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Coexistence of Two Strains in an Avian Influenza Model

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Title:
Coexistence of Two Strains in an Avian Influenza Model
Creator:
Ponce, Joan
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English

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Subjects / Keywords:
Biological evolution ( jstor )
Birds ( jstor )
Diseases ( jstor )
Eigenvalues ( jstor )
Influenza ( jstor )
Mortality ( jstor )
Parametric models ( jstor )
Pathogens ( jstor )
Poultry ( jstor )
Simulations ( jstor )
Avian influenza
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Undergraduate Honors Thesis

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Abstract:
The coexistence of strains of avian influenza arise from interactions between the strains, such as co-infection, cross immunity, mutation and super-infection. We use a mathematical model to describe the behavior of two strains of avian influenza, one high pathogenic and one low pathogenic on a population of birds without considering interactions between the strains. The SI model includes a nonlinear culling term for the HPAI and recovery coefficient for LPAI and does not include a recovery rate of the domestic birds infected with HPAI. We computed reproduction numbers and invasion numbers. We find that besides the disease-free equilibrium there is a dominance equilibrium for each strain which is locally asymptotically stable under certain conditions. We show that the culling rate alone generates coexistence of the LPAI and the HPAI strains. The coexistence equilibrium is locally asymptomatically stable whenever it exists. ( en )
General Note:
Awarded Bachelor of Science; Graduated May 7, 2013 magna cum laude. Major: Mathematics, Emphasis/Concentration: Bachelor of Science
General Note:
College/School: College of Liberal Arts and Sciences
General Note:
Advisor: Maia Marcheva

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University of Florida
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University of Florida
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Copyright Joan Ponce. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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COEXISTENCEOFTWOSTRAINSINANAVIANINFLUENZA MODEL JOANPONCE Abstract. Thecoexistenceofstrainsofavianinuenzaarisefrominteractionsbetweenthestrains,suchasco-infection,crossimmunity,mutationandsuper-infection. Weuseamathematicalmodeltodescribethebehavioroftwostrainsofavianinuenza, onehighpathogenicandonelowpathogeniconapopulationofbirdswithoutconsideringinteractionsbetweenthestrains.TheSImodelincludesanonlinearcullingtermfor theHPAIandrecoverycoecientforLPAIanddoesnotincludearecoveryrateofthe domesticbirdsinfectedwithHPAI.Wecomputedreproductionnumbersandinvasion numbers.Wendthatbesidesthedisease-freeequilibriumthereisadominanceequilibriumforeachstrainwhichislocallyasymptoticallystableundercertainconditions. WeshowthatthecullingratealonegeneratescoexistenceoftheLPAIandtheHPAI strains.Thecoexistenceequilibriumislocallyasymptomaticallystablewheneverit exists. 1. Introduction OutbreaksofhighlypathogenicavianinuenzaH5N1indomesticpoultry,aswellas wildmigratorybirdspresentagreatthreattohumansduetothepossibilityofH5H1 mutatingintohighlycontagioushuman-tohumantransmissiblestrain[2].Therehave beenseveralpandemicoutbreaksofinuenzainthepast.Inthe20thcenturythreepandemicsoccurred,thespanishu,theasianuandthehongkongu.Since2003,there havebeenhumanfatalitiesassociatedtoavianuthatresultedfromtheinteractions betweendomesticbirdsandhumansinAsia[4]. TheknownvirusesthatcauseinuenzainbirdsbelongtotheInuenzaAvirus.Only threesubtypesofinuenzaAarehighlypathogenicinhumans,namelyH5N1,H7N3, H7N7,andH9N2[1].Thesubtyperesponsibleforcausingwidespreadoutbreaksofavian uinbirdsisthesubtypeH5N1,morespecicallyabird-adaptedstraincalledHPAI H5N1.Mathematicalmodelshavebeenusedtodescribethemultiplemechanismsthat allowthecoexistenceofpathogenstrains.InthisarticleweconsideranSIepidemic modelwithtwostrains,withoutinvolvinganyotherfactorsthatmaycausecoexistence, andaimtoshowthatnon-linearcullingalonecausescoexistenceoftwostrains,onehigh pathogenicandonelowpathogenicstrainofthesamevirus. Amongthefactorsthatallowthecoexistenceofvariouspathogenspeciesthereis cross-immunity[4],mutationofasinglestrain[5],super-infection[6]andco-infection[7]. Thepreviouslymentionedmechanismsofcoexistencearisefromtheinteractionsbetween strains.Externalfactorsthatpromotecoexistenceofstrainsinclude,densitydependent hostmortality[8],verticaltransmission[9],andsaturatingcontactrates[1].Inamodel Date :April24,2013. 1

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2 involvingsaturatingcontactrates,causedbythevariationofthesocialactivityofinfectedindividualswithdierentstrains,coexistenceispossibleundercertainconditions, givenbytheinvasionreproductionnumbers.Inthisarticle,weareconsideringcoexistenceoftwostrainsinthesamepopulationofbirds.Themodelanalyzedinthispaper doesnotincludeanyofthepreviouslymentionedmechanismsofcoexistenceofstrains. Theyareintentionallyleftouttobringtolightanotherpossiblemechanismthatcan increasepathogensgeneticdiversity,namelynon-linearculling. Weconsideratwostrainmodelwithalowpathogenicstrainandahighlypathogenic strain.Inaddition,weintroduceselectivecullingrate.Inselectiveculling,theperson whodoesthecullingcanaccuratelyidentifythebirdsinfectedwiththehighlypathogenic strainandcullsonlythosebirds.Inthismodelanindividualinthebirdpopulationcannotbeinfectedwithbothstrainsofthevirus,whichmeansthatthereisnoco-infection. Thereproductionnumbershavebeencomputedaswellastheinvasionnumbersofthe strains.Thecoexistenceofthestrainsisshowntooccurundercertainconditionswhich arebiologicallyinterpretable.Theresultsinthearticlewouldextendtoshowsaturating treatmentcanalsoleadtocoexistenceinmultistrainmodels.Insection2weintroduce themodel.Insection3 ; wecomputethedisease-freeequilibriaandproveitsglobalstability,aswellasthedominanceequilibriumofboththehighlypathogenicstrainandthe lowpathogenicstrain.Thestabilitiesofboththeseequilibriaareveriedandsimulations areusedtoshowtheexistenceoftheseequilibria.Insection4 ; weproveanalytically thatthereexistsauniquecoexistenceequilibriumanditislocallystablewheneverit exists.Insection5 ; wepresentsomeofthesimulations.Finallyinsection6 ; wediscuss theresultsobtainedfromthemodel. 2. ModelingLowandHighPathogenicAvianInfluenzaStrains Inthissection,weintroduceamodelofAvianInuenzawithtwodistinctstrains: lowpathogenicAIandH5N1highlypathogenicAIvirusstrains.Themodelisbased ontheSIRmodelproposedbyO.KermackandA.GrayMcKendrick.Inthemodel S t denotesthenumberofsusceptibledomesticbirdsattime t I t and J t represent thenumberofdomesticbirdsinfectedbythelowpathogenicandhighlypathogenicAI virusstrain,respectively.Moreover R t refersthenumberofdomesticbirdsrecovered fromthelowpathogenicavianinuenzavirusstrain.Inthismodel,susceptibledomestic birdscanbecomeinfectedwithLPAIvirusstrain I atatransmissionrate 1 orbyH5N1 HPAIvirusstrain J atarate 2 .H5N1HPAIvirusstrainhasahighmortalityrate. Henceinthismodel,weassumethatthebirdsinfectedwithH5N1HPAIvirusexitthe infectedcompartmentonlythroughdeathwithadeathrate .Thebirth/recruitment rateforsusceptibleindividualsisgivenasaconstantrateanditisassumedthat susceptiblebirdsleavethecompartmentwithaconstantdeathrate .Moreovera portionofdomesticbirdsinfectedwithHPAIvirusstrainisremovedfromthesystem throughselectivecullingwithcullingrate J= A + J ,cullingcoecient andculling constant A .Itisassumedthatfarmerscanperfectlydistinguishthedomesticbirds infectedwithHPAIvirusfromtherestpartofthepoultry.Saturationinthepercapita cullingrateoccursthroughresourcelimitation.Thegeneralmodeltakesthefollowing form:

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3 .1 8 > > > > > > < > > > > > > : dS dt = )]TJ/F34 11.9552 Tf 11.955 0 Td [( 1 SI )]TJ/F34 11.9552 Tf 11.956 0 Td [( 2 SJ )]TJ/F34 11.9552 Tf 11.956 0 Td [(S + I; dI dt = 1 SI )]TJ/F15 11.9552 Tf 11.956 0 Td [( + I dJ dt = 2 SJ )]TJ/F15 11.9552 Tf 11.955 0 Td [( + J A + J J withnonnegativeparametersandinitialconditions: S ;I ;J 0 : Thetotal numberofdomesticbirdsisdenotedby N ,where N = S + I + J .Modelvariablesand parametersalongwiththeirdenitionsarelistedinTable1. Table1. Denitionofthevariablesinthemodelingframework Variable/ParameterMeaning S Susceptibledomesticbirds I BirdsinfectedwithLPAI J BirdsinfectedwithHPAI Birth/recruitmentrateofdomesticbirds 1 TransmissionrateofLPAIamongdomesticbirds 2 TransmissionrateofHPAIamongdomesticbirds CullingcoecientforpoultryinfectedwithHPAI A CullingconstantforpoultryinfectedwithHPAI RecoveryrateofdomesticbirdsinfectedwithLPAI Naturaldeathratefordomesticbirds Thesystemalwayshasadisease-freeequilibrium E 0 ,where E 0 = =; 0 ; 0 ; 0.The reproductionnumbersofthediseaseforLPAIandHPAIvirusstrainsaredenedas R 1 = 1 + ; R 2 = 2 2 ; respectively.Thesystemhasareproductionnumberdenedas R 0 =max fR 1 ; R 2 g Notethattheregionofattractionofthesystemis )-278(= f S;I;J 2 R 3 + :0 S + I + J = g : 3. TrivialandSemitrivialEquilibriaandtheirstability Theorem3.1. If R 1 < 1 and R 2 < 1 ,thenthedisease-freeequilibrium E 0 islocally asymptoticallystable.Itisunstableif R 1 > 1 or R 2 > 1 .

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4 Proof. FirstnotethattheJacobianmatrixofthesystemevaluatedattheDFEisas following: .1 J j E 0 = ; 0 ; 0 ; 0 = 0 B B B B B B B B B @ )]TJ/F34 11.9552 Tf 9.298 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( 1 + )]TJ/F34 11.9552 Tf 9.298 0 Td [( 2 0 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( + 0 00 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 11.955 0 Td [( 2 AJ + J 2 A + J 2 1 C C C C C C C C C A Itisanuppertriangularmatrix.Hencetheeigenvaluesofthismatrixarelocatedalong thediagonal: 1 = )]TJ/F34 11.9552 Tf 9.299 0 Td [(; 2 = 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( + ; 3 = 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [(: Notethatif R 1 < 1and R 2 < 1,thentheeigenvalues 1 ; 2 ; 3 < 0arenegative realnumbers.Therefore,thediseasefreeequilibriumislocallyasymptoticallystable. Howeverwhen R 1 > 1or R 2 > 1,wehave 1 )]TJ/F15 11.9552 Tf 12.063 0 Td [( + > 0or 2 )]TJ/F34 11.9552 Tf 12.063 0 Td [(> 0. Hencethediseasefreeequilibriumisunstable,if R 1 > 1or R 2 > 1. Moreover,weobtainthefollowingresult: Theorem3.2. If max fR 1 ; R 2 g < 1 ,thenthediseasefreeequilibriumisgloballyasymptoticallystable. Proof. Fortheglobalstabilityanalysisofthediseasefreeequilibrium E 0 ,wewilluse LasalleInvariancePrinciple.Letconsiderthefunction V = I + J .Notethatthe derivativeofitalongthesolutionsofthesystem2 : 1is dV dt = 1 SI )]TJ/F15 11.9552 Tf 11.955 0 Td [( + I + 2 SJ )]TJ/F34 11.9552 Tf 11.955 0 Td [(J )]TJ/F34 11.9552 Tf 19.02 8.088 Td [(J 2 A + J [ 1 ] + I + 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [( J )]TJ/F34 11.9552 Tf 21.387 8.087 Td [(J 2 A + J 2 + R 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 I + R 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 J )]TJ/F34 11.9552 Tf 21.386 8.088 Td [(J 2 A + J 2 < 0 : sincemax fR 1 ; R 2 g 1.HencebyLasalleInvariancePrinciple,foranysolution S;I;J 2 ,theomegalimitsetofthissolutionisasubsetofthelargestinvariantsetin n x 2 )-278(: V x =0 o Notethatthelargestinvariantsetin n x 2 )-278(: V x =0 o isthesingletonsetof =; 0 ; 0 ; 0, whichisthediseasefreeequilibrium.Thenanysolutionin)-261(convergestotheDFEwhen max fR 1 ; R 2 g < 1.

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5 Furthermore,thefollowingresultcanbeexpectedandisnothardtoestablish: Theorem3.3. Themodel2.1hasauniquedominanceequilibriumofstrain I and auniquedominanceequilibriumofstrain J .Theuniquedominanceequilibrium E 1 = S 1 ;I 1 ; 0 ofstrain I existsi R 1 > 1 .Similarly,theuniquedominanceequilibrium E 2 = S 2 ; 0 ;J 2 ofstrain J existsi R 2 > 1 Proof. Thedominanceequilibriumofstrain I istheequilibriumwhichhas J infected equilibriumcomponentzeroand I infectedequilibriumdierentthanzero.Hence,to ndthedominanceequilibriumofstrain I ,welet J =0andassume I 6 =0.Thena dominanceequilibrium E 1 = S 1 ;I 1 ; 0ofstrain I mustsatisfythefollowingequation system: .2 8 < : 0= )]TJ/F34 11.9552 Tf 11.955 0 Td [( 1 S 1 I 1 )]TJ/F34 11.9552 Tf 11.955 0 Td [(S 1 + I 1 ; 0= 1 S 1 I 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( + I 1 Bythesecondequationinthesystemabove,wehave .3 S 1 = + 1 Alsobytherstequationinthesystem3.2,weget 1 S 1 I 1 )]TJ/F34 11.9552 Tf 11.955 0 Td [(I 1 = )]TJ/F34 11.9552 Tf 11.955 0 Td [(S 1 Substitutingtheequation3.3intotheequationabove,weobtain I 1 = 1 )]TJ/F15 11.9552 Tf 17.656 8.088 Td [(1 R 1 : ThenthedominanceequilibriumofIstrainis E 1 = S 1 ;I 1 ; 0 ; where S 1 = + 1 ;I 1 = )]TJ/F15 11.9552 Tf 17.657 8.088 Td [(1 R 1 .Notethatitexistsi R 1 > 1. Tondthedominanceequilibriumofstrain J ,weuseasimiliarargument.Weobtain aquadraticpolynomialof J 2 fromtheequationsystemconsidered. J 2 isadominance equilibriumofstrain J anditmustsatisfy: 0= 2 + [ J 2 ] 2 + 2 A + + )]TJ/F15 11.9552 Tf 11.956 0 Td [( 2 [ J 2 ]+ 2 A )-222(R 2 : Thenbyquadraticformula,wehave [ J 2 ] 1 ; 2 = )]TJ/F34 11.9552 Tf 9.298 0 Td [(a 2 p a 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 a 1 a 3 2 a 1 ; where a 1 = 2 + a 2 = 2 A + + )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 and a 3 = 2 A )-222(R 2 Itisalwaystruethat a 1 > 0.Thenthereexistsauniquepositiveroot J 2 ofthisquadratic polynomiali a 3 < 0i R 2 > 1.Specically,thedominanceequilibriumofstrain J is E 2 = S 2 ; 0 ;J 2 ;

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6 where S 2 = [ 2 J 2 + ] and J 2 = )]TJ/F34 11.9552 Tf 9.299 0 Td [(a 2 + p a 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 a 1 a 3 2 a 1 ,with a 1 ;a 2 ;a 3 givenabove. Nowdeneinvasionnumbers ^ R I and ^ R J forstrain I and J ,respectively,asfollows: ^ R I = 2 S 1 and ^ R J = 1 + 1 2 + J 2 A + J 2 : Bytheequilibriumcondition,wehave 2 S 2 = + J 2 A + J .Hence ^ R J = 1 S 2 + : Thenthefollowingresultisestablished: Theorem3.4. Thedominanceequilibriumofstrain I islocallyasymptoticallystable if ^ R I < 1 andunstableif ^ R I > 1 .Similarly,thedominanceequilibriumofstrain J is locallyasymptoticallystableif ^ R J < 1 andunstableif ^ R J > 1 Proof. Firstnotethat J j E 1 = S 1 ;I 1 ; 0 = 0 @ )]TJ/F34 11.9552 Tf 9.299 0 Td [( 1 I 1 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( 2 S 1 1 I 1 00 00 2 S 1 )]TJ/F34 11.9552 Tf 11.955 0 Td [( 1 A Considerthefollowingreducedmatrix: J 1 = )]TJ/F34 11.9552 Tf 9.298 0 Td [( 1 I 1 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( 2 S 1 1 I 1 0 Ithaseigenvalues 1 ; 2 withnegativerealpart,since Tr J 1 < 1and Det J 1 > 0. Theniftheeigenvalue 3 = 2 S 1 )]TJ/F34 11.9552 Tf 12.28 0 Td [( isnegative,thedominanceequilibrium E 1 islocallyasymptoticallystable.Notethatitholdsi ^ R I < 1 : Moreover E 1 isunstableif ^ R I < 1 : Bysimiliarargumentabove,onecanalsoanalysethestabilityofthedominanceequilibrium E 2 : TheJacobianmatrixevaluatedat E 2 isasfollows: J j E 2 = S 2 ; 0 ;J 2 = 0 B B @ )]TJ/F34 11.9552 Tf 9.299 0 Td [( 2 J 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( 1 S 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.298 0 Td [( 2 S 2 0 1 S 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( + 0 2 J 2 0 2 S 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( + J 2 A + J 2 + AJ 2 A + J 2 2 1 C C A Notethatbytheequilibriumcondition,wehave 2 S 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( + J 2 A + J 2 + AJ 2 A + J 2 2 = )]TJ/F34 11.9552 Tf 21.261 8.088 Td [(AJ 2 A + J 2 2 Toobtainsomeoftheeigenvalues,wecanreducedthematrixabovetothefollowing 2 2matrix: J 2 = 0 @ )]TJ/F34 11.9552 Tf 9.299 0 Td [( 2 J 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.299 0 Td [( 2 S 2 2 J 2 )]TJ/F34 11.9552 Tf 21.261 8.088 Td [(AJ 2 A + J 2 2 1 A Ithasbotheigenvalues 1 ; 2 withnegativerealpart,since Tr J 2 < 0and Det J 2 > 0 Thenthedominanceequilibrium E 2 islocallyasymptoticallystableiftheeigenvalue 3 = )]TJ/F34 11.9552 Tf 9.299 0 Td [( 1 S 2 )]TJ/F15 11.9552 Tf 12.02 0 Td [( + isnegativewhichholdsi ^ R J < 1.Furthermoreitisunstableif ^ R J > 1 :

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7 4. CoexistenceofLPAIandHPAIvirusstrainsinthemodel Theorem4.1. If min fR 1 ; R 2 g > 1 ,thenthereexistsauniquecoexistenceequilibrium if min n ^ R I ; ^ R J o > 1 Proof. Anequilibrium ^ E = ^ S; ^ I; ^ J ofthesystem2.1mustsatisfythefollowing equationsystem: .1 8 > > > > < > > > > : 0= )]TJ/F34 11.9552 Tf 11.955 0 Td [( 1 ^ S ^ I )]TJ/F34 11.9552 Tf 11.955 0 Td [( 2 ^ S ^ J )]TJ/F34 11.9552 Tf 11.956 0 Td [( ^ S + ^ I 0= 1 ^ S )]TJ/F15 11.9552 Tf 11.955 0 Td [( + 0= 2 ^ S )]TJ/F15 11.9552 Tf 11.955 0 Td [( + ^ J A + ^ J Bythesecondequationofthesystem4.1,weobtain .2 ^ S = + 1 Aftersubstitutingtheequationaboveintothethirdequationin4.1,weobtain 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( + ^ J A + ^ J =0 : Itisequivalenttotheequation: R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 A + ^ J = ^ J: Solvingfor ^ J ,weobtain .3 ^ J = A R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F34 11.9552 Tf 11.956 0 Td [( R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Firstnoticethatif R 1 R 2 ,thenthereisnopositivesolution ^ J: Nowsuppose R 2 > R 1 .Then ^ J> 0i )]TJ/F34 11.9552 Tf 11.955 0 Td [( R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 > 0i + > ^ R I Letmin n ^ R I ; ^ R J o > 1 : Then ^ R J > 1 : Noticethat ^ R J = 1 ^ R I J 2 A + J 2 + : Hence 1+ J 2 A + J 2 > ^ R I since ^ R J > 1 : Thenitconcludesthat + > ^ R I .Therefore ^ J> 0 : Next,wesolvetheequationsystem4.1for ^ I andshowthat ^ I> 0 : Bytherstequationin4.1,wehave = ^ S 1 ^ I + 2 ^ J + )]TJ/F34 11.9552 Tf 11.955 0 Td [( ^ I: Substituting4.2intotheequationabove,weobtain = + 1 1 ^ I + 2 ^ J + )]TJ/F34 11.9552 Tf 11.955 0 Td [( ^ I:

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8 Aftermultiplyingbothsideoftheequalityby 1 ; weobtain = + ^ I + R 2 R 1 ^ J + 1 R 1 )]TJ/F34 11.9552 Tf 13.151 8.088 Td [( ^ I : Then ^ I + )]TJ/F34 11.9552 Tf 13.15 8.087 Td [( = )]TJ/F15 11.9552 Tf 13.151 8.087 Td [( 1 R 1 )]TJ 13.151 8.087 Td [(R 2 R 1 ^ J: Substitutingtherighthandsideoftheequality4.3intotheequalityabove,weget ^ I = 1 )]TJ/F15 11.9552 Tf 17.657 8.088 Td [(1 R 1 )]TJ/F34 11.9552 Tf 24.285 8.088 Td [(A + R 2 R 1 R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F34 11.9552 Tf 11.955 0 Td [( R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Nowwewanttoshowthat ^ I> 0. Bythedenitionof ^ R J ,wehave ^ R J = S 2 ^ S : Then ^ R J > 1impliesthat S 2 > ^ S: Moreoverwehave 2 S 2 = + J 2 A + J 2 and 2 ^ S = + ^ J A + ^ J : Then 2 S 2 )]TJ/F15 11.9552 Tf 13.952 3.022 Td [(^ S > 0.Thisimplies J 2 > ^ J Nowdene F I;J := )]TJ/F15 11.9552 Tf 13.952 3.022 Td [(^ S 1 I + 2 J + + I and G I;J := )]TJ/F34 11.9552 Tf 11.955 0 Td [(I )]TJ/F34 11.9552 Tf 11.955 0 Td [(S 2 2 J + : Firstnotethat F I;J = )]TJ/F34 11.9552 Tf 12.71 0 Td [(I )]TJ/F15 11.9552 Tf 14.706 3.022 Td [(^ S 2 J + ,byequilibriumcondition.Thenfor all I;J ,wehave G I;J 0.Since F ^ I; ^ J =0,wehave ^ I = F ; ^ J > 0.Thenwecanconclude that ^ I> 0 : Theorem4.2. Thecoexistenceequilibrium ^ E islocallyasymptoticallystablewhenever itexists. Proof. Toanalyzethestabilityofthecoexistenceequilibrium,werstconsidertheJacobianmatrixevaluatedatthisequilibrium: .4 J j ^ E = ^ S; ^ I; ^ J )]TJ/F34 11.9552 Tf 11.956 0 Td [(I = 0 B B B B @ )]TJ/F34 11.9552 Tf 9.299 0 Td [( 1 ^ I )]TJ/F34 11.9552 Tf 11.955 0 Td [( 2 ^ J )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 11.955 0 Td [( )]TJ/F34 11.9552 Tf 9.298 0 Td [( 1 ^ S + )]TJ/F34 11.9552 Tf 9.298 0 Td [( 2 ^ S 1 ^ I )]TJ/F34 11.9552 Tf 9.298 0 Td [( 0 2 ^ J 0 )]TJ/F34 11.9552 Tf 9.298 0 Td [( )]TJ/F34 11.9552 Tf 26.505 8.088 Td [(A ^ J A + ^ J 2 1 C C C C A Nextweobtainthefollowingcharacteristicequation: 3 + b 1 2 + b 2 + b 3 =0 ;

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9 Figure1. ThesolutionsSt,ItandJtwithrespecttotime.The parametersconsideredare=1 : 77 ; 1 =4 : 95 10 )]TJ/F32 7.9701 Tf 6.587 0 Td [(5 ; 2 =0 : 000229 ; = 0 : 001678 ; =0 : 034 ; =0 : 012 where b 1 = 1 ^ I + 2 ^ J + + A ^ J A + ^ J 2 b 2 = 1 ^ I + 2 ^ J + A ^ J A + ^ J 2 + 2 2 ^ S ^ J + 1 ^ I b 3 = 1 ^ I A ^ J A + ^ J 2 andapplyRouth-Hurwitzcriteria.Noticethat b 1 > 0 ;b 2 > 0 ;b 3 > 0and b 1 b 2 )]TJ/F34 11.9552 Tf 11.955 0 Td [(b 3 > 0 : ThenbytheRouth-Hurwitzcriteria,thecoexistenceequilibrium ^ E islocallyasymptoticallystable. 5. Simulations Fig.1wasplottedusingrealdatafromttinganavianinuenzamodeltohuman avianinuenzacases[11]andwecanclearlyobservethatthelowpathogenicstrain vanishesquicklywhilethehighpathogenicprevails.However,thereisaverysmallarea inwhichbothstrainscoexistveryclosetozero. InFig.2,wecanclearlyobservethatthereexistsaregionofcoexistencewithbiologicallysignicantparametersinthemodel,howeverthelowpathogenicstraintends tozeroveryquickly.Variationoftheparametersgivesusabiggerareaofcoexistence betweenthetwostrains. Fig.3showscoexistenceofthistwostrains.InFig.3weuseadierentsetof parameters,namelyincreasing 2 anddecreasing 1 .Thisresultsareconsistentwith theanalyticalresultsprovingcoexistenceofthetwostrains. InFig.4,weplot onthex-axisand 1 onthey-axis.Wecanseethatforaxed the regionofcoexistencefordierentvaluesof 1 increasesas increases.Thecoexistence

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10 Figure2. ThesolutionItwithrespecttotimewiththesameparametersconsideredinFig.1. Figure3. ThesolutionsItandJtwithrespecttotime.The parametersconsideredare=1 : 77 ; 1 =12 : 2 4 : 95 10 )]TJ/F32 7.9701 Tf 6.587 0 Td [(5 ; 2 = 0 : 00023 ; =0 : 001678 ; =0 : 034 ; =0 : 012 Figure4. Regionofcoexistencebetweenthetwostrains ofthestrainsisdirectlyrelatedtoculling,thusthehigherthecullingcoecientthe morelikelythecoexistenceofstrains.

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11 6. Discussion Inthispaper,westudyatwo-strainSImodel,onestrainisHPAIandtheother oneisLPAI.Weincludeaselectivecullingrate.Weshowthatifthereproduction numbers R 1 and R 2 aresmallerthanone,thenthedisease-freeequilibriumislocally andgloballyasymptomaticallystable.Furthermore,thedominanceequilibrium E 1 E 2 ofLPAIHPAIvirusstrainexistsandisuniquewhenthereproductionnumber ^ R 1 ^ R 2 isbiggerthanone.Moreover,thedisease-freeequilibriumislocallyasymptomatically stablewhenbothreproductionnumbersarelessthanoneandunstableifatleastoneof themisgreaterthanone.Thesimulationshavebeendonefordierentparametersand theresultsareconsistentwiththeanalyticalresultsobtainedinsection3. Boththelowpathogenicstrainandthehighpathogenicstrainhaveauniquedominanceequilibriumwhenthereproductionnumbersaregreaterthan1.Theyarelocally asymptoticallystableiftheinvasionnumbers ^ R I and ^ R J aresmallerthanone,respectively.Theuniquenessofthecoexistenceequilibriumisveriedanalyticallywhenthe invasionnumbers ^ R I and ^ R J arebothlargerthanone.Itwasalsoveriedthatthe coexistenceequilibriumislocallyasymptomaticallystablewheneveritexists. CoexistenceofstrainsispresentedinFig.3withthegivenparameters.Whenweconsiderrealisticparameterstakenfromttinganavianinuenzamodeltohumanavian inuenzacases[11]thelowpathogenicstraininthegraphtendstozero. Theregionofcoexistenceplottedagainst and 1 ispresentedinFig.4.Foraxed ,theregionofcoexistencefordierentvaluesof 1 increasesas increases.Culling playsanimportantroleinthecoexistenceofthepathogens.Inparticular,thelarger themorelikelythecoexistence. References [1] Xue-ZhiLi,Xi-ChaoDuan,MiniGhosh,Xiu-YingRuan ,Pathogencoexistenceinducedby saturatingcontactrates, NonlinearAnalysis-realWorldApplications 10 ,pp.3298-3311. [2] WebsterRG,PeirisM,ChenH,GuanY ,H5N1outbreaksandenzooticinuenza.Emerg InfectDis.2006;12:38.doi:10.3201/eid1201.051024. [3] M.Martcheva ,Anevolutionarymodelofinuenzawithdriftandshift, J.Biol.Dynamics. Vol 6,2012,p.299-332. [4] M.Martcheva,M.Nuno,Z.Feng,C.Castillo-Chavez ,MathematicalModelsofInuenza: TheRoleofCross-Immunity,QuarantineandAge-Structure,LectureNotesinMathematics,Vol. 1945,MathematicalEpidemiologyFredBrauer,PaulinevandenDriessche,JianhongWu,Eds., Springer-Verlag,Berlin,2008,p.349-364. [5] De-LianQian,Xue-ZhiLi,M.Ghosh ,CoexistenceofStrainsinducedbyMutation, InternationalJournalofBiomathematics ,05:03. [6] M.A.Nowak,R.M.May ,Superinfectionandtheevolutionofparasitevirulence, Proceedings: BiologicalSciences Vol.255,issue1342,p.81-89. [7] MayRM,MANowak ,Coinfectionandtheevolutionofparasitevirulence, ProcRSoc B261 ,p.209-215. [8] Andreasen,Viggo;Pugliese,A. ,Pathogencoexistenceinducedbydensitydependenthost mortality, JournalofTheoreticalBiology Vol.177 ,p.159-165. [9] Lipsitch,S.Siller,M.Nowak. ,Theevolutionofvirulenceinpathogenswithverticaland horizontaltransmission, Evolution 50 ,p.1729-1741. [10] LeongHK,GohCS,ChewST,etal ,PreventionandcontrolofavianinuenzainSingapor, Ann.Acad.Med.Singap. 37 ,p.5049.

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12 [11] M.Martcheva,J.Lucchetti,M.Roy ,AnAvianInuenzaModelanditsFittoHumanAvian InuenzaCases, AdvancesinDiseaseEpidemiology J.M.Tchuenche,Z.Mukandavire,Eds.,Nova SciencePublishers,NewYork,2009,p.1-30.