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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.
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