]TJ /F7 11.955 Tf 13.15 8.08 Td[(H11 L11IHw. 37

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TodeterminerLw,wesolvethedifferentialequationdrLw d=)]TJ /F10 11.955 Tf 9.29 0 Td[(qw()H11IHwrLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wrLw,rLw(0)=LwILwandobtainrLw=LwILwe)]TJ /F14 7.97 Tf 6.59 0 Td[(w)]TJ /F14 7.97 Tf 6.59 0 Td[(H11IHwR0qw(s)ds.SubstitutingtheexpressionsforSwandrLwintothefourthequation,weseethatIHwsatisestheequationF(IHw)=0whereF(x)isthefollowingmonotonedecreasingfunction F(x)=H11 L11(w+Lw)+H11 L11Lw)]TJ /F7 11.955 Tf 5.47 -9.69 Td[(w(RLw0)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F7 11.955 Tf 11.95 0 Td[(H11xZ10qw()e)]TJ /F14 7.97 Tf 6.59 0 Td[(w)]TJ /F14 7.97 Tf 6.59 0 Td[(H11xR0qw(s)dsd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw).(3)ThereexistsauniquepositiveIHwintheinterval(0,^ILw)suchthatF(IHw)=0,sinceF(0)=H11 L11(w+Lw)+H11 L11)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(w(RLw0)]TJ /F6 11.955 Tf 11.95 0 Td[(1)Z10qw()e)]TJ /F14 7.97 Tf 6.59 0 Td[(wd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)=H11^Sw+H11Z10qw()^rLw()d)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)=(w+Hw+Hw)(^RHw)]TJ /F6 11.955 Tf 11.95 0 Td[(1)>0andF(^ILw)=H11 L11(w+Lw))]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)=(w+Hw+Hw)1 ^RLw)]TJ /F6 11.955 Tf 11.95 0 Td[(1<0. 38

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3.3.5StabilityoftheCoexistenceEquilibrium:Toinvestigatethestabilityofcoexistenceequilibrium(Sw,ILw,rLw,IHw,RHw)westartbylinearizingthesystem( 3 ).WesetSw(t)=Sw+u(t),ILw(t)=ILw+v(t),rLw(,t)=rLw+x(,t),IHw(t)=IHw+y(t),RHw(t)=RHw+z(t).Substitutingintotheequationsofthesystem( 3 )andusingtheequation( 3 )forcoexistenceequilibrium,wegetthefollowinglinearsystemafterdroppingthequadratictermsinperturbations:du dt=)]TJ /F7 11.955 Tf 9.3 0 Td[(L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(L11ILwu)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwu)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F7 11.955 Tf 11.95 0 Td[(wu (3)dv dt=L11Swv+L11ILwu)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)v (3)@x @t+@x @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()H11rLwy)]TJ /F10 11.955 Tf 11.95 0 Td[(qw()H11IHwx)]TJ /F7 11.955 Tf 11.95 0 Td[(wx (3)x(0,t)=Lwvdy dt=H11IHwu+H11Swy+H11IHwZ10qw()x(,t)d (3)+H11yZ10qw()rLwd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)ydz dt=Hwy)]TJ /F7 11.955 Tf 11.95 0 Td[(wz (3)Welookforexponentialsolutionsofthesystem( 3 )-( 3 ).Substituting( 3 )into( 3 )-( 3 ),wegetthefollowingeigenvalueproblem: 39

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u=)]TJ /F7 11.955 Tf 9.3 0 Td[(L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(L11ILwu)]TJ /F7 11.955 Tf 11.95 0 Td[(H11IHwu)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F7 11.955 Tf 11.96 0 Td[(wu (3)v=L11Swv+L11ILwu)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)v (3)x+dx d=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()H11rLwy)]TJ /F10 11.955 Tf 11.95 0 Td[(qw()H11IHwx)]TJ /F7 11.955 Tf 11.95 0 Td[(wx (3)x(0)=Lwvy=H11IHwu+H11Swy+H11IHwZ10qw()x()d (3)+H11yZ10qw()rLwd)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)yz=Hwy)]TJ /F7 11.955 Tf 11.96 0 Td[(wz (3)Wesolvethenon-homogeneousdifferentialequation( 3 ),byrstmultiplyingwiththeintegratingfactor;e(+w)+H11IHwR0qw(s)ds.Wethenintegratebothsidesfrom0toandusetheinitialconditionx(0)=Lwvtoobtain: x()=Lwve)]TJ /F5 7.97 Tf 6.58 -.12 Td[((+w))]TJ /F14 7.97 Tf 6.58 0 Td[(H11IHwR0qw(s)ds)]TJ /F7 11.955 Tf 11.95 0 Td[(H11yf(),(3)where f()=Z0qw(s)rLw(s)e)]TJ /F5 7.97 Tf 6.59 -.11 Td[((+w)()]TJ /F3 7.97 Tf 6.59 0 Td[(s))]TJ /F14 7.97 Tf 6.59 0 Td[(H11IHwRsqw()d.(3)SinceL11Sw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)=0,equation( 3 )reducesto v=L11ILwu.(3)Similarlyusingtheequationofcoexistenceequilibrium( 3 ),from( 3 )weseethatH11Sw+H11R10qw()rLwd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)=0.Hence,( 3 )reducesto y=H11IHwu+H11IHwZ10qw()x()d.(3) 40

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Substitutingthesolutionforx(),( 3 ),intothereducedequationfory( 3 ),weobtainthefollowingequationforywhichinvolvesonlyuandv y=H11IHwu+A()v)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwH11B()y(3)where A()=LwH11IHwZ10qw()e)]TJ /F5 7.97 Tf 6.59 -.11 Td[((+w))]TJ /F14 7.97 Tf 6.59 0 Td[(H11IHwR0qw(s)dsd.(3)and B()=Z10qw()f()d.(3)Combiningthesereducedequationsfor( 3 )and( 3 )togetherwith( 3 ),wegetthefollowingeigenvalueproblem;u=)]TJ /F7 11.955 Tf 9.3 0 Td[(L11Swv)]TJ /F7 11.955 Tf 11.96 0 Td[(L11ILwu)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwu)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swy)]TJ /F7 11.955 Tf 11.95 0 Td[(wuv=L11ILwu (3)y=H11IHwu+A()v)]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwH11B()y.Thissystemwillhavenon-zerosolutionfor(u,v,y)ifthedeterminantofthissystemiszero,thatis,if )]TJ /F7 11.955 Tf 9.3 0 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(L11ILw)]TJ /F7 11.955 Tf 11.95 0 Td[(H11IHw)]TJ /F7 11.955 Tf 11.96 0 Td[(w)]TJ /F7 11.955 Tf 9.29 0 Td[(L11SwH11SwL11ILw)]TJ /F7 11.955 Tf 9.3 0 Td[(0H11IHwA())]TJ /F7 11.955 Tf 9.3 0 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwB()=0.(3) 41

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Fromthedeterminant,weobtainthecharacteristicequationfortheeigenvalueproblem( 3 ):3+2)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(H11IHwH11B()+L11ILw+H11IHw+w+)]TJ /F7 11.955 Tf 10.46 -9.69 Td[(L11ILw+H11IHw+w)]TJ /F7 11.955 Tf 12.95 -9.69 Td[(H11IHwH11B()+H11SwH11IHw+L11SwL11ILw+H11SwL11ILwA()+H11IHwH11B()L11SwL11ILw=0.Thischaracteristicequationhasrootswithpositiverealparts.Thusinstabilityandoscillationsoccur.Weshowthatsustainedoscillationsarepossibleinthesystem( 3 )bydemonstratingitwithaspecicexampleinthenextsection. 3.4NumericalSimulationsInthissection,weperformseveralnumericalsimulationsofthemodel( 3 ).Weobtaintheapproximatesolutionofthesystem( 3 )byconstructinganimplicitnitedifferencemethod.WediscretizethedomainD=f(,t):0A,0tTgbytakingequalstepsizesinbothtanddirection.Thus,t=.Inallsimulations,wechoosethecross-immunityfunctionqw()tobethefollowingstepfunction; qw()=8>><>>:0ifaqif>a(3)whereaisanarbitraryconstant.Sincethevariableisthetime-since-recoveryfromlowpathogenicavianinuenza,thiscrossimmunityfunctionqw()meansthatthewildbirdsarefullyprotectedfromHPAIforaperiodoftimea.Afterthatperioda,theprotectionwanes.Thisisareasonableassumptionforlowpathogenicandhighpathogenicinuenzaintheavianpopulation,anditisinagreementwiththeresultsofthestudyby( 54 ). 42

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Therstquestionthatweaddresswithsimulationsiswhethersystemcan( 3 )canexhibitoscillationsinwhichboththeLPAIandtheHPAIoscillateatnon-zerovalues.Toaddressthisquestion,weanalyzethecharacteristicequation. Figure3-1. ThenumberofwildbirdsinfectedwithLPAI(thinline)andthenumberofwildbirdsinfectedwithHPAI(thickline)exhibitoscillations. Withtheabovechoiceofqw(),wecomputetheintegralsinf(),A(),andB()andget;f()=qLwILw1 )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F10 11.955 Tf 11.96 0 Td[(eae)]TJ /F5 7.97 Tf 6.59 -.11 Td[((+w))]TJ /F14 7.97 Tf 6.59 0 Td[(H11IHwq()]TJ /F3 7.97 Tf 6.59 0 Td[(a)andA()=H11IHwLwqe)]TJ /F5 7.97 Tf 6.58 0 Td[((+w)a +w+H11ILwqandB()=q2LwILwe)]TJ /F14 7.97 Tf 6.58 0 Td[(wa (+w+H11ILwq)(w+H11ILwq).Afteranalyzingthecharacteristicsequationwendparametersofthemodel( 3 ),whosesolutionsexhibitsoscillations.TheseparametersaregiveninTable 3-1 .Toillustratetheoscillations,wesimulatethesolutionsofthewildbirdsystem( 3 )usinganimplicitnitedifferencemethod.Timeismeasuredinyears,andthenaltimeforthesimulationsareT=40andA=30.ThenumberofwildbirdsinfectedwithLPAI 43

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andthenumberofwildbirdsinfectedwithHPAI,whichexhibitsustainedoscillations,areplottedinFigure 3-1 .TheFigureshowsthattheoscillationsinLPAIhaveamuchlargeramplitudethantheoscillationsofHPAIsuggestingthattheinstabilityofthedynamicsofLPAIismorepronounced.Furthermore,theoscillationsinHPAIfollowtheoscillationsofLPAIwiththepeakinHPAIoccurringrightafterthedropinLPAI.ThisbehaviorpartlyremindsofaLotka-Volterrapredator-preydynamicswheretheoscillationsinthepredatorfollowtheoscillationsofthepreywith1/4ofaturn.ThisanalogyisperhapsnotsurprisingsinceHPAIinfectsindividualsrecoveredfromLPAI. Table3-1. Parametervaluesofthewildbirdsmodel( 3 )whichexhibitsoscillations. ParameterValueParameterValueParameterValue w1020L110.1278rLw(,0)28=Tw1=2H110.7140IHw(0)2Hw460.925Sw(0)2000q1Hw365=7ILw(0)10a0.25Lw365=7 ThenextquestionweaddressishowchangesinthetransmissionofLPAIaffectthecompetitionofthestrains.ToinvestigatetheimpactofL11,weobtaintheapproximatesolutionsofthesystem( 3 )forthreedifferentvaluesofL11.WexallotherparametersandchangeonlyL11.TheresultsshowthatincreasingL11increasestheLPAIprevalenceanddecreasestheHPAIprevalence.WeplotILw(t)andIHw(t)inFigure 3-2 .System( 3 )canmodelthedynamicsofLPAIandHPAIinbothwildanddomesticbirds.TheobservationthatincreasingtransmissionincreasesprevalenceofLPAIanddecreasesprevalenceofHPAImayshedlightonwhywildbirdshavelowerprevalenceofHPAIcomparedtodomesticbirds.TheimmunitycreatedbycirculatingLPAIprotectswildbirdsfromHPAI.Incontrast,domesticbirdsareprotectedfromLPAIandconsequently,theyexperiencemuchmoreserioussymptomsfromHPAIandHPAIhasmuchhigherprevalenceindomesticbirds,particularlyinthepoultrypopulationofsomecountries. 44

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Figure3-2. LPAIandHPAIprevalenceforthreedifferentvaluesofL11 Anotherquestionofinterestishowthedurationofcross-immunityaffectstheLPAIandHPAIprevalence.SinceweassumethatLPAIprovidesfullprotectionagainstHPAIfor2[0,a],itisinterestingtoknowhowthelengthofthisprotectionaffectsthedynamics.WetakethreedifferentvaluesofaandplotILw(t)andIHw(t)inFigure 3-3 .Weobservethatincreasingdurationofcross-immunitydecreasestheprevalenceofHPAIamongwildbirdpopulations.Whatisunexpectedisthatcross-immunityalsoincreasestheprevalenceofLPAI,eventhoughitdoesnotdirectlyaffectLPAI.ThechangeinLPAIisatleastaspronouncedasthechangeinHPAI. Figure3-3. Effectofcross-immunityonHPAIprevalence AnotherquestionthatweconsideristheimpactofLPAIandHPAItransmissionratesH11andL11ontheHPAIprevalenceatcoexistenceequilibrium.Toinvestigatethis 45

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topic,weconsidertheequationfortheIHw.TheHPAIprevalenceinthecoexistenceequilibriumsatisestheequationF(IHw)=0givenin( 3 )whichinvolvesthefollowingintegralC=Z10qw()e)]TJ /F14 7.97 Tf 6.59 0 Td[(w)]TJ /F14 7.97 Tf 6.58 0 Td[(H11IHwR0qw(s)dsd.WecomputetheintegralC()bytakingqw()tobethestepfunctionin( 3 )andobtainC=qe)]TJ /F14 7.97 Tf 6.58 0 Td[(wa w+H11IHwq.Thus,theHPAIcoexistenceequilibrium,IHwsatises H11 L11(w+Lw)+H11 L11Lw)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(w(RLw0)]TJ /F6 11.955 Tf 11.96 0 Td[(1))]TJ /F7 11.955 Tf 11.96 0 Td[(H11IHwqe)]TJ /F14 7.97 Tf 6.58 0 Td[(wa w+H11IHwq)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)=0(3)Weplot( 3 )forseveralvaluesofL11inFigure 3-4 . Figure3-4. HPAIcoexistenceequilibrium WeconsiderhowthetwotransmissionratesimpactIHw.WenoticethatforlowvaluesofHPAItransmissionrateandlowHPAIprevalence,theHPAIprevalencedoesnotdependonthetransmissionrateofLPAI.However,astheHPAIprevalenceincreasesitbecomesmoreandmoresensitivetotheLPAItransmissionrate.AsurprisingconclusionisthatforhigherHPAIprevalences,theHPAIprevalencedepends 46

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signicantlyontheLPAItransmissionrateL11.ItisfurtherclearthatincreaseinH11increasestheHPAIprevalenceIHwwhileincreaseintheLPAItransmissionrateL11decreasesHPAIprevalenceIHw.ThisobservationagainsuggeststhathighertransmissionofLPAIinwildbirdsmayberesponsibleforlowerprevalenceofHPAI.Finally,werecallthatHPAIislocallystableif^RLw<1,andLPAIislocallystableif^RHw<1.Wenoticethatthetwoinvasionnumberscanbewrittenasfunctionofthetworeproductionnumbers:^RLw=h(RLw0,RHw0)^RHw=g(RLw0,RHw0).Therefore,wecanplotthecurvesh(RLw0,RHw0)=1andg(RLw0,RHw0)=1inthe(RLw0,RHw0)plane.ThegureweobtainisgiveninFigure 3-5 . Figure3-5. Theregionsofcoexistenceanddominance Thegureshowsthattheareainthe(RLw0,RHw0)planewhereHPAIdominatesisthelargest.HPAIdominateswheneverRHw0>RLw0.TheareawhereLPAIdominatesisbelowthelowercurve.TheareawhereLPAIandHPAIcoexististheareabetweenthecurves.Thisareaislargerifqislargerandwhenaissmaller.ThatistosaythatcoexistenceismorelikelyifthefullimmunityprovidedbyLPAIisshorter. 47

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3.5DiscussionThischapterintroducesaLPAI-HPAIavianinuenzamodel.HPAI-infectedindividualscaninfectindividualsrecoveredfromLPAIatsomereducedinfectivity.However,weassumethatLPAI-infectedindividualscannotinfectHPAI-recoveredindividuals.Furthermore,LPAI-recoveredindividualsarestructuredbytime-since-recoveryandtheimmunitytoHPAIcreatedbypriorinfectionbyLPAIwanesasthetime-since-recoveryincreases.ThismodeldescribestheHPAI-LPAIdynamicsinbothdomesticandwildbirds.Wendthatthesystemhasauniquedisease-freeequilibrium.WedenethereproductionnumbersfortheLPAIandHPAI.Ifbothreproductionnumberaresmallerthanone,thenweshowthatthedisease-freeequilibriumisbothlocallyandgloballystable.Ifoneofthereproductionnumbersisgreaterthanone,thenthedisease-freeequilibriumisunstable.Furthermore,wendthatifthereproductionnumberofLPAIisgreaterthanone,thereisauniqueLPAI-onlyequilibrium.Similarly,ifthereproductionnumberofHPAIisgreaterthanone,thereisauniqueHPAI-onlyequilibrium.WedenetheinvasionreproductionnumbersofLPAIandHPAI.TheinvasionreproductionnumberofLPAIisgreaterthanoneifandonlyifthereproductionnumberofLPAIisgreaterthenthereproductionnumberofHPAI.WeprovethattheLPAI-onlyequilibriumislocallyasymptoticallystableiftheHPAIinvasionnumberissmallerthanone,thatiftheHPAIcannotinvadetheequilibriumoftheLPAI.SimilarlyweprovethattheHPAI-onlyequilibriumislocallyasymptoticallystableiftheLPAIinvasionnumberissmallerthanone,thatistheLPAIcannotinvadetheequilibriumoftheHPAI.ThisistosaythatHPAIdominatesinthepopulationanddrivesLPAItoextinctionifandonlyifHPAIreproductionnumberislargerthantheLPAIreproductionnumber.HPAIdominatesformorevaluesofthereproductionnumbersthanLPAIandeverytimethereproductionnumberofHPAIislargerthanthereproductionnumberofLPAI;however,webelievethatforrealisticparametervaluesLPAIhasahigherreproductionnumber,particularlyinwildbirds. 48

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Finallyweshowthatifbothinvasionnumbersarelargerthanonecoexistenceequilibriumispresent.Themechanismofcoexistenceiscross-immunity.Weshowthatthecoexistenceequilibriumisunique.Inaddition,weshowthroughsimulationsthatthecoexistenceequilibriummaylosestabilityandcoexistenceintheformofsustainedoscillationsispossible.InthiscaseHPAI'speakfollowsrightaftertheLPAI'speak.SimulationsalsosuggestthatincreasingthetransmissioncoefcientofLPAIincreasestheprevalenceofLPAIanddecreasestheprevalenceofHPAI.Furthermore,increasingthetransmissioncoefcientofHPAIanddecreasingthetransmissioncoefcientofLPAIbothincreasetheHPAIprevalence.BasedonthesesimulationsweconcludethathighertransmissionofLPAIinwildbirdsmayberesponsibleforthelowerprevalenceofHPAIcomparedtopoultry.Finally,assumingthatLPAIprotectsagainstHPAIcompletelyforaperiodoftime,weinvestigatetheeffectthedurationofprotectionhasontheLPAIandHPAIprevalence.WendthatincreasingthedurationofprotectionhasanimpactonboththeLPAIprevalenceandtheHPAIprevalence.Inparticular,itincreasestheLPAIprevalenceanddecreasestheHPAIprevalence.SeveralarticleshaveinvestigatedthedynamicsofLPAIandHPAIbefore( 38 ; 5 ).However,theeffectsofcross-immunityanddurationofprotectionhavenotbeenstudied.InthisarticleweinvestigatetheseeffectsaswellashowtransmissionimpactstheprevalenceofLPAIandHPAI.Furthermore,priorstudieshavefoundthatODEmodelswithcross-immunityhavenotebeenabletoexhibitsustainoscillations( 6 ; 7 ).Togenerateoscillationsinthemulti-straininuenzadynamics,aquarantinestatewasintroducedin( 45 ).Here,wehaveusedadifferentapproachtomodeltheoscillationsinavianinuenzadynamics,namely,wehaveintroducedatime-since-recoveryindependentvariable.Weshowviasimulationsthatthecoexistenceequilibriumcanbedestabilizedandcoexistenceintheformofsustainedoscillationispossible. 49

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CHAPTER4DYNAMICSOFLOWANDHIGHPATHOGENICAVIANINFLUENZAINWILDANDDOMESTICBIRDPOPULATIONS 4.1IntroductionInfectiousdiseasesoftenemergefromcomplexecologicalcommunities( 22 ).Moreover,manyimportanthost-pathogensystemsconsistofmultiplepathogenstrains,circulatingamongmultiplespeciesofhosts.Understandinghowmulti-speciestransmissionaffectspersistenceofagivenpathogenstraincanhelpinformpredictionandmanagementofinfectiousdiseaseoutbreaks,andunderstandinghowsuchtransmissionamonghostsmodulatesthecoexistenceofpathogenstrainsandthusthemaintenanceofgeneticvariationwithinpathogensisessentialforgauginghowpathogensarelikelytoevolve.Thiscommunitydimensionofepidemiologyiswidelyrecognizedasbeingasignicantfrontierinquantitativeepidemiologyandthepublichealthsciences( 29 ).Theseissuesarisewithparticularurgencyinthecaseoftheavianinuenzaviruses(AIVs),whichpresentaglobaleconomicprobleminthepoultryindustrycostinghundredsofmillionsofdollars( 49 ),andposeaseriouspublichealthriskduetothethreatofemergenceofanovelpathogenstraincirculatingamonghumanhosts,withpotentiallydevastatingconsequences( 63 ).InuenzaAvirusescaninfectmanyspeciesofwarm-bloodedvertebrates( 65 ),butthegreatmajorityofviralstrainsappeartobefoundinwildwaterbirds,suchasshorebirdsandgulls(Charadriiformes)andducksandgeese(Anseriformes)( 33 ).Thesespeciescancomeintocontactwithdomesticpoultry,whichcanposeadirectthreattothepoultryindustry,andalsoprovidesaconduitforpotentialtransmissiontohumans.Mathematicalmodelscanprovideessentialtoolsforunderstandingmanyaspectsofinfectiousdiseasedynamics( 29 ),andbecomeparticularlyimportantwhengrapplingwiththecomplexitiesofmulti-pathogen,multi-hostsystems,forinstancewhenhoststhemselvesmaymountstrain-specicimmuneresponsestoinfection.Arealistic 50

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modelofavianinuenzawouldbehighlycomplex,sinceitwouldhavetoaccountfortransmissionwithinandamongmultiplepotentialspeciesofwildhosts,manyofwhicharemigratory( 60 )andoccupyseasonallyforcedenvironments(seerefs.in( 63 )).Asaway-stationtowardssucharealisticmodel,hereweconsiderasysteminwhichtherearetwohostpopulations,whichwecalldomesticandwildbirdpopulations,eachofwhichhasrelativelysimpleintrinsicdynamics.Thesetwohostpopulationsareinturninfectedbytwostrainsofavianinuenza,oneofwhichisastrainofLPAI,andtheotherastrainofHPAI.HPAIvirusesaredenedbythefactthattheycauseatleast75%mortalityin4-8weekchicken,infectedintavenously( 58 ).Thebasicdynamicsofeachhostconsistsofasteadyowoffreshsusceptiblesintoeachhostpopulation,andaconstantrateofmortality.Intheabsenceofthevirus,thehostshaveverystabledynamics.(Thisassumptionwouldneedtoberelaxedwhenconsideringthedetaileddynamicsofnaturalpopulations,whichuctuateseasonallyandamongyears.)Transmissionofthevirusoccursinadensity-dependentfashion,bothwithinandbetweenthesetwopopulations.HostscanrecoverfrominfectionwithLPAI,andwhentheydorecover,areimmuneforlifefromfurtherinfectionbythisviralstrain.Consistentwithempiricalevidence,thereisadegreeofcross-protectionintheimmuneresponse,soinfectionbyLPAIcanprotectagainstHPAI.However,thiscross-immunityfadeswithtime,andincorporatingthedynamicsofsuchtime-dependentfadeoutinimmuneprotectionisoneofthemathematicalcomplexitiesofthemodel.Bycontrast,infectionwithHPAIisassumedtoalwaysleadtodeath(possiblybyculling)indomesticbirds;inwildbirds,HPAIleadstodeathorrecoverywithpermanentimmunitytobothstrains.Ourfocuswillbeontheimplicationsofpartialcross-immunity,buttoputourresultsintocontext,itisusefultoconsiderwhatmightbeexpectedwhencross-immunityiscomplete.Ifcross-immunityiscomplete,thenLPAIandHPAIsimplycompeteforsusceptiblehosts.Ifthereisonlyonepopulation,withinwhicheachstraincouldpersist 51

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alone,whicheverstraincanpersistatthelowestlevelofsusceptibleswilleliminatetheotherstrain.Withtwopopulations,therearetworesources(thesusceptiblesinthetwopopulations),sothereareotherpossibilities.Oneisthatthetwostrainscoexist,forexampleifLPAIisbetteratexploitingwildsusceptiblesandHPAIisbetteratexploitingdomesticsusceptibles.Anotherpossibilityisthateachstraincanexcludetheother,inwhichcasetherststraintoarrivepersistsandthesecondstraincannotinvade(alternativeequilibria).Ifcross-immunityisnotcomplete,HPAIcaninfectatleastsomeLPAI-recoveredbirds,andsoithasanadditionalresource.Therefore,coexistenceispossibleinasinglepopulationifLPAIisbetteratexploitingsusceptibles.Withcompletecross-immunity,LPAIwouldeliminateHPAI,butotherwiseitispossibleforHPAItoinvadeandpersistbyinfectingLPAI-recoveredbirds.Withtwopopulations,ofcourse,thereisadditionalscopeforcoexistence.Theanalysesandsimulationspresentedbelowhelpilluminatetheconditionsthatpermitsuchcoexistence.Werstpresentthebasicmodel(foraowchartofthemodel,seeFigure 4-1 ).Then,wecharacterizetheconditionsforeachviralstraintobeabletoincreasewhenrare,andalone.Wederiveexpressionsforthebasicreproductionnumberforeachstrain,whicharefunctionsofthejointdensitiesofthedomesticandwildbirdpopulations.Next,weconsidertheconditionsforincreaseofeachstrain,whentheotherstrainispresent,andaimatcharacterizingconditionsforthecoexistenceofthetwostrains.Suchcoexistenceisnotguaranteed.Thetwoviralstrainscanbeviewedasinteractingintwodistinctways.First,theycompeteexploitativelyforhealthyhosts.Giventhattherearetwohostpopulations,asnotedabove,thereisthepotentialforadegreeofnichepartitioningthatcouldfacilitatetheircoexistence( 22 ).Secondly,thelossofpartialimmunitymeansthereisapartial,time-laggedfacilitationofthedynamicsofHPAI,emergingfromhostswhogetinfectedwithLPAI,butrecover.ThismeansthatevenifallhostshavebeeninfectedbyLPAI(sonofullysusceptiblehostsareavailableatall),somehostscanbecomeavailableforinfectionbyHPAI. 52

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ThisreplenishmentofhostsforHPAIinvolvesalag,relativetoLPAIinfection.Wewillusenumericalsimulationstodemonstratethatthispermitstheentiresystemtopersist,butwithsustained,large-scaleoscillationsininfectionbyeachviralstrain.Suchoscillationscanemergeevenifeachviralstrainonitsowntendstowardsastableequilibriumwhenitaloneisinfectingthetwohostpopulations. 4.2TheModelWeconsideratime-sincerecoverystructuredmodeltostudythedynamicsoflowandhighpathogenicavianinuenzainwildanddomesticbirdpopulations.Thewildbirdpopulationisdividedintononintersectingclassesofsusceptibles(Sw),infectedwithHPAI(IHw),infectedwithLPAI(ILw),recoveredfromLPAI(rLw),andrecoveredfromHPAI(RHw).Similarly,thedomesticbirdpopulationisdividedintosusceptible(Sd),infectedwithHPAI(IHd),infectedwithLPAI(ILd)andrecoveredfromLPAI(rLd)classes.SincethedetectionofevenoneHPAIinfecteddomesticbirdresultsincullingoftheentirefarmorthedeathofthedomesticbird,wedonotincludearecoveredclassfromHPAIforthedomesticbirdpopulation.TherecoveredclassesrLw(,t),rLd(,t)denotethedensityofrecoveredbirdswithtime-sincerecoveryparameter.Thesusceptiblebirdpopulationsaregeneratedbytherecruitment/birthrates(wandd),reducedbythenaturaldeathrates(w,andd)andbyinfectionwithHPAIorLPAI.ThenewinfectionswithLPAIandHPAIperunittimearemodeledbyHwandLwinwildbirds,andbyHdandLdindomesticbirds.TheforceofinfectionforLPAIandHPAIinthewildbirdpopulationaregivenbyLw=L11ILw+L12ILdHw=H11IHw+H12IHdSimilarly,theforceofinfectionforLPAIandHPAIinthedomesticbirdpopulationaregivenbyLd=L21ILw+L22ILdHd=H21IHw+H22IHd 53

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Theaggregateparameterscanbeinterpretedastheproductofnumberofcontactsmadebysusceptible(wildordomestic)birdwithaninfected(LPAIorHPAI)birdandtheprobabilitythatthecontactresultedintransmission.ForinstanceH12istheHPAItransmissionrateofwildbirdsincontactwithdomesticbirds,similarly,L21istheLPAItransmissionrateofdomesticbirdsincontactwithwildbirds.Thus,therateofchangeofthepopulationofsusceptiblewildanddomesticbirdpopulationsaregivenby;dSw dt=w)]TJ /F7 11.955 Tf 11.96 0 Td[(LwSw)]TJ /F7 11.955 Tf 11.95 0 Td[(HwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,dSd dt=d)]TJ /F7 11.955 Tf 11.96 0 Td[(LdSd)]TJ /F7 11.955 Tf 11.96 0 Td[(HdSd)]TJ /F7 11.955 Tf 11.96 0 Td[(dSd.TheinfectedwildbirdsrecoverfromLPAIinfectionatarateLwandthedomesticbirdsrecoveratarated.LPAIcausesmildinfectionindomesticandwildbirds,henceweneglecttheLPAI-induceddeathrate.TheLPAI-infectedwildanddomesticbirdpopulationsincreasebythenewincidencesLwSwandLdSdrespectively.ThusthewildanddomesticbirdpopulationsinfectedwithLPAIsatisfythefollowingequationsdILw dt=LwSw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)ILw,dILd dt=LdSd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+d)ILd.TheHPAI-infectedwildanddomesticbirdpopulationincreasesbythenewincidencesHwSwandHdSdrespectively.WildbirdsinfectedwithHPAIcanrecoveratarateHw;domesticbirdsdonotrecoverfromHPAI.StudiesshowthatanearlierinfectionwithLPAIprovidestemporaryimmunitytowardHPAIandthisimmunityfadeswithtime-since-recoveryfromLPAI( 15 ; 54 ).LetbethetimeelapsedsincetherecoveryfromlastLPAIinfection,thentheadditionalnewHPAIinfectionsperunittimefromwildbirdsthathaverecoveredfromLPAIaregivenbythetermHwZ10qw()rLw(,t)d, 54

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whereqw()isthesusceptibilitytoHPAIofawildbirdthatrecoveredfromLPAItimeunitsagorelativetothatofanaivewildbird.Similarly,thenewHPAIinfectionsperunittimeofthedomesticbirdsrecoveredfromLPAIinfectionsaregivenbythetermHdZ10qd()rd(,t)d,whereqd()istherelativesusceptibilitytoHPAIofanLPAI-recovereddomesticbird.ThusthewildanddomesticbirdpopulationsinfectedwithHPAIsatisfythefollowingequationsdIHw dt=HwSw+HwZ10qw()rLw(,t)d)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)IHw,dIHd dt=HdSd+HdZ10qd()rd(,t)d)]TJ /F6 11.955 Tf 11.95 0 Td[((d+Hd)IHd,whereHwandHdareadditionaldeathratesinducedbyHPAIinwildanddomesticbirdsrespectively.WecombinethesedifferentialequationswiththoseforLPAI-recoveredclasses,rLw(,t)andrLd(,t),whichhaverelativesusceptibilitiestoHPAIofqw()andqd(),respectively,where0qw()1,0qd()1forevery>0.Thusthedifferentialequationsmodelingrecoveredclassesare@rLw @t+@rLw @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()HwrLw)]TJ /F7 11.955 Tf 11.96 0 Td[(wrLw,rLw(0,t)=LwILw,@rd @t+@rd @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qd()Hdrd)]TJ /F7 11.955 Tf 11.96 0 Td[(drd,rd(0,t)=dILd.Withtheabovenotations,westudythefollowingtime-since-recoverystructured,multi-strain,multi-populationmodel 55

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dSw dt=w)]TJ /F7 11.955 Tf 11.96 0 Td[(LwSw)]TJ /F7 11.955 Tf 11.95 0 Td[(HwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,dILw dt=LwSw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)ILw@rLw @t+@rLw @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()HwrLw)]TJ /F7 11.955 Tf 11.96 0 Td[(wrLwrLw(0,t)=LwILwdIHw dt=HwSw+HwZ10qw()rLw(,t)d)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)IHwdRHw dt=HwIHw)]TJ /F7 11.955 Tf 11.95 0 Td[(wRHwdSd dt=d)]TJ /F7 11.955 Tf 11.96 0 Td[(LdSd)]TJ /F7 11.955 Tf 11.96 0 Td[(HdSd)]TJ /F7 11.955 Tf 11.96 0 Td[(dSd,dILd dt=LdSd)]TJ /F6 11.955 Tf 11.95 0 Td[((d+d)ILd@rd @t+@rd @=)]TJ /F10 11.955 Tf 9.3 0 Td[(qd()Hdrd)]TJ /F7 11.955 Tf 11.95 0 Td[(drdrd(0,t)=dILddIHd dt=HdSd+HdZ10qd()rd(,t)d)]TJ /F6 11.955 Tf 11.96 0 Td[((d+Hd)IHd.(4)Aschematicowdiagramofmodel( 4 )isgiveninFigure 4-1 ,andtheassociatedmodelvariablesandparametersaredenedinTable 4-1 andTable 4-2 ,respectively. Table4-1. Denitionofthevariablesofmodel( 4 ) VariableMeaningVariableMeaning SwSusceptiblewildbirdsSdSusceptibledomesticbirdsILwLPAI-infectedwildbirdsILdLPAI-infecteddomesticbirdsIHwHPAI-infectedwildbirdsIHdHPAI-infecteddomesticbirdsrLwWildbirdsthathaverecoveredfromLPAIrdDomesticbirdsthathaverecoveredfromLPAIRHwWildbirdsthathaverecoveredfromHPAI 56

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. . Sw . rLw . IHw . ILw . RHw . . . . . . . ILd . IHd . rd . Sd . . . . . . w . w . w . w . w+Hw . w . Lw . Hw . Lw . qw()Hw . Hw . d . d . d . d . d+Hd . Ld . Hd . d . qd()Hd . . . . . . Figure4-1. Flowchartofmodel( 4 ) 57

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Table4-2. Denitionoftheparametersofmodel( 4 ) VariableMeaning dBirth/recruitmentrateofdomesticbirdswBirth/recruitmentrateofwildbirdsdNaturaldeathrateofdomesticbirdswNaturaldeathrateofwildbirdsHdHPAI-inducedmortalityratefordomesticbirdsHwHPAI-inducedmortalityrateforwildbirdsdRecoveryrateofdomesticbirdsfromLPAILwRecoveryrateofwildbirdsfromLPAIHwRecoveryrateofwildbirdsfromHPAIL11=H11LPAI/HPAItransmissionratetosusceptiblewildbirdsfrominfectedwildbirdsL12=H12LPAI/HPAItransmissionratetosusceptiblewildbirdsfrominfecteddomesticbirdsL22=H22LPAI/HPAItransmissionratetosusceptibledomesticbirdsfrominfecteddomesticbirdsL21=H21LPAI/HPAItransmissionratetosusceptibledomesticbirdsfrominfectedwildbirdsqw()RelativesusceptibilityofLPAI-recoveredwildbirdstowardHPAIqd()RelativesusceptibilityofLPAI-recovereddomesticbirdstowardHPAI 4.3LPAI-HPAIdynamicsinwildanddomesticbirdpopulationsWestudytheexistenceandstabilityofequilibriaofsystem( 4 ).Model( 4 )has4equilibria:thediseasefreeequilibrium(DFE);twoboundaryequilibria,LPAI-onlyandHPAI-only;andacoexistenceequilibrium. 4.3.1Disease-FreeEquilibriumThesystem( 4 )hasadisease-freeequilibrium"0givenby"0=(Sw,0,0,0,0,Sd,0,0,0),whereSw=w w,andSd=d d. 58

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TheLPAIandHPAIbasicreproductionnumbersforinthewildbirdpopulationaredenotedbyRL11andRH11respectively,andaregivenbyRL11=L11w w(w+Lw),RH11=H11w w(w+Hw+Hw).TheepidemiologicalmeaningofbasicreproductionnumberRL11(RH11)isthenumberofsecondarycasesproducedbyoneLPAI(HPAI)infectedwildbirdduringitsinfectiousperiodinanentirelysusceptiblepopulationofwildbirds.Similarly,thebasicreproductionnumbersforLPAIandHPAIinthedomesticbirdpopulationaredenotedbyRL22andRH22,respectively,andaregivenbyRL22=L22d d(d+d),RH22=H22d d(d+Hd).Wealsodenethereproductionnumbersbetweenpopulations.Inparticular,theLPAIandHPAIreproductionnumbersofdomesticbirdsinthewildbirdpopulationaredenotedbyRL12andRH12,respectively,andaregivenbyRL12=L12w w(d+d),RH12=H12w w(d+Hd).ThereproductionnumberRL12(RH12)givesthenumberofsecondarycasesoneLPAI(HPAI)infecteddomesticbirdwillproduceduringitslifetimeasinfectiousinanentirelysusceptiblewildbirdpopulation.Similarly,wedenotetheLPAIandHPAIreproductionnumberofwildbirdsinthedomesticbirdpopulationasRL21andRH21,respectively,whicharegivenbyRL21=L21d d(w+Lw),RH21=H21d d(w+Hw+Hw).ThereproductionnumberRL21(RH21)givesthenumberofsecondarycasesoneLPAI(HPAI)infectedwildbirdwillproduceduringitslifetimeasinfectiousinanentirelysusceptibledomesticbirdpopulation. 59

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WecallthereproductionnumbersRL11,...,RH22population-specicreproductionnumbersandthereproductionnumbersRL12,...,RH21cross-populationreproductionnumbers.WedenotethebasicreproductionnumberofLPAIforthefullsystem( 4 )asRL,whichisgivenbyRL=RL11+RL22+q )]TJ /F2 11.955 Tf 5.48 -9.68 Td[(RL11)-221(RL222+4RL12RL21 2.SimilarlythebasicreproductionnumberofHPAIforthefullsystem( 4 )isgivenbyRH=RH11+RH22+q )]TJ /F2 11.955 Tf 5.48 -9.69 Td[(RH11)-222(RH222+4RH12RH21 2.ThesebasicreproductionnumbersRL,RHarethresholdvalueswhichdeterminewhetherLPAIorHPAIcaninvadethedisease-freeequilibrium.ThebasicreproductionnumberR0ofthefullsystem( 4 )isthemaximumoftheLPAIandHPAIreproductionnumbers:thatis,R0=maxfRL,RHg. Theorem4.1. IfRL<1andRH<1thentheDFE,"0,islocallyasymptoticallystable. Proof. Let(uw,vw,xw,yw,zw,ud,vd,xd,yd)=(Sw,ILw,rLw,IHw,RHw,Sd,ILd,rd,IHd))]TJ /F7 11.955 Tf 12.54 0 Td[("0denotetheperturbationsaroundtheDFE;thenweobtainthefollowinglinearized 60

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system. duw dt=)]TJ /F7 11.955 Tf 9.3 0 Td[(L11Swvw)]TJ /F7 11.955 Tf 11.96 0 Td[(L12Swvd)]TJ /F7 11.955 Tf 11.96 0 Td[(H11Swyw)]TJ /F7 11.955 Tf 11.95 0 Td[(H12Swyd)]TJ /F7 11.955 Tf 11.96 0 Td[(wuw,dvw dt=L11Swvw+L12Swvd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)vw@xw @t+@xw @=)]TJ /F7 11.955 Tf 9.3 0 Td[(wxwxw(0,t)=Lwvwdyw dt=H11Swyw+H12Swyd)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)ywdzw dt=Hwyw)]TJ /F7 11.955 Tf 11.96 0 Td[(wzwdud dt=)]TJ /F7 11.955 Tf 9.3 0 Td[(L21Sdvw)]TJ /F7 11.955 Tf 11.96 0 Td[(L22Sdvd)]TJ /F7 11.955 Tf 11.95 0 Td[(H21Sdyw)]TJ /F7 11.955 Tf 11.95 0 Td[(H22Sdyd)]TJ /F7 11.955 Tf 11.95 0 Td[(dud,dvd dt=L21Sdvw+L22Sdvd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+d)vd@xd @t+@xd @=)]TJ /F7 11.955 Tf 9.3 0 Td[(dxdxd(0,t)=dvddyd dt=H21Sdyw+H22Sdyd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+Hd)yd(4)Supposethattheperturbationsxw(t,)andxd(t,)haveexponentialformssuchasxw=etxw()andxd=etxd().Afterdroppingthebars,weobtainthefollowingrstorderODEs:xw+dxw d=)]TJ /F7 11.955 Tf 9.3 0 Td[(wxw,xw(0)=Lwvw,andxd+dxd d=)]TJ /F7 11.955 Tf 9.3 0 Td[(dxd,xd(0)=dvd.Solvingtheabovedifferentialequations,weobtain:xw()=Lwvwe)]TJ /F5 7.97 Tf 6.59 0 Td[((+w),xd()=dvde)]TJ /F5 7.97 Tf 6.58 0 Td[((+d).Theinfectedcompartmentsx=(vw,vd,yw,yd)ofthelinearizedsystem( 4 )aredecoupledfromtheremainingequations.Usingthenextgenerationmatrixapproach,thelinearizedsystemfortheinfectedcompartmentx=(vw,vd,yw,yd)canberewrittenasx0=(F)]TJ /F10 11.955 Tf 11.96 0 Td[(V)x 61

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whereF=0BBBBBBB@L11SwL12Sw00L21SdL22Sd0000H11SwH12Sw00H21SdH22Sd1CCCCCCCAandV=0BBBBBBB@w+Lw0000d+d0000w+Hw+Hw0000d+Hd1CCCCCCCAThenextgenerationmatrixK=FV)]TJ /F5 7.97 Tf 6.58 0 Td[(1isamatrixofreproductionnumbers:K=0BBBBBBB@RL11RL1200RL21RL220000RH11RH1200RH21RH221CCCCCCCATheLPAIbasicreproductionnumberRListheprincipaleigenvalueofthematrixKL=0B@RL11RL12RL21RL221CARL=RL11+RL22+q )]TJ /F2 11.955 Tf 5.48 -9.68 Td[(RL11)-221(RL222+4RL12RL21 2.Similarly,theHPAIreproductionnumberRHistheprincipaleigenvalueofthematrixKH=0B@RH11RH12RH21RH221CARH=RH11+RH22+q )]TJ /F2 11.955 Tf 5.48 -9.69 Td[(RH11)-222(RH222+4RH12RH21 2. 62

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ThereproductionnumberR0isgivenbytheprincipaleigenvalueofthenextgenerationmatrixK.Thusthebasicreproductionnumberofthefullsystem( 4 )isR0=maxfRL,RHg.NotethatifR0<1,thenalleigenvaluesofthesubsysteminvolvinginfectedcompartments(vw,vd,yw,yd)havenegativerealparts( 13 )(Theorem2,page33).Forvaluesofdifferentfromtheeigenvaluesofthesubsystem,wehave(vw,vd,yw,yd)=(0,0,0,0),whichleadstoxw()=xd()=0.Theremainingeigenvaluesofthefullsystemare5=)]TJ /F7 11.955 Tf 9.3 0 Td[(w,6=)]TJ /F7 11.955 Tf 9.3 0 Td[(wand7=)]TJ /F7 11.955 Tf 9.29 0 Td[(d.Hencealltheeigenvaluesarenegativeorhavenegativerealparts.Thus,theDFEislocallyasymptoticallystablewhenR0<1.IfR0>1,thenthe(vw,vd,yw,yd)subsystemhasaneigenvaluewithapositiverealpart,thustheDFEisunstable. 4.3.2LPAI-onlyandHPAI-onlyEquilibriaThesystem( 4 )hastwoboundaryequilibria:theLPAI-onlyequilibriumdenotedby"L=(SLw,ILLw,rLLw,0,0,SLd,ILLd,rLd,0)andtheHPAI-onlyequilibriumdenotedby"H=(SHw,0,0,IHHw,RHHw,SHd,0,0,IHHd).TheinvasionnumberofHPAIwhenthesystemisattheLPAI-onlyequilibriumis^RHLanditisgivenby ^RHL=aRH11+bRH22+q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(aRH11)]TJ /F10 11.955 Tf 11.96 0 Td[(bRH222+4abRH12RH21 2,(4)where a=w(SLw+Bw) wb=d(SLd+Bd) dBw=Z10qw()rLLw()dBd=Z10qd()rLd()d(4) 63

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Similarly,theinvasionnumberofLPAIwhenthesystemisattheHPAI-onlyequilibriumis^RLHand ^RLH=cRL11+dRL22+q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(cRL11)]TJ /F10 11.955 Tf 11.96 0 Td[(dRL222+4cdRL12RL21 2,(4)where c=wSHw w,d=dSHd d.(4)Aswiththereproductionnumbers,theinvasionreproductionnumbersarealsoobtainedthroughthenextgenerationapproach( 13 )wherethenextgenerationoperatorofHPAIinvadingtheequilibriumofLPAIisgivenbyKHL=0B@aRH11aRH12bRH21bRH221CA.Correspondingly,thenextgenerationoperatoroftheLPAIinvadingtheequilibriumofHPAIisgivenbyKLH=0B@cRL11cRL12dRL21dRL221CA.Wecallthemaindiagonalentriesofnextgenerationmatricespopulation-specicinvasionnumbers,anddenotethemby^RL11,H,...,^RH22,Lwhere^RH11,L=aRH11,^RH22,L=bRH22,^RL11,H=cRL11,^RL22,H=dRL22.Wecalltheoffdiagonalentriesthecross-populationinvasionnumbers,anddenotethemby^RL12,H,...,^RH21,L,where^RH12,L=aRH12,^RH21,L=bRH21,^RL12,H=cRL12,^RL21,H=dRL21.WedenotetheforcesofinfectionofLPAIwhenwildanddomesticbirdpopulationsareatthe"LequilibriumbyLLwandLLdrespectively: 64

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LLw=L11ILLw+L12ILLd,LLd=L21ILLw+L22ILLd.(4)ThentheLPAI-onlyequilibrium"Lsatisesthefollowingsteadystateequation. 0=w)]TJ /F7 11.955 Tf 11.95 0 Td[(LLwSLw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSLw,0=LLwSLw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)ILLw,drLLw d=)]TJ /F7 11.955 Tf 9.3 0 Td[(wrLLw,rLLw(0)=LwILLw,0=d)]TJ /F7 11.955 Tf 11.95 0 Td[(LLdSLd)]TJ /F7 11.955 Tf 11.95 0 Td[(dSLd,0=LLdSLd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+d)ILLd,drLd d=)]TJ /F7 11.955 Tf 9.3 0 Td[(drLd,rd(0)=dILLd.(4)First,wesolvetheODEsandobtain;rLLw()=LwILLwe)]TJ /F14 7.97 Tf 6.58 0 Td[(wandrLd()=dILLde)]TJ /F14 7.97 Tf 6.58 0 Td[(d.Then,solvingsystem( 4 )wegetSLw=w LLw+w,ILLw=wLLw (LLw+w)(w+Lw)SLd=d LLd+d,ILLd=dLLd (LLd+d)(d+d).WeshowtheexistenceanduniquenessofanLPAI-onlyequilibriumbyshowingtheexistenceanduniquenessofLLwandLLd.Solvingtheequationsin( 4 )forILLwandILLd,weseethatifLLwandLLd,areunique,soareILLwandILLdifandonlyifL11L226=L12L21.WethensubstitutetheexpressionsforILLwandILLdinto( 4 )andobtain LLw=1LLw LLw+w+2LLd LLd+d,LLd=3LLw LLw+w+4LLd LLd+d,(4) 65

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where1=RL11w,2=RL12wd w,3=RL21dw dand4=RL22d.Basedontheseequationsin( 4 ),settingu1=LLwandu2=LLdwedeneanonlinearoperatorPinthefollowingway.Letu=(u1,u2);thenP(u)=(1u1 u1+w+2u2 u2+d,3u1 u1+w+4u2 u2+d)=u.Foranytwou=(u1,u2)andv=(v1,v2),wesaythatu>vprovidedthatu1>v1andu2>v2.Then,K=fu2R2s.t.u>0g,isapositiveconeinR2.IfwesetC=[0,1+2][0,3+4],thentheoperatorPmapsCintoitself. Proposition4.1. LetDP(u)denotethederivativeoftheoperatorP;thenthespectralradiusofDP(u)islessthan1. Proof. ThederivativeoftheoperatorPisDP(u)=0B@1w (u1+w)22d (u2+d)23w (u1+w)24d (u2+d)21CA.NotethatDP(u)isapositivematrix,sinceallitsentriesarepositive.LetAbea22squarematrixgivenas:A=0B@1 (u1+w)2 (u2+d)3 (u1+w)4 (u2+d)1CA.Clearly,DP(u)A.SinceP(u1,u2)=(u1,u2),dividingbyu1weobtain1=1 u1+w+2 u2+dz,u2 u1=3 u1+w+4 u2+dz,wherez=u2 u1.Letv=0B@1z1CA,thenAv=v.Thus1isaneigenvalueofAcorrespondingtoapositiveeigenvector.ByPerron-FrobeniusTheorem,thespectralradiusofAis(A)=1.Furthermore,(DP(u))<(A)sinceDP(u)1. 66

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Proof. Letu=(u1,u2)andv=(v1,v2)s.t.u>v,thenbytheMeanValueTheorem,P(u1,u2))]TJ /F10 11.955 Tf 11.96 0 Td[(P(v1,v2)=1w (u1+w)2(u1)]TJ /F10 11.955 Tf 11.96 0 Td[(v1)+4d (u2+d)2(u2)]TJ /F10 11.955 Tf 11.95 0 Td[(v2)>0HencePismonotoneinK.Ifu1andu2arelessthan>0,thentheoperatorP(u)satisesP(u)>Au,whereA=0B@1 +w2 +d3 +w4 +d1CA.Noticethatwhen=0,theprincipaleigenvalueofthematrixA=0isRL>1.Determine>0suchthattheprincipaleigenvalueofAisRL=1.LetvbetheeigenvectorcorrespondingtotheprincipaleigenvalueRLofA.Therefore,Av=v,suchthatv>0.Rescalevsothatitscomponentsarelessthan,thatisv=(v1,v2)wherev1v.ToshowtheexistenceofLPAI-onlyequilibrium,wedeneanincreasingsequence;v0=vandvj=P(vj)]TJ /F5 7.97 Tf 6.59 0 Td[(1).Notethatkvjk0,thenDP(v)uDP(w)u.Thus,wehaveDP(u2)(u2)]TJ /F10 11.955 Tf 11.96 0 Td[(u1)DP()(u2)]TJ /F10 11.955 Tf 11.95 0 Td[(u1)DP(u1)(u2)]TJ /F10 11.955 Tf 11.96 0 Td[(u1).Repeatingntimes,weobtain;)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(DP(u2)n(u2)]TJ /F10 11.955 Tf 11.95 0 Td[(u1)u2)]TJ /F10 11.955 Tf 11.96 0 Td[(u1)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(DP(u1)n(u2)]TJ /F10 11.955 Tf 11.95 0 Td[(u1),since,DP()(u2)]TJ /F10 11.955 Tf 12.87 0 Td[(u1)=u2)]TJ /F10 11.955 Tf 12.87 0 Td[(u1.Since(DP(u1))<1and(DP(u2))<1(seeProposition 4.1 ),therefore)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(DP(u2)n!0and)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(DP(u1)n!0.Thus,wehaveu1=u2. 67

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Now,supposethattherearetwoxedpointsu1andu2orderedasu1Ku2,whichmeansu11u21andu12u22.Thenu=u1)]TJ /F10 11.955 Tf 11.95 0 Td[(u2=P(u1))]TJ /F10 11.955 Tf 11.96 0 Td[(P(u2)=DP()(u1)]TJ /F10 11.955 Tf 11.95 0 Td[(u2)=DP()u,whereu=(u1,u2)withu1<0andu2>0,andu1Ku2.NoticethatforanywKv,wehaveDP(w)uDP(v)usinceu1<0andu2>0.Thatiswehave,DP(u1)(u2)]TJ /F10 11.955 Tf 11.96 0 Td[(u1)DP()(u2)]TJ /F10 11.955 Tf 11.95 0 Td[(u1)DP(u2)(u2)]TJ /F10 11.955 Tf 11.96 0 Td[(u1).Applyingthesamestepsasbefore,wearriveatu1=u2.Soineitherorder,thereexistsauniquexedpoint,andthereforeauniqueequilibrium. Theorem4.3. TheLPAI-onlyequilibriumislocallyasymptoticallystableiff^RHL<1. Proof. Weobtainthefollowinglinearsystemforperturbations. duw dt=)]TJ /F6 11.955 Tf 11.96 0 Td[((LLw+w)uw)]TJ /F7 11.955 Tf 11.95 0 Td[(L11SLwvw)]TJ /F7 11.955 Tf 11.95 0 Td[(L12SLwvd)]TJ /F7 11.955 Tf 11.95 0 Td[(H11SLwyw)]TJ /F7 11.955 Tf 11.95 0 Td[(H12SLwyd,dvw dt=LLwuw+(L11SLw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw))vw+L12SLwvd@xw @t+@xw @=)]TJ /F10 11.955 Tf 11.96 0 Td[(qw()(H11yw+H12yd)rLLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wxwxw(0,t)=Lwvwdyw dt=H11SLwyw+H12SLwyd+(H11yw+H12yd)Bw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)ywdzw dt=Hwyw)]TJ /F7 11.955 Tf 11.96 0 Td[(wzwdud dt=)]TJ /F6 11.955 Tf 11.96 0 Td[((LLd+d)ud)]TJ /F7 11.955 Tf 11.96 0 Td[(L21SLdvw)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLdvd)]TJ /F7 11.955 Tf 11.96 0 Td[(H21SLdyw)]TJ /F7 11.955 Tf 11.96 0 Td[(H22SLdyd,dvd dt=LLdud+L21SLdvw+(L22SLd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+d))vd@xd @t+@xd @=)]TJ /F10 11.955 Tf 11.96 0 Td[(qd()(H21yw+H22yd)rLd)]TJ /F7 11.955 Tf 11.95 0 Td[(dxdxd(0,t)=dvddyd dt=H21SLdyw+H22SLdyd+(H21yw+H22yd)Bd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+Hd)yd(4) 68

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whereBwandBdareasdenedin( 4 ).Consideringtheexponentialsolutionssuchasxw(,t)=etxw(),xd(,t)=etxd(),yw=etywandyd=etydweobtaintwonon-homogeneouslinearrstorderdifferentialequations.Solvingthem,weget:xw()=Lwvwe)]TJ /F5 7.97 Tf 6.59 0 Td[((+w))]TJ /F6 11.955 Tf 11.95 0 Td[((H11yw+H12yd)Z0qw(s)rLLw(s)e)]TJ /F5 7.97 Tf 6.59 0 Td[((+w)()]TJ /F3 7.97 Tf 6.58 0 Td[(s)ds.xd()=dvde)]TJ /F5 7.97 Tf 6.59 0 Td[((+d))]TJ /F6 11.955 Tf 11.95 0 Td[((H21yw+H22yd)Z0qd(s)rLLd(s)e)]TJ /F5 7.97 Tf 6.59 0 Td[((+d)()]TJ /F3 7.97 Tf 6.59 0 Td[(s)ds.Fortheremainingequations,whichdonotdependonxw()andxd(),wesupposethattheperturbationsareexponentialfunctionsoftheformuw=etuw,ud=etud,vw=etvw,vd=etvd,zw=etzw.Wegetthefollowingeigenvalueproblemafterdroppingthebars, 0B@AB0C1CA=0B@xy1CA(4)wherex=(uw,ud,vw,vd,zw),y=(yw,yd),A=0BBBBBBBBBB@)]TJ /F6 11.955 Tf 9.3 0 Td[((LLw+w)0)]TJ /F7 11.955 Tf 9.3 0 Td[(L11SLw)]TJ /F7 11.955 Tf 9.3 0 Td[(L12SLw00)]TJ /F6 11.955 Tf 9.3 0 Td[((LLd+d))]TJ /F7 11.955 Tf 9.3 0 Td[(L21SLd)]TJ /F7 11.955 Tf 9.3 0 Td[(L22SLd0LLw0L11SLw)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Lw)L12SLw00LLdL21SLdL22SLd)]TJ /F6 11.955 Tf 11.95 0 Td[((d+d)00000)]TJ /F7 11.955 Tf 9.3 0 Td[(w1CCCCCCCCCCA,B=0BBBBBBBBBB@)]TJ /F7 11.955 Tf 9.3 0 Td[(H11SLw)]TJ /F7 11.955 Tf 9.3 0 Td[(H12SLw)]TJ /F7 11.955 Tf 9.3 0 Td[(H21SLd)]TJ /F7 11.955 Tf 9.3 0 Td[(H22SLd0000Hw01CCCCCCCCCCAC=0B@H11(SLw+Bw))]TJ /F6 11.955 Tf 11.95 0 Td[((w+Hw+Hw)H12(SLw+Bw)H21(SLd+Bd)H22(SLd+Bd))]TJ /F6 11.955 Tf 11.95 0 Td[((d+Hd)1CA. 69

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Theequationsinvolvinghighpathogenicavianinuenza,thatisywandydintheaboveeigenvalueproblem,decouple.Thus,twoeigenvaluesofthesystemwillbedeterminedbythesubsysteminvolvingequationsofywandyd(matrixC;theothereigenvaluesaretheeigenvaluesofA).TheeigenvaluesoftheJacobianmatrixChavenegativerealpartifandonlyifthespectralradiusofthenextgenerationmatrixislessthan1( 13 )(Theorem2,page33).Followingthenextgenerationmatrixapproach,weobtainthefollowingnextgenerationmatrixKHL=FV)]TJ /F5 7.97 Tf 6.59 0 Td[(1whereF=0B@H11(SLw+Bw)H12(SLw+Bw)H21(SLd+Bd)H22(SLd+Bd)1CAandV=0B@w+Hw+Hw00d+Hd1CA.TheprincipaleigenvalueofthenextgenerationmatrixKHLgivestheinvasionnumberofHPAIwhichisdenotedby^RHL;ifthisisgreaterthanorequalto1,thenatleastoneeigenvalueofChasapositiverealpart,sotheLPAI-onlyequilibriumisunstable.ThustheeigenvaluesofChavenegativerealpartif^RHL<1.Bycontradiction,weshowthatiftheeigenvaluesofthematrixAdonothavenon-negativerealparts.ThecharacteristicequationofAis: )]TJ /F7 11.955 Tf 9.3 0 Td[(L12SLwL21SLd(w+)(d+)+[(d+d+))]TJ /F7 11.955 Tf 5.48 -9.69 Td[(LLd+d+)]TJ /F7 11.955 Tf 11.95 0 Td[(L22SLd(d+)][(w+Lw+))]TJ /F7 11.955 Tf 5.48 -9.69 Td[(LLw+w+)]TJ /F7 11.955 Tf 11.96 0 Td[(L11SLw(w+)]=0.(4) 70

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Werewritethe( 4 )as: )]TJ /F6 11.955 Tf 5.47 -9.85 Td[((d+d+))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(LLd+d+)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLd(d+))]TJ /F6 11.955 Tf 12.96 -9.85 Td[((w+Lw+))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(LLw+w+)]TJ /F7 11.955 Tf 11.95 0 Td[(L11SLw(w+) (d+)(w+)=L12SLwL21SLd.(4)If<()0,then (d+d+))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(LLd+d+)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLd(d+) d+=(d+d+))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(LLd+d+ (d+))]TJ /F7 11.955 Tf 11.95 0 Td[(L22SLdjd+d+jLLd+d+ jd+j)]TJ /F7 11.955 Tf 11.95 0 Td[(L22SLd>jd+d+j)]TJ /F7 11.955 Tf 17.93 0 Td[(L22SLdd+d)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLd.(4)Similaranalysisyields, jw+Lw+jLLw+w+ w+)]TJ /F7 11.955 Tf 11.96 0 Td[(L11SLww+Lw)]TJ /F7 11.955 Tf 11.96 0 Td[(L11SLw.(4)Sothecharacteristicequation( 4 )leadsthefollowinginequality L12SLwL21SLd>)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(d+d)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLd)]TJ /F7 11.955 Tf 12.96 -9.68 Td[(w+Lw)]TJ /F7 11.955 Tf 11.96 0 Td[(L11SLw.(4)FromtheequationsfortheLPAI-onlyequilibriumweobtainw+Lw)]TJ /F7 11.955 Tf 12.08 0 Td[(L11SLw=L12ILLdSLw ILLwandd+d)]TJ /F7 11.955 Tf 11.96 0 Td[(L22SLd=L21ILLwSLd ILLd.Thustheinequality( 4 )becomes L12SLwL21SLd>L21ILLwSLd ILLdL12ILLdSLw ILLw=L12SLwL21SLd.(4)Thiscontradictioncompletestheproof.Hence,thecharacteristicequation( 4 )cannothaverootswithnon-negativerealparts. Theorem4.4. ThereexistsauniqueHPAI-onlyequilibriumwhenRH>1.TheHPAI-onlyequilibriumislocallyasymptoticallystableif^RLH<1andunstableif^RLH>1. Proof. ProofofTheorem 4.4 isverysimilartotheproofofTheorem 4.2 andTheorem 4.3 ,andwillbeomitted. 71

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4.3.3CoexistenceEquilibriumInthissubsection,weinvestigatetheexistenceofthecoexistenceequilibrium(i.e.,interiorequilibrium),thatis,theequilibriuminwhichbothlowpathogenicandhighpathogenicavianinuenzaarepresentinwildanddomesticbirdpopulations.Wesupposethatalltheparameters,L11,L12,...,H21,H22arepositive.Specialcasescanbeobtainedbysettingsomeorallthecross-coefcientstozero.Forinstance,theLPAIandHPAImightcoexistonlyinthewildbirdpopulation,andonlyHPAIpersistinthedomesticbirdpopulation.Inthispaper,wewillonlyconsiderthecasewhenbothpathogencoexistinbothpopulations.Thus,thecoexistenceequilibrium"=(Sw,ILw,rLw,IHw,RHw,Sd,ILd,rd,IHd),satisesthefollowingequations 0=w)]TJ /F7 11.955 Tf 11.96 0 Td[(LwSw)]TJ /F7 11.955 Tf 11.95 0 Td[(HwSw)]TJ /F7 11.955 Tf 11.96 0 Td[(wSw,0=LwSw)]TJ /F6 11.955 Tf 11.95 0 Td[((w+Lw)ILwdrLw d=)]TJ /F10 11.955 Tf 9.3 0 Td[(qw()HwrLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wrLw,rLw(0)=LwILw,0=HwSw+HwZ10qw()rLw()d)]TJ /F6 11.955 Tf 11.96 0 Td[((w+Hw+Hw)IHw,0=HwIHw)]TJ /F7 11.955 Tf 11.96 0 Td[(wRHw,0=d)]TJ /F7 11.955 Tf 11.95 0 Td[(LdSd)]TJ /F7 11.955 Tf 11.96 0 Td[(HdSd)]TJ /F7 11.955 Tf 11.95 0 Td[(dSd,0=LdSd)]TJ /F6 11.955 Tf 11.95 0 Td[((d+d+m)ILd,drd d=)]TJ /F10 11.955 Tf 9.3 0 Td[(qd()Hdrd)]TJ /F7 11.955 Tf 11.95 0 Td[(drd,rd(0)=dILd,0=HdSd+HdZ10qd()rd()d+mILd)]TJ /F6 11.955 Tf 11.96 0 Td[((d+Hd)IHd.(4)WestudytheexistenceoftheinteriorequilibriumbyshowingtheexistenceoftheforcesofinfectionsLw,Ld,HwandHd.Wesolveequationsoftheequilibriumfor 72

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Sw,ILw,rLw,IHw,Sd,ILd,rLd,andIHd,andobtain:Sw=w Lw+Hw+w,ILw=LwSw w+Lw,IHw=HwSw w+Hw+Hw+HwR10qw()rLw()d w+Hw+HwSd=d Ld+Hd+d,ILd=LdSd d+d,IHd=HdSd d+Hd+HdR10qd()rd()d d+Hd.Setting,w()=e)]TJ /F14 7.97 Tf 6.59 0 Td[(HwR0qw(s)ds)]TJ /F14 7.97 Tf 6.58 0 Td[(wandd()=e)]TJ /F14 7.97 Tf 6.58 0 Td[(HdR0qd(s)ds)]TJ /F14 7.97 Tf 6.58 0 Td[(d,weobtainrLw()=LwILww(),rLd()=dILdd().Usingaboveexpressionsandthedenitionsofforcesofinfections,wearriveatthefollowingequationsLw=dL12Ld (d+d)(d+Hd+Ld)+wL11Lw (Lw+w)(w+Hw+Lw), (4)Ld=dL22Ld (d+d)(d+Hd+Ld)+wL21Lw (Lw+w)(w+Hw+Lw), (4)Hw=wH11Hw (w+Hw+Hw)(w+Hw+Lw)1+LwLw w+LwZ10qw()w()d (4)+dH12Hd (d+Hd)(d+Hd+Ld)1+dLd d+dZ10qd()d()d,Hd=wH21Hw (w+Hw+Hw)(w+Hw+Lw)1+LwLw w+LwZ10qw()w()d (4)+dH22Hd (d+Hd)(d+Hd+Ld)1+dLd d+dZ10qd()d()d.Notethatw()andd()dependonHw,Hd.Using( 4 )-( 4 ),wedeneanon-linearoperatorTinthefollowingway.Letu=(Lw,Ld,Hw,Hd),then T(u)=(T1(u),T2(u),T3(u),T4(u))=u(4) 73

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Forvectors,u1=(1Lw,1Ld,1Hw,1Hd)andu2=(2Lw,2Ld,2Hw,2Hd),wedeneapartialorderandsaythatu1Ku2ifandonlyif1Lw2Lw,1Ld2Ld,1Hw2Hw,1Hd2Hd.WiththispartialorderK,KT=fu2R4uK0gisapositiveconeinR4.WedenethesetCTtobeCT:=[0,K1][0,K2][0,K3][0,K4],whereK1=L11w w+L12d d,K2=L21w w+L22d d,K3=2H11w w+H12d d,K4=H21w w+H22d d.Thenon-linearoperatorTmapsCTintoitself,anditismonotoneintheconeKTasseeninthefollowingproposition. Proposition4.2. 1) DerivativesofthenonlinearoperatorTsatisfythefollowinginequalities:@Ti @Lw>0@Ti @Ld>0@Ti @Hw<0@Ti @Hd<0i=1,2.@Ti @Lw<0@Ti @Ld<0@Ti @Hw>0@Ti @Hd>0i=3,4. 2) TismonotoneinKT,thatisu1Ku2=)T(u1)KT(u2). 3) TmapsthesetCTintoitself.T:CT!CT. Proof. 1) Weonlyprovetheinequalities@T1 @Lw>0and@T1 @Hw<0,sincetheinequalitiesofotherderivativeswheni=1,2canbederivedbyapplyingthesamesteps.NotethatT1(Lw,Ld,Hw,Hd)=dL12Ld (d+d)(d+Hd+Ld)+wL11Lw (Lw+w)(w+Hw+Lw).Thus;@T1 @Lw=(wL11)(w+Hw) (Lw+w)(w+Hw+Lw)2>0.And,@T1 @Hw=)]TJ /F5 7.97 Tf 6.58 0 Td[(wL11Lw (Lw+w)(w+Hw+Lw)2<0.Next,weprovetheinequalities@T3 @Lw<0and@T3 @Hw>0.Inequalitiesfortheotherderivativeswheni=3,4canbeshowninasimilarway.Notethat,T3(Lw,Ld,Hw,Hd)=Hw 74

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asin( 4 ).DerivativeofT3withrespecttoLwis@T3 @Lw=)]TJ /F6 11.955 Tf 9.3 0 Td[(wH11Hw (w+Hw+Hw)(w+Hw+Lw)21+LwLw w+LwZ10qw()w()d+wH11Hw (w+Hw+Hw)(w+Hw+Lw)Lw w+LwZ10qw()w()d.Combiningtheterms,weobtain@T3 @Lw=)]TJ /F6 11.955 Tf 9.29 0 Td[(wH11Hw (w+Hw+Hw)(w+Hw+Lw)21)]TJ /F7 11.955 Tf 13.15 8.77 Td[(Lw(w+Hw) w+LwZ10qw()w()d.Clearly,@T3 @Lw<0isnegative,providedthat (w+Hw)Z10qw()w()d<1.(4)Since0qw()1,theintegral( 4 )islessthanthefollowingintegralZ10(Hwqw()+w)w()d=1,(notethatZ10qw(s)ds=1).DerivativeofT3withrespecttoHwis@T3 @Hw=wH11(w+Lw) (w+Hw+Hw)(w+Hw+Lw)21+LwLw w+LwZ10qw()w()d)]TJ /F6 11.955 Tf 78.71 8.76 Td[(wH11Hw (w+Hw+Hw)(w+Hw+Lw)LwLw w+LwZ10qw()Z0qw(s)dsw()d=wH11(w+Lw) (w+Hw+Hw)(w+Hw+Lw)21+LwLw w+LwZ10qw()w()d)]TJ /F7 11.955 Tf 14.35 8.77 Td[(LwLwHw(w+Hw+Lw) (w+Lw)(w+Lw)Z10qw()Z0qw(s)dsw()d.Reorganizingtheterms,weobtain@T3 @Hw=wH11(w+Lw) (w+Hw+Hw)(w+Hw+Lw)2"1+LwLw w+LwZ10qw()w()d (4))]TJ /F7 11.955 Tf 13.15 0 Td[(HwZ10qw()Z0qw(s)dsw()d)]TJ /F7 11.955 Tf 24.55 8.76 Td[(Hw2 w+LwZ10qw()Z0qw(s)dsw()d!#. 75

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Thederivative@T3 @Hwispositiveiftheterminsidethesquarebracketsin( 4 )ispositive.Thus@T3 @Hw>0if,Z10qw()w()d)]TJ /F7 11.955 Tf 11.95 0 Td[(HwZ10qw()Z0qw(s)dsw()d>0, (4)and1)]TJ /F7 11.955 Tf 51.84 8.77 Td[(LwLw (w+Lw)(w+Lw)Hw2Z10qw()Z0qw(s)dsw()d>0. (4)Applyingintegrationbyparts,( 4 )becomesZ10qw()w()d+Z0qw(s)dse)]TJ /F14 7.97 Tf 6.59 0 Td[(ww()10)]TJ /F11 11.955 Tf 11.96 16.27 Td[(Z10w()qw()e)]TJ /F14 7.97 Tf 6.59 0 Td[(w)]TJ /F7 11.955 Tf 11.96 0 Td[(wZ0qw(s)dse)]TJ /F14 7.97 Tf 6.58 0 Td[(wd=wZ10Z0qw(s)dsw()d>0.SinceLw w+LwLw w+Lw<1,theexpressionin( 4 )isgreaterthanthefollowing 1)]TJ /F7 11.955 Tf 11.96 0 Td[(Hw2Z10qw()Z0qw(s)dsw()d.(4)Byintegrationbyparts,( 4 )becomes1)]TJ /F7 11.955 Tf 11.95 0 Td[(HwZ10qw()w()d+wHwZ10qw()Z0qw(s)dsw()d,whichispositive,sinceHwR10qw()w()d<1. 2) WeprovethemonotonicityoftheoperatorT,byshowingthatT3(u1)T3(u2)whenever(u1)K(u2).Becauseofthesymmetry,thestepsforprovingtherestoftheinequalitiesT1(u1)T1(u2),T2(u1)T2(u2)andT4(u1)T4(u2)aresimilar.T3(u1))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(u2)=T3(1Lw,1Ld,1Hw,1Hd))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(2Lw,2Ld,2Hw,2Hd)=T3(1Lw,1Ld,1Hw,1Hd))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(2Lw,1Ld,1Hw,1Hd)+T3(2Lw,1Ld,1Hw,1Hd))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(2Lw,2Ld,1Hw,1Hd)+T3(2Lw,2Ld,1Hw,1Hd))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(2Lw,2Ld,2Hw,1Hd)+T3(2Lw,2Ld,2Hw,1Hd))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(2Lw,2Ld,2Hw,2Hd). 76

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UsingtheMeanValueTheoremweobtainT3(u1))]TJ /F10 11.955 Tf 11.96 0 Td[(T3(u2)=@T3 @Lw(1,1Ld,1Hw,1Hd)(1Lw)]TJ /F7 11.955 Tf 11.96 0 Td[(2Lw)+@T3 @Ld(2Lw,2,1Hw,1Hd)(1Ld)]TJ /F7 11.955 Tf 11.96 0 Td[(2Ld)+@T3 @Hw(2Lw,2Ld,3,1Hd)(1Hw)]TJ /F7 11.955 Tf 11.96 0 Td[(2Hw)+@T3 @Hd(2Lw,2Ld,2Hw,4)(1Hd)]TJ /F7 11.955 Tf 11.95 0 Td[(2Hd).Wejustprovedthat@T3 @Lw<0@T3 @Ld<0@T3 @Hw>0@T3 @Hd>0.Since(u1)K(u2),wehave1Lw)]TJ /F7 11.955 Tf 12.19 0 Td[(2Lw0,1Ld)]TJ /F7 11.955 Tf 12.19 0 Td[(2Ld0,1Hw)]TJ /F7 11.955 Tf 12.19 0 Td[(2Hw0,1Hd)]TJ /F7 11.955 Tf 12.19 0 Td[(2Hd0.ThusT3(u1))]TJ /F10 11.955 Tf 11.95 0 Td[(T3(u2)0. 3) Next,weshowthatTmapsthesetCTintoitselfbyshowingitforT1:CT!CT.SinceLd d+Hd+Ld<1andLw w+Hw+Lw<1,itisclearthatT1(u)L12d d+L11w w. Withthissetting,let"L=(LLw,LLd,0,0)denotetheLPAI-onlyequilibrium,"H=(0,0,HHw,HHd)denotetheHPAI-onlyequilibriumand"=(Lw,Ld,Hw,Hd)denotethecoexistenceequilibrium.Intheprevioussection,weshowedthatifbothinvasionnumbersaregreaterthanunity,thenbothLPAI-onlyandHPAI-onlyequilibriaareunstable.Next,weshowthatinsuchasituation,thereexistsacoexistenceequilibrium,".Werstlinearizethenon-linearoperatorTaroundtheLPAI-onlyandtheHPAI-onlyequilibria,anddenotethelinearizationbyDT("j)forj=L,H.Foranyu=(Lw,Ld,Hw,Hd),wehave T("j+u)="j+DT("j)u+N(u)j=L,H.(4) 77

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LetjbethespectralradiusofDT("j)forj=L,H,thenbythePerron-FrobeniusTheoremjisaneigenvalueofthelinearoperatorDT("j).ByProposition 4.2 ,DT("j)isapositivematrixintheordercreatedbytheconeKT.Thus,thespectralradiusisasimpleeigenvaluetowhichtherecorrespondsa“positive”eigenvectorintheconeKT.Inparticular;DT("L)v=Lv,DT("H)u=Hu,wherevK0anduK0. Proposition4.3. ThespectralradiusL>1ifandonlyif^RHL>1,andthespectralradiusH>1ifandonlyif^RLH>1. Proof. WeonlyshowthatL>1iff^RHL>1,sincetheothercaseissimilar.WehaveDT("L)v=Lvwherevisthepositiveeigenvector,vK0.ThelinearizationmatrixDT("L)attheLPAI-onlyequilibriumisgivenasfollows;DT("L)=0BBBBBBBB@@T1 @Lw("L)@T1 @Ld("L)@T1 @Hw("L)@T1 @Hd("L)@T2 @Lw("L)@T2 @Ld("L)@T2 @Hw("L)@T2 @Hd("L)@T3 @Lw("L)@T3 @Ld("L)@T3 @Hw("L)@T3 @Hd("L)@T4 @Lw("L)@T4 @Ld("L)@T4 @Hw("L)@T4 @Hd("L)1CCCCCCCCAwhichisequivalenttothefollowingblockdiagonalmatrix,DT("L)=0B@DTL1,2DTH1,20DTH3,41CA 78

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The22blockdiagonalmatricesareasfollows;DTL1,2=0B@wL11w (Lw+w)(w+LLw)2dL12d (Ld+d)(d+LLd)2wL21w (Lw+w)(w+LLw)2dL22d (Ld+d)(d+LLd)21CADTH1,2=0B@)]TJ /F5 7.97 Tf 6.59 0 Td[(wL11LLw (Lw+w)(w+LLw)2)]TJ /F5 7.97 Tf 6.59 0 Td[(dL12LLd (Ld+d)(d+LLd)2)]TJ /F5 7.97 Tf 6.59 0 Td[(wL21LLw (Lw+w)(w+LLw)2)]TJ /F5 7.97 Tf 6.59 0 Td[(dL22LLd (Ld+d)(d+LLd)21CAandthecomponentsofthe22matrixDTH3,4areasfollows;@T3 @Hw("L)=aRH11@T3 @Hd("L)=aRH12@T4 @Hw("L)=bRH21@T4 @Hd("L)=bRH22.TheprincipaleigenvalueofDTH3,4is^RHL>1.ItremainstoshowthattheeigenvaluesofDTL1,2aresmallerthanone,whichweshowedinProposition 4.1 . Theorem4.5. Assume^RLH>1and^RHL>1,thenthereexistsatleastonecoexistenceequilibrium"=(Lw,Ld,Hw,Hd). Proof. Since^RLH>1and^RHL>1,Proposition 4.3 impliesthatL>1andH>1.Notethatwealsohave;"H0and>0s.t."H+u
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Notethat(H)]TJ /F6 11.955 Tf 12.7 0 Td[(1)uK0and(H)]TJ /F6 11.955 Tf 12.7 0 Td[(1)u+N(u)K0forsmallenough.Thus,T("H+u)K"H+u.Similarly,T("L)]TJ /F7 11.955 Tf 11.73 0 Td[(v)="L)]TJ /F7 11.955 Tf 11.73 0 Td[(v)]TJ /F7 11.955 Tf 11.73 0 Td[((L)]TJ /F6 11.955 Tf 11.74 0 Td[(1)v+2N(v).Sinceforsmallenough,wehave)]TJ /F6 11.955 Tf 9.3 0 Td[((L)]TJ /F6 11.955 Tf 11.95 0 Td[(1)v+N(u)K0.Thus,wehaveT("L)]TJ /F7 11.955 Tf 11.96 0 Td[(v)K"L)]TJ /F7 11.955 Tf 11.96 0 Td[(v.Tisamonotoneoperator,soweapplytheoperatorTtotheaboveinequalityrepeatedlyandobtain;Tn("L)]TJ /F7 11.955 Tf 11.95 0 Td[(v)KTn)]TJ /F5 7.97 Tf 6.58 0 Td[(1("L)]TJ /F7 11.955 Tf 11.96 0 Td[(v)KK"L)]TJ /F7 11.955 Tf 11.96 0 Td[(v.Hence,Tn("L)]TJ /F7 11.955 Tf 11.95 0 Td[(v)isadecreasingsequence.Inadditionwehave;"H+uKT("H+u)KT("L)]TJ /F7 11.955 Tf 11.95 0 Td[(v)Similarly,applyingthenon-linearoperatorTntimes,wehave"H+uKTn("L)]TJ /F7 11.955 Tf 11.96 0 Td[(v)HenceTn("L)]TJ /F7 11.955 Tf 11.23 0 Td[(v)isadecreasingsequenceboundedbelowbysomethingstrictlylargerthan"H.Thus,thesequenceconvergestosomethingwithstrictlypositivecomponents.Tn("L)]TJ /F7 11.955 Tf 11.96 0 Td[(v)!"K"H+uasn!1Thus"=(Lw,Ld,Hw,Hd)issuchthatLw>0,Ld>0,Hw>0andHd>0.Hence,thereexistsacoexistenceequilibrium.Ournumericalsimulationshavenotrevealedalternativeequilibria. 4.4SimulationsUnderstandinghowLPAIandHPAIcompeteandcoexistinwildanddomesticbirdpopulationscanfurtherbeapproachedthroughsimulations.Todoso,itisnecessarytoassesssomereasonablevaluesforparametersinthemodels.Theparametervalueswe 80

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chooseareforillustrativepurposes,groundedinempiricalstudies,buttoascertainmoreaccuratevaluesrequiresmoredetailedempiricalstudiesinthefuture. 4.4.1EstimatingParameterValuesDeterminingrealisticoratleastplausibleparametervaluesisobstructedbytheenormousdiversityofwildanddomesticbirdspeciesthatcanbeaffectedbyavianinuenzaandthelackoftimeseriesdata.AvianinuenzaALPAIviruseshavebeenisolatedfrommorethan100differentspeciesofwildbirds.AvianinuenzaAvirusesarepredominantlyfoundingulls,ternsandshorebirdsorwaterfowlsuchasducks,geeseandswans( 8 ).Thesewildbirdsareconsideredasreservoirs(hosts)forLPAIviruses.HPAIvirusesalsoinfectthesespeciespredominantly,killingsomespecieswithindaysandinfectingotherswithoutsymptoms.Averagelifespanvariesdramaticallyfromspeciestospecies.Mallardshavealifespanof3years( 67 )whilealbatrossescanliveupto38years.Atableofvariousbirds'lifespanisgivenin( 57 ).WeassumeLPAIisnotvirulenttowildbirds( 32 ).WefurthertakewildbirdstobeinfectedwithLPAIforarangeof2-21days.WeassumethesamedurationfortheHPAIinfection.Hence,Lw,HwandHwrangefrom365=2)]TJ /F6 11.955 Tf 12.2 0 Td[(365=21.Therecruitmentrateofwildbirdsisunknown.Wetakewintherange1000)]TJ /F6 11.955 Tf 12.21 0 Td[(3000birdsperyear.Thisimpliesacarryingcapacityofwildbirdsfrom500to15,000.Weuseasimilarparameterrangefordomesticfowl.ThismightliterallypertaintosaythewildwaterfowlpopulationsfoundinasinglesmalllakeinChina,interactingwithalocalpopulationofdomesticwaterfowl.Alternatively,thiscouldrefertopopulation”units”,andthuslargerspatialareas.PoultryisinfectedwithLPAIvirusesmainlythroughcontactwithinfectedwildbirdsorcontaminatedsurfacesand/orwater.LPAIisamildillnessinpoultrytypicallyleadingtorecovery.WeassumeaninfectionperiodforLPAIof2)]TJ /F6 11.955 Tf 12.48 0 Td[(21daysinpoultry.HPAIisextremelyvirulentinpoultryandcausessevereillnessanddeath,typicallywithin48hours.WeassumenorecoveryfromHPAIinpoultrysinceaffectedindividualseitherdieoraredestroyedforsecurityreasons.Poultryisusuallykeptfor2years( 42 );wetakea 81

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Table4-3. Parameterranges ParameterRangeAverage w0.027-1year)]TJ /F5 7.97 Tf 6.59 0 Td[(10.1w1000-3000birds/year1500Lw365/21-365/2year)]TJ /F5 7.97 Tf 6.59 0 Td[(136.5Hw365/21-365/2year)]TJ /F5 7.97 Tf 6.59 0 Td[(136.5Hw365/21-365/2year)]TJ /F5 7.97 Tf 6.59 0 Td[(136.5qw0-10.5d0.2-2year)]TJ /F5 7.97 Tf 6.58 0 Td[(10.5d1000-3000birds/year1500Ld365/21-365/2year)]TJ /F5 7.97 Tf 6.59 0 Td[(136.5Hd365/5-365/2year)]TJ /F5 7.97 Tf 6.58 0 Td[(180.0qd0-10.5 range0.5)]TJ /F6 11.955 Tf 12.29 0 Td[(5years,sothatd=0.2to2year)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Thereare20.4billionpoultryunitsintheworld( 42 ).Wetakedintherange1000)]TJ /F6 11.955 Tf 12.49 0 Td[(3000withaveragevalueof1500.Thisisconsistentwiththenumberofpoultryunitsestimatedfromliteraturevaluesiftheyaremeasuredinunitsof107. 4.4.2MainquestionsAI'srichecologyandevolutionisasourceofnovelmathematicalmodelscapableofaddressingnewquestionsinbiology.Theoretically,eachpopulationmaybeasourceforapathogen,wheretheintra-populationtransmissionofthepathogenallowsthepathogentosustainitselfwithinthefocalpopulation,orasink,wheretheintra-populationtransmissionisnotsufcienttosustainthepathogenbuttransmissioninthesinkpopulationismaintainedbyspilloverinfectionfromasourcepopulation( 12 ).Naturally,thepathogenpersistsifatleastoneofthehostpopulationsisasource.However,asinglepathogenmightalsopersistifbothhostpopulationsaresinks(basicallybecausecross-transmissionineffectincreasesthethenumberofavailablehosts).Inthecasewhentwohostpopulationsandtwopathogensarepresent,thesituationismorecomplex.WewillcallpopulationAasinkforpathogenpifpathogenpcannotpersistinpopulationAifpopulationAisisolatedfrompopulationB.Couldapathogenpersistinsink-sinkhostpopulationswhenundercompetitionfromanother 82

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pathogen?If“yes”,underwhatconditions?Couldtwopathogenspersistifbothhostpopulationsaresinkpopulationsforeachoneofthem?Thestatusofwildbirdsanddomesticbirdsassource/sinksforLPAIandHPAIvirusesinsomecasesisknown.WildbirdsareasourcehostpopulationforLPAIviruses,assomespeciesofwildbirdsareanaturalreservoirforthem.ThereislittlediscussionintheliteratureaboutwhetherLPAIvirusesareendemicindomesticbirdpopulations.Basedonthedata,however,ourresultsin( 38 )concludedthatdomesticbirdsareasinkhostpopulationfortheLPAIviruses.AlthoughweestimatedtheLPAIvirusreproductionnumbertobeaboveone,LPAIcannotpersistonitsowninpoultrybecauseitisout-competedbyHPAI.Ontheotherhand,HPAIvirusesarenowendemicindomesticbirdpopulationsinsomecountriesinAsiaandAfrica( 52 ),andourmodelcapturesthatscenario( 38 ).Thesource/sinkstatusofwildanddomesticbirdsforHPAIandLPAIaresummarizedinTable 4.4.2 . Table4-4. Source-sinkstatusofbirdstoAIviruses. LPAIHPAI wildbirdssource?domesticbirdssinksource Thesource/sinkstatusofwildbirdsforHPAIvirusesisanopenquestionofsignicantinterest( 58 ; 55 ).IstheHPAIviruscapableofsustainedtransmissioninwildbirdpopulation?Whatistheroleofcross-immunity?WeaddressthesequestionsaswellasthequestionofoscillatorycoexistenceofLPAIandHPAIthroughtheODEversionofmodel( 4 )(inwhichqwandqdareconstantsratherthanfunctionsoftime-since-infection)inthenextsubsection. 4.4.3SimulationswiththefullODEsystemWeexploredconditionsforcoexistencebyconductingsimulationsoftheordinarydifferentialequation(ODE)systemcorrespondingtomodel( 4 ).IntheODEsystem,therelativesusceptibilitiesofLPAI-recoveredbirds,whichin( 4 )wereqw()andqd(), 83

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aresettoconstantsqwandqd,meaningthatcross-immunitydoesnotfadewithtime.Therefore,allLPAI-recoveredbirdsineachpopulationarethesame,andsocanbecombinedintovariablesRLwandRLd,withtherateofchangeforthewildpopulationgivenbydRLw dt=LwILw)]TJ /F10 11.955 Tf 11.95 0 Td[(qwHwRLw)]TJ /F7 11.955 Tf 11.95 0 Td[(wRLw(andananalogousequationforthedomesticpopulation).IntheHPAI-infectedequations,theintegralsarereplacedbyqwRLworqdRLd,givingasystemofnineODEs.WeinvestigatescenariosofcoexistenceofLPAIandHPAIinwildanddomesticbirdsintheformofanequilibriumorintheformofsustainedoscillations.Wewillcalltheorderofprevalences“realistic”ifinthewildbirdsLPAIprevalenceishigherthanHPAIprevalence,andindomesticbirdsHPAIprevalenceishigherthanLPAIprevalence.Wewillexpectourprevalencesinthesimulationtobeinthisrealisticorder.Figure 4-2 showsacoexistenceequilibriumwithrealisticparametervaluesandrealisticprevalenceorder,thatisHPAI'sprevalenceindomesticbidsishigherthanLPAIindomesticandLPAIprevalenceishigherthantheHPAIprevalenceinwildbirds.Thesolutionstabilizestoanequilibrium.ForFigure 4-2 theLPAIreproductionnumbersareRL11=2.05,RL12=0.91.RL21=0.835,RL22=0.984.Inaddition,theHPAIreproductionnumbersareRH11=0.546,RH12=0.432,RH21=0.0863andRH22=2.7.Theinvasionnumbersare^RHL=1.75and^RLH=1.98.Weseethat,asweexpect,thepopulation-specicreproductionnumbersofLPAIinwildbirdsandHPAIindomesticbirdsarehigherthanone;allothernumbersarelowerthanone.Withtheseparameters,wildbirdsareasinkforHPAIwithrealisticparametervaluesandarealisticorderofprevalences.WenotethatwecanobtainwithrealisticparametersandrealisticprevalenceorderacasewhereHPAIinwildbirdsisasource.However,theIHwwouldbelargerandalargerIHwshouldbemoredetectableinpractice.ThuswiththeavailableinformationwecannotdeduceforsurewhetherHPAI 84

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willpersistonitsowninwildbirds;however,themodelsuggeststhatthesituationisclosesttorealityifHPAIisasinkforwildbirds. Figure4-2. Figureshowscoexistencewithrealisticparametervalues.Theparametervaluesusedinthegureare:w=2000,w=0.25,Hw=36.5,Hw=36.5,Lw=73,qw=0.5,L11=.018776,H11=0.005,d=1020,d=0.5,Hd=36.5,d=52.14,qd=0.5,L22=.02539,H22=0.04897,L12=0.006,L21=0.03,H12=0.002,H21=0.031.ThereproductionnumbersareRL=2.54andRH=2.71.Theinvasioncoefcientsareasfollows:^RHL=1.75and^RLH=1.98.TheredlineshowsHPAIinwildbirds,theorangedashedlineshowsHPAIindomesticbirds,thebluelineshowsLPAIinwildbirds,thegreendashedlineshowsLPAIindomesticbirds. Figure 4-3 showsthatthefullsystemcanexhibitsustained,complexoscillations.Wenotethattheprevalencesaregenerallyinrealisticorder.ForwildbirdsLPAIisgenerallyhigherthanHPAI.Thereversedorderisobservedfordomesticbirds.TheoscillationsofLPAIandHPAIareshiftedhalfaperiodbothinwildanddomesticbirds.Thatis,whenLPAIisathighvalues,HPAIisatlowvaluesandviceversa.ThisisamanifestationofthecompetitionofLPAIandHPAIforsusceptiblehostsinbothwildanddomesticbirds.Wenotethatinthefullsystemoscillationscanbeobtainedforrelativelyintermediateorlowvaluesforqwandqd,whichshowsthatevenhighlevelsofcross-immunitytoHPAIcandestabilizethesystem.TheparametersHdandHwchangetheshapeoftheoscillations.Ingeneral,oscillations,wheneverfound,areobservedinamodestneighbourhoodoftheparametersforwhichtheyoccur. 85

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Figure4-3. Figureshowsoscillationswithrealisticparametervalues.Theparametervaluesusedinthegureare:w=2000,w=0.25,Hw=36.5,Hw=36.5,Lw=73,qw=0.45,L11=.018776,H11=0.015,d=1020,d=0.5,Hd=36.5,d=52.14,qd=0.5,L22=.025,H22=0.04897,L12=0.006,L21=0.03,H12=0.0,H21=0.031.ThereproductionnumbersareRL=2.54andRH=2.7.Theinvasioncoefcientsareasfollows:^RL=1.37and^RH=1.86.TheredlineshowsHPAIinwildbirds,theorangedashedlineshowsHPAIindomesticbirds,thebluelineshowsLPAIinwildbirds,thegreendashedlineshowsLPAIindomesticbirds. Furthermore,wenotethatoscillationandpersistenceofHPAIoccursinthecasewhenH12=0,thatiswhentransmissionfromdomestictowildbirdsofHPAIdoesnotoccur.InthiscasepersistenceofHPAIisonlypossibleifRH11>1.WenotethatHPAIinwildbirdsemerges(orislikelydetectable)onlyfromtimetotime.Figure 4-4 isanillustrationofasink-sinkscenarioforbothpathogens.Asink-sinkscenarioisascenariowherebothpathogensaresinksforeachofthepopulationsbuttheycanpersisttogetherinacoexistenceequilibrium.Wesaythatasink-sinkscenariooccursifthefollowingissatisedineachofthepopulationsiftheyareisolated(nocross-transmission): Thereproductionnumbersandtheinvasionnumbersofbothpathogensaresmallerthanone.Wewereabletoproduceanexampleofthisscenario,whereallintraandcross-populationcomponentsofthereproductionnumbersandinvasionreproductionnumbersare 86

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smallerthanone.ThecoexistenceofLPAIandHPAIunderasink-sinkscenarioisshowninFigure 4-4 .Allcomponentsofthereproductionnumbersandtheinvasion Figure4-4. Figureshowscoexistencewithrealisticparametervalues.Theparametervaluesusedinthegureare:w=2000,w=0.25,Hw=36.5,Hw=36.5,Lw=73,qw=0.426,L11=.0086,H11=0.005,d=1020,d=0.5,Hd=36.5,d=52.14,qd=1,L22=.02539,H22=0.0166,L12=0.0043,L21=0.0131,H12=0.0014,H21=0.0332.ThereproductionnumbersareRL=1.45andRH=1.29.Theinvasioncoefcientsareasfollows:^RL=1.17and^RH=1.22.TheredlineshowsHPAIinwildbirds,theorangedashedlineshowsHPAIindomesticbirds,thebluelineshowsLPAIinwildbirds,thegreendashedlineshowsLPAIindomesticbirds. reproductionnumbersaresmallerthanone,andarelistedintable 4-5 . Table4-5. Sink-Sink ReproductionnumbersValuesInvasionNumbersValues RL110.94^RH11,L0.46RL120.65^RH12,L0.26RL210.36^RH21,L0.92RL220.98^RH22,L0.91RH110.55^RL11,H0.86RH120.3^RL12,H0.6RH210.92^RL21,H0.26RH220.91^RL22,H0.69 Inthiscase,ifallcross-coefcientsp12=p21=0wherep=L,H,thenbothLPAIandHPAIwilldieout.Persistenceofbothpathogensoccursonlythroughthe 87

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cross-populationtransmission.Thisscenarioiseasytondforanyparametersbutinourexampletheparametersareplausableandwehavearealisticprevalenceorderinwildanddomesticbirds. 4.4.4LPAIandHPAIdynamicsinthewildbirdsystemonlyWesawthatthefullODEsystemcorrespondingtosystem( 4 )canexhibitoscillationswhereLPAIandHPAIcoexist.Aninterestingquestionoccurswhetherthecoexistenceequilibriumcanlosestabilityifrestrictedtojustthewildbirdsystem.ThisquestionisofparticularimportanceintheODEcaseasitiswellknownthatsimpleODEmodelswithcrossimmunitydonotleadtooscillations.Forinstance,Castillo-Chavezetal.foundthatagestructureorquarantineneedstobeintroducedforacross-immunitymodeltoshowoscillations( 6 ; 7 ).However,itturnsoutthatthisisnotthecasewithsystem( 4 ).Thecharacteristicequationofthecoexistenceequilibriumlooks“almost”stablebutforsomeparametervaluesthecoexistenceequilibriumcanbedestabilized.Figure 4-5 showssustainedoscillationsforbothLPAIandHPAI.TheoscillationsinLPAIhavemuchlargeramplitude.HPAIpeaksfollowLPAIpeaksbyabout1/4periodwhichistypicalforclassicalpredator-preydynamics.Theparameterschosenincludingthereproductionnumbersandinvasionreproductionnumbershaveplausiblevalues.Toobtainoscillationswiththeseparameterchoices,oursimulationssuggestedthatweneedtochooseqw1.Thatsuggeststhatoscillations,whichoftenmimicoutbreaks,occuriftheLPAIcross-immunitytoHPAIisnearlyorcompletelynon-existent.Figure 4-6 showssustainedoscillations.Lookingmorecloselyatthegurewecanseetwooscillationpatternssuperimposed,differinginperiod.Withtheshortperiodoscillations,thepeakofLPAIisfollowedbyapeakofHPAI,somewhatresemblingpredator-preyoscillations.Theunstableequilibriumvaluesaregivenby(Sw,ILw,RLw,IHw,RHw)=(5301.83,38.2707,12316.7,16.3273,26426.1).InthesimulationinFigure 4-6 thereproductionnumberofLPAIissomewhathightoberealistic.Decreasingqwto0.9fromtheparameterlistedinFigure 4-6 allowsthe 88

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Figure4-5. Figureshowssustainedoscillationsinthewildbirdsonlysystem.Parametervaluesaretakenasfollows:w=2000,w=0.14,Hw=49.5,Hw=51.6,Lw=73,qw=0.98,L11=.018776,H11=0.015,Sw(0)=3449.72,ILw(0)=14.684,RLw(0)=3366.78,IHw(0)=7,RHw(0)=769.5.ThereproductionnumbersareRL=3.66andRH=2.116.Theinvasioncoefcientsareasfollows:^RL=1.73and^RH=2.08.TheredlineshowsHPAIandthebluelineshowsLPAI. oscillationsofLPAIandHPAItobeshiftedsotheyarehalftheperiodoutofphase,sothatthemaximumofHPAIoccursatthesamemomentastheminimumofLPAI.Inthiscasewesaythethesystemexhibitsfullycompetitiveoscillation.Itisusefultodevelopsomeintuitiveunderstandingforwhyoscillationsariseinthissystem.Biologically,thesystemisnotreallyanalogoustoapredator-preysystem.RecallthatLPAIandHPAIbothattacksusceptiblehosts.Ifqw=0,thereiscompletecross-immunity,andtherelationbetweenLPAIandHPAIissimplythatofbeingcompetitorsforsusceptiblehosts.Onedoesnotndcoexistenceinthiscaseinasinglepopulation.Inthismodel,infectionbyHPAIalwaysgivescompleteimmunitytoLPAI.However,ifqw>0,thereisonlypartial(orno)immunitytoHPAIconferredbypriorinfectionbyLPAI,soLPAI-recoveredhostscanbeinfectedbyHPAI.AdirectpredationanalogueinthissystemwouldbeifHPAIcouldinfectLPAI-infectedhostsandeliminatetheLPAIinfection,therebydirectlyreducingthenumberofLPAI-infectedhosts.Inourmodel,HPAIdoesnothavethisdirecteffectbecauseitjustattacksLPAI-recovered 89

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Figure4-6. Figureshowssustainedoscillationsinthewildbirdsonlysystem.Parametervaluesaretakenasfollows:w=3810,w=0.054,Hw=87.5,Hw=87.4,Lw=69.4,qw=0.99,L11=.0131,H11=0.01,Sw(0)=5000,ILw(0)=40,RLw(0)=12000,IHw(0)=15,RHw(0)=25000.ThereproductionnumbersareRL=13.3andRH=4.0.Theinvasioncoefcientsareasfollows:^RL=3.3and^RH=4.TheredlineshowsHPAIandthebluelineshowsLPAI. hosts.However,attackingLPAI-recoveredhostsincreasestheprevalenceofHPAI,andallowsittoinfectmoresusceptiblehosts,forwhichitiscompetingwithLPAI.Itwouldthereforebeanalogoustoasysteminwhichonecompetitorcanconsumethecarcassesoftheother.FortheparametersofFigure 4-6 ,thenumberofLPAI-infectedhostsincreaseswheneverSw>(w+Lw)=Lww=5302anddecreasesotherwise.AsILwincreases,itdecreasesSwuntilitisbelowthisvalue(HPAIalsohelpsdecreaseSw,butitislesscommon,especiallywhenILwisnearitspeak).ForHPAItoincrease,itrequiresSw+qwRLw>(w+Hw+Hw)=Hww=17495.Eventhoughthisthresholdishigher(duetothehighdeathrate),itappliestothesumofsusceptibleandLPAI-recoveredhosts(thelatterdiscountedbyqw).Becausemost 90

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LPAI-infectedbirdsrecover,asthepeakinILwdrawsdownSw,italsoincreasesRLw,sothattheconditionforHPAItocontinueincreasingcansometimescontinuetobemetafterLPAIhasstartedtodecrease,asinthegure.Fortheparametersofthegure,HPAIreliesmostlyonLPAI-recoveredbirds,thepeakofwhichisafterthepeakinILw.HPAIthereforeisincreasingmostrapidlyaftertheLPAIpeak.Eventually,HPAIdepletesthehostsitattacks,andstartstodecrease.Bythistime,thesusceptiblehostshavestartedtoincrease(becauseofthelowlevelofILw),buttheythenincreasefasteruntiltheyarehighenoughforILwtostarttoincrease.Sooscillationsinthissystemarisebecauseofacombinationofcompetition,andaphenomenonanalogousto“scavenging”amongcarnivores.Wenextaddressthequestionofwhetherwecanreduceqwandstillobtainoscillations.Themostinuentialparameterforthattooccurisw,whichneedstobefairlylow(0.14inFigure 4-5 and0.054inFigure 4-6 ,bothreasonableforwildbirds)toproduceoscillationswithsmallerqw.Raisingwallowsoscillationswithoutwbecomingexcessivelysmallandthereforeunrealisticforwildbirdpopulations.RaisingthesumHw+Hwalsoallowsforloweringqw.Stillwithnearlyrealisticotherparameters,qwneedstostayabove0.9foroscillationstooccur.LPAIpersistsathigherlevelsthanHPAI,whichistherealisticscenarioforwildbirdpopulations.However,raisingqwleadstooscillationsbutalsoincreasestheprevalenceofHPAIattimestolevelshigherthanLPAIwhichinwildbirdsisunrealistic.Lackofcross-immunityfromLPAIindomesticbirdsmayexplainwhyHPAIpersistsindomesticbirdsathigherprevalencelevels.Forrealisticparametervalues,itappearsthatinmostcasesoscillationsofLPAIhavelargeramplitudeandgotohighervaluescomparedtooscillationsinHPAI. 4.5DiscussionAvianinuenzacontinuestobeathreattohumanhealth.Recently,strainsofHPAIH7N9havestartedinfectinghumansandholdpotentialtoturnpandemicwith 91

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deadlyconsequences.Studyingavianinuenzainbirdsandhumansisofparamountimportanceifwearetobepreparedforthenextdeadlypandemic.Inthispaperweintroduceanavianinuenzamodelformultiplebirdpopulations.Themodelincorporatestwostrains,onelowpathogenic(LPAI)andonehighpathogenic(HPAI).WeareinterestedinstudyingthedynamicsofLPAIandHPAIinwildanddomesticbirds.WecomputethereproductionnumbersRLandRHandtheinvasionreproductionnumbers^RHLand^RLH.Themodelhasauniquedisease-freeequilibriumwhichislocallystableifbothreproductionnumbersaresmallerthanone.TherearealsoauniqueLPAI-onlyandauniqueHPAI-onlyequilibriawhichexistiftheLPAI(HPAI)reproductionnumberislargerthanone.TheLPAI-onlyequilibriumislocallyasymptoticallystablewheneveritexistsif^RHL<1.TheHPAI-onlyequilibriumislocallyasymptoticallystablewheneveritexistsif^RLH<1.Weshowthatif^RLH>1and^RHL>1thereexistsacoexistenceequilibrium.Thequestionabouttheuniquenessofthecoexistenceequilibriumremainsopen.Simulationssuggestthatthecoexistenceequilibriumisnotstableforallparameterregimes.Infact,thecoexistenceequilibriumcanbedestabilizedeveninthecorrespondingODEsysteminwhichqwandqdareassumedconstant.Sincethesemi-trivialequilibriaarelocallystable,thisclearlysuggeststhattheinteractionbetweenthestrains,thatisqw6=0and/orqd6=0,isnecessaryforthedestabilizationofthecoexistenceequilibrium.Next,weaskedwhetherthepresenceofbothpopulationsandtransmissionbetweenthepopulationswasnecessaryforinstability.Investigatingthewildbirdsystemonly(see( 62 )),wendthattheODEmodelofwildbirdswithLPAIandHPAIalsoexhibitsoscillationsinwhichbothLPAIandHPAIpersist.Inthewildbirdsystem,oscillationsarefoundwithhighvaluesofqw1,whichmeansthatdestabilizationofthesystemoccursifcross-immunityislow.Inthefullsystemoscillationscanbefoundforlargerrangesofqwandqd.Thus,transmissionbetweenthetwopopulationsallowsfordestabilizationofthesystemforavarietyofcross-immunitylevels.Forsustainedoscillationsinasingle 92

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populationconsideredalone,LPAI-recoveredbirdsmustbealmostassusceptibletoHPAIinfectionasarenaivebirds.Simulationssuggestthatforplausibleparametervalueswecanalsoproducerealisticprevalences.Inparticular,inwildbirdstheLPAIprevalenceishigherthantheHPAIprevalence,whileindomesticbirdsitisviceversa.Ofparticularinterestisthecasewhenapopulationisasinkforapathogenbutpersistenceinamulti-populationmulti-pathogensystemisstillpossible.WecallpopulationAasinkforpathogenp,wherep=LPAIorp=HPAI,ifpathogenpcannotpersistaloneinpopulationA,ifisolated.Itiswell-knownthat,inasystemwithtwosinkhabitats,apopulationcansometimespersistbyusingbothhabitats.Weinvestigatethisquestioninthecaseofcompetitionofpathogens.Inthecaseofcompetition,wesaythatapopulationAisasinkforpathogenpifitswithin-populationreproductionnumberislessthanone,orifitsreproductionnumberisgreaterthanone,itswithin-populationinvasionreproductionnumberissmallerthanone(andtheotherpathogenispresent).Weshowthroughsimulationsthatcoexistenceofbothpathogensispossible,ifalltheirwithin-populationandcross-populationreproductionnumbersaresmallerthanone.ThisobservationisveryimportantsinceestimatesofthereproductionnumberofHPAIH5N1inpoultryvaryaroundone(( 63 ; 42 ; 47 ))butourresultsimplythatevenifthereproductionnumberisbelowone,HPAImaypersistinthewild-domesticbirdssystem,evenundercompetitionwithLPAI.Futureempiricalstudieswillberequiredtoreneparameterestimationandascertainthelikelihoodofobservingthecomplexdynamicsrevealedbythismodel.Also,inthefutureitwouldbeusefultoexplorealternativemodelsofrecruitmentthantheconstantrateofinputassumedinmodel( 4 ).Finally,itislikelythatspatialdynamicsaresignicantinthissystem.Manywildwaterfowlaremigratoryandcanmoveoverlargeareas.Somebirdsmayreturntothesameareaeachwinter,butothersmaymoveamongregions.Domesticfowlareconcentratedinmorediscretelocations,withlessmobility,oneexpects.Dealingwithspatialpatchiness,migration,andheterogeneity 93

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willlikelybeimportantinmorerealisticfuturecharacterizationsofcross-populationtransmissioninavianinuenza. 94

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REFERENCES [1] D.J.ALEXANDER,Anoverviewofepidemiologyofavianinuenza,Vaccine,25(2006),p.5637-5644. [2] A.APISORNTHANARAK,M.D.,L.MUNDY,M.D.InfectioncontrolforAvianInuenza(H5N1)inHealtcareSettings.Chapter3.AvianInuenzaREsearchProgress.E.P.Allegra(Editor),(2008)p.73-78. [3] F.B.AGUSTO,A.B.GUMMEL,Theoreticalassessmentofavianinuenzavaccine,Dis.Cont.D.Sys.B13(1)(2010),p.1-25. [4] R.B.BELSHE.Theoriginsofpandemicinuenza-lessonsdemothe1918virus.N.Engl.J.Med.353(2005)2209-11. [5] L.BOUROUIBA,A.TESLYA,J.WU,Highlypathogenicavianinuenzaoutbreakmitigatedbyseasonallowpathogenicstrains:Insightsfromdynamicmodeling,JTB271(2011),p.181-201. [6] C.CASTILLO-CHAVEZ,H.HETHCOTE,V.ANDREASEN,S.LEVIN,W.M.LIU,Epidemiologicalmodelswithagestructure,proportionatemixingandcross-immunity,J.Math.Biol.27(1989),p.159-165. [7] C.CASTILLO-CHAVEZ,H.HETHCOTE,V.ANDREASEN,S.LEVIN,W.M.LIU,Cross-immunityinthedynamicsofhomogeneousandheterogeneouspopulations,MathematicalEcology(Trieste,1986),WorldSci.Publishing,Teaneck,NJ,(1988),p.303-316. [8] CDC,AvianInuenzainBirds, http://www.cdc.gov/u/avianu/avian-in-birds.htm [9] CDC,KeyFactsAboutAvianInuenza(BirdFlu)andHighlyPathogenicAvianInuenzaA(H5N1)Virus, http://www.cdc.gov/u/avian/gen-info/facts.htm [10] L.CLARK,J.HALL,Avianinuenzainwildbirds:statusasreservoirs,andriskstohumansandagriculture,Ornithol.Monogr.60(2006),p.3-29. [11] E.CLASS,etal.,HumaninuenzaAH5N1virusrelatedtohighlypathogenicavianinuenzavirus,Lancet351(1998),p.472-477. [12] J.J.DENNEHY,N.A.FRIEDENBERG,R.C.MCBRIDE,R.D.HOLT,P.E.TURNER,Experimentalevidencethatsourcegeneticvariationdrivespathogenemergence,Proc.R.Soc.B277(1697)(2010),p.3113-3121. [13] P.VANDENDRIESSCHE,J.WATMOUGH,Reproductionnumbersandsub-thresholdendemicequilibriaforcompartmentalmodelsofdiseasetransmission,Mathe-maticalBiosciences180(2002),p.29-48. [14] A.DODSON,Populationdynamicsofpathogenswithmultiplehostspecies,TheAmericanNaturalist164,(2004),p564-576. 95

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[15] E.O'NEILL,J.M.RIBERDY,R.G.WEBSTER,D.L.WOODLAND,HeterologousprotectionagainstlethalA/HongKong/156/97(H5N1)inuenzavirusinfectioninC57BL/6mice,J.Gen.Virol.81,(2000),p.2689-96. [16] S.R.FEREIDOUNI,E.STARICK,M.BEER,D.KALTHOFFetal.,Highlypathogenicavianinuenzavirusinfectionofmallardswithhomo-andheterosubtypicimmunityinducedbyLPAIviruses,PLoSOne4(8),e6705. [17] N.M.FERGUSON,D.A.T.CUMMINGS,S.CAUCHEMEZ,C.FRASER,S.RILEY,A.MEEYAI,S.IAMSIRITHAWORN,D.S.BURKE,StrategiesforcontaininganemerginginuenzapandemicinSoutheastAsia,Nature437(2005),p.209-214. [18] B.N.FIELDS,D.M.KNIPE,P.M.HOWLEY(eds),FieldsVirology,3rdEd.,Lippincott–Raven,Philadelphia,1996. [19] T.C.GERMANN,K.KADAU,I.M.LONGINI,JR.,C.A.MACKEN,MitigationstrategiesforpandemicinuenzaintheUnitedStates,PNAS103(15)(2006),p.5935-5940. [20] R.F.GRAIS,J.H.ELLIS,G.E.GLASS,Assessingtheimpactofairlinetravelonthegeographicspreadofpandemicinuenza,Eur.J.Epidem.18(2003),p.1065-1072. [21] A.B.GUMEL,Globaldynamicsofatwo-strainavianinuenzamodel,Int.J.Comp.Math.86(1)(2009),p.85-108. [22] R.D.HOLT,A.P.DOBSON,Extendingtheprinciplesofcommunityecologytoaddresstheepidemiologyofhost-pathogensystems,in“EcologyofEmergingInfectiousDiseases”(S.K.CollingeandC.Ray,eds.),OUP,Oxford,2005,p.6-27. [23] H.INABA,Endemicthresholdandstabilityinanevolutionaryepidemicmodel,inMathematicalApproachesforEmergingandReemergingInfectiousDiseases:Models,MethodsandTheory,IMAVol.Math.Appl.126,Springer,NewYork,(2002),p.337-359. [24] S.IWAMI,Y.TAKEUCHI,X.LIU,Avian-humaninuenzaepidemicmodel,Math.Biosci.207(2007),p.1-25. [25] S.IWAMI,Y.TAKEUCHI,X.LIU,Avianupandemic:Canwepreventit?,JTB257(2009),p.181-190. [26] S.IWAMI,Y.TAKEUCHI,A.KOROBEINIKOV,X.LIU,Preventionofavianinuenzaepidemic:Whatpolicyshouldwechoose?,JTB252(2008),p.732-741. [27] V.A.A.JENSEN,J.YOSHIMURA,Populationscanpersistinanenvironmentconsistingofsinkhabitatsonly,PNAS95(7)(1998),p.3696-3698. 96

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[28] D.KALTHOFF,A.BREITHAUPT,J.P.TEIFKE,A.GLOBIG,etal.Pathogenicityofhighlypathogenicavianinuenzavirus(H5N1)inadultmuteswans,EID14(2008),p.1267-1270. [29] M.J.KEELING,P.ROHANI,Modelinginfectiousdiseasesinhumansandinanimals,PrincetonUniversityPress,Princeton,2008. [30] K.I.KIM,Z.LIN,L.ZHANG,Avian-humaninuenzaepidemicmodelwithdiffusion,Nonlin.Analysis:RWA11(2010),p.313-322. [31] S.KRAUSS,D.WALKER,S.P.PRYOR,L.NILES,L.CHENGHONG,V.S.HINSHAW,R.G.WEBSTER,InuenzaAvirusesofmigratingwildaquaticbirdsinNorthAmerica,VectorBorneZoonoticDis.4(3)(2004),p.177-189. [32] T.KUIKEN,Islowpathogenicavianinuenzavirusvirulentforwildwaterbirds?ProcBiolSci.280(1763)(2013):20130990.doi:10.1098/rspb.2013.0990. [33] LATTORE-MARGALEF,N.ETAL.,Long-termvariationininuenzaAvirusprevalenceandsubtypediversityinmigratorymallardsinnorthernEurope,Proc.Roy.Soc.B281(2014),20140098 [34] A.LEMENACH,E.VARGU,R.F.GRAIS,D.L.SMITH,A.FLAHAULT,Keystrategiesforreducingspreadofavianinuenzaamongcommercialpoultryholdings:lessonsfortransmissiontohumans,Proc.R.Soc.B273(2006),p.2467-2475. [35] X.Z.LI,J.X.LIU,M.MARTCHEVA,Anage-structuredtwo-strainmodelwithsuper-infection,Math.Biosci.Eng.7(1)(2010),p.125-149. [36] R.LIU,V.R.S.K.DUVVURI,J.WU,Spreadpatternformationanditsimplicationsforcontrolstrategies,Math.Model.Nat.Phenom.3(7)(2008),p.161-179. [37] I.M.LONGINI,A.NIZAM,S.XU,K.UNGCHUSAK,W.HANSHAOWORAKUL,D.A.T.CUMMINGS,M.E.HALLORAN,Containingpandemicinuenzaatthesource,Science309(2005),p.1083-1087. [38] J.LUCCHETTI,M.ROY,M.MARTCHEVA,Anavianinuenzamodelanditsttohumanavianinuenzacases,in“AdvancesinDiseaseEpidemiology”(J.M.Tchuenche,Z.Mukandavire,eds.),NovaSciencePublishers,NewYork,NY,2009,p.1-30. [39] B.LUPIANI,S.REDDY,Thehistoryofavianinuenza,Comp.Immunol.Microbiol.Infect.Dis.32(4)(2009),p.311-323. [40] P.MAGAL,S.RUANSustainedoscillationsinanevolutionaryepidemiologicalmodelofinuenzaAdrift,Proc.R.Soc.A466(2116)(2010),p.965-992. [41] M.MARTCHEVA,AnevolutionarymodelofinuenzaAwithdriftandshift,J.Biol.Dynamics(inpress). 97

PAGE 98

[42] M.MARTCHEVA,Avianinuenza:modelingandimplicationsforcontrol,J.Biol.Systems,(toappear). [43] C.E.MILLS,J.M.ROBINS,C.T.BERGSTROM,M.LIPSITCH,Pandemicinuenza:riskofmultipleintroductionsandtheneedtoprepareforthem,PLoSMedicine3(6)(2006),p.1-5. [44] V.J.MUNSTER,A.WALLENSTEN,C.BAAS,G.F.RIMMELZWAAN,M.SCHUTTEN,B.OLSEN,A.D.OSTERHAUS,R.A.FOUCHIER,Mallardsandhighlypathogenicavianinuenzaancestralviruses,NorthernEurope,EID11(10)(2005),p.1545-1551. [45] M.NUNO,Z.FENG,M.MARTCHEVA,C.CASTILLO-CHAVEZ,Dynamicsoftwo-straininuenzawithisolationandpartialcross-immunity,SIAMJ.Appl.Math.65(3)(2005),p.964-982. [46] J.OTTE,J.HINRICHS,J.RUSHTON,D.ROLAND-HOLST,D.ZILBERMAN,Impactsofavianinuenzavirusonanimalproductionindevelopingcountries,Perspec-tivesinAgr.,Vet.Sci.,Nutrition,Nat.Resources3(2008),No.080. [47] P.S.PANDIT,D.A.BUNN,S.A.PANDE,S.S.ALY,ModelinghighlypathogenicavianinuenzatransmissioninwildbirdsandpoultryinWestBengal,India,ScienticReports3,2013,2175. [48] C.M.PEASE,AnEvolutionaryepidemiologicalmechanismwithapplicationstotypeAinuenza,Theor.Pop.Biol.31(1987),p.422-452. [49] PEPIN,K.M.ETAL.UsingquantitativediseasedynamicsasatoolforguidingresponsetoavianinuenzainpoultryintheUnitedStatesofAmerica,Preven-tiveVeterinaryMedicine113(2014),p376-397. [50] R.S.SCHRIJVER,G.KOCH(EDS.),AvianInuenza:PreventionandControl,Springer,TheNetherlands,2005. [51] I.SCONES,Theinternationalresponsetoavianinuenza:science,policy,politics,AvianInuenza:Science,PolicyandPolitics,(I.Scoones,ed.),EarthScan,London,(2010),p.1-18. [52] I.SCONES,P.FORSTER,Unpackingtheinternationalresponsetoavianinuenza:actors,networksandnarratives,AvianInuenza:Science,PolicyandPolitics,(I.Scoones,ed.),EarthScan,London,(2010),p.19-64. [53] K.F.SHORTRIDGE,PoultryandinuenzaH5N1outbreaksinHongKong,1997:abridgedchronologyandvirusisolation,Vaccine17(1999)S26-S29. [54] S.H.SEO,R.G.WEBSTER,Cross-reactive,cell-mediatedimmunityandprotectionofchickensfromlethalH5N1inuenzavirusinfectioninHongKongpoultrymarkets,J.Virol.75(2001),p.2516-2525. 98

PAGE 99

[55] E.SPACKMAN,Abriefintroductiontotheavianinuenzavirus,inAvianInuenzaVirus(E.Spackman,eds.),HumanaPress,(2008),p.1-6. [56] D.E.STALLKNECHT,J.D.BROWN,Ecologyofavianinuenzainwildbirds,AvianInuenza,(D.E.Swayne,ed.),p.43-58,(2008). [57] STANFORD,StanfordBirds, http://web.stanford.edu/group/stanfordbirds/text/essays/How Long.html [58] D.L.SUAREZ,InuenzaAvirus,inAvianInuenza,(D.E.Swayne,ed.),p.3-17,2008. [59] D.E.SWAYNE,Epidemiologyofavianinuenzainagriculturalandotherman-madesystem,inAvianInuenza,(D.E.Swayne,ed.),p.59-85,2008. [60] TAKEKAWA,J.Y.ETAL.,MovementsofwildRuddyShelducksintheCentralAsianFlywayandtheirspatialrelationshiptooutbreaksofhighlypathogenicavianinuenzaH5N1,Viruses5(2013),p2129-2152. [61] T.TIENSIN,M.NIELEN,H.VERNOOIJ,et.al,TransmissionofthehighlypathogenicavianinuenzavirusH5N1withinocksduringthe2004epidemicinThailand,JID196(2007),p.1679-1684. [62] N.TUNCER,J.TORRES,M.MARTCHEVA,DynamicsofLowandHighPathogenicAvianInuenzainWildBirdPopulation,inDynamicalSystems:Theory,ApplicationsandFutureDirections,(J.Thuenche,ed.),NovaPublishers,NewYork,2013,p.235-259. [63] N.TUNCER,M.MARTCHEVA,ModelingseasonalityinavianinuenzaH5N1,J.Biol.Systems,21(4)2013,1450009. [64] USDA,AnEarlyDetectionSystemforHighlyPathogenicH5N1AvianInuenzainWildMigratoryBirds, ww.usda.gov/documents/wildbirdstrategicplanpdf.pdf [65] R.G.WEBSTER,W.J.BEAN,O.T.GORMAN,T.M.CHAMBERS,Y.KAWAOKA,EvolutionandecologyofinuenzaAviruses,Microbiol.Rev.56(1)(1992)p.152-179. [66] WHO,CumulativenumberofconrmedhumancasesofavianinuenzaA(H5N1)reportedtoWHO, http://www.who.int/inuenza/human animal interface/H5N1 cumulative table archives/en/ [67] WIKIPEDIA,Mallard, http://en.wikipedia.org/wiki/Mallard [68] WORLDORGANIZATIONFORANIMALHEALTH,Avianinuenza2.7.12,Terrestrialanimalhealthcode-2006,WorldOrganizationforAnimalHealth,Paris,France,2006. 99

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BIOGRAPHICALSKETCH JuanTorresreceivedaBachelorofScienceinmathematicswithaminorinphysicsfromtheUniversityofFloridain2008.HewentontograduateschoolattheUniversityofFloridain2008,andcompletedhisPhDinmathematicsin2015,specializinginmathematicalbiology. 100