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1TIMEEVOLUTIONOFNON-LETHALINFECTIOUSDISEASES:ASEMI-CONTINUOUSAPPROACH.A.NovielloDipartimentodiMatematicaedApplicazioni“RenatoCaccioppoli”UniversitàdegliStudidiNapoli“FedericoII”I-80100Napoli,ItalyF.RomeoDipartimentodiFisica“E.R.Caianiello”UniversitàdegliStudidiSalernoI-84081Baronissi(SA),ItalyR.DeLucaINFMandDIIMA,UniversitàdegliStudidiSalernoI-84084Fisciano(SA),ItalyABSTRACTAmodeldescribingthedynamicsrelatedtothespreadingofnon-lethalinfectiousdiseasesinafixed-sizepopulationisproposed.Themodelconsistsofanon-lineardelay-differentialequationdescribingthetimeevolutionoftheincrementinthenumberofinfectiousindividualsanddependsuponalimitednumberofparameters.Predictionsareingoodqualitativeagreementwithdataoninfluenza.PACS:87.23.Cc2IINTRODUCTIONThedynamicallawsgoverningthespreadingofinfectiousdiseasesamongaclosedpopulationareofinterestmainlyinthefieldofmedicalscience.However,owingtothestirringofpublicopiniongeneratedbythediffusionoflethalviruses,socialissuesarealsotobeconsidered.Indeed,muchattentionhasbeenlatelydevotedtothestudyoftheHIVvirus,givenitsstronglylethalcharacter,sothattraditionalmathematicalmodels[1,2]havebeenmodifiedaccordingtothecharacteristicfeaturesofthisillness[3,4].Morerecently,SARShashadagreatsocialandeconomicimpactinthewholeworld[5].Eventhemostcommonnon-lethalviralinfectionsas,forexample,influenza,arecauseofsomesocialdistress,giventheirperiodicappearanceandtheirpotentialityofbeingharmfultotheweakexposedpopulation.Classicmodels[6]giveafairlygooddescriptionofthetimeevolutionofinfectiousdiseases.Mostoftheassumptionsmadeinthistraditionaltypeofapproachgreatlysimplifytheanalysisofthecomplexproblemofthespreadingofsuchillnessesamongacertainnumberofindividuals.However,inordertogiveaccountofthegreatvarietyofresponsestothesameviralinfectionandtodistinguishamongdifferentsocialhabitsofdifferentindividuals,anetworkapproachcanbeadopted[7,9].Thetopologicalissuesaddressedbythenetworkapproachareveryimportantperse,sincetheyfindapplicationinotherfields,suchas,forexample,sociology[10].Inaddition,theyareusefulindescribing,morerealistically,theproblemandindefiningthelimitsofvalidityoftraditionalmodels.Inthepresentwork,wepropose,underthesamebasichypothesesoftraditionalmodels,amodifiedSIR(Susceptible-Infectious-Recovered)model.Thetimeevolutionofnon-lethalinfectiousillnessesinacommunityofhighlymobileindividualsor,equivalently,inapopulationwithdifferentspeciesuniformlydistributedovertheobservationlandscape,willbeanalyzed.Thepaperwillbeorganizedintwomainparts.InthefirstpartwereconsidertheequationsgoverningthetimeevolutionofSIRmodelsbyassumingthatanindividual,infectedattimet,recoversafteranintervaloftimeτ,whichistakentobethesameforeverypopulationmember.The3interactionbetweentheS-Ispeciesistakentoberegulatedbyaconstantstatisticalparameterπ,whileotherhorizontalcross-interactionsareneglected.ThetimeintervalinwhichtheS-speciespopulationismonitored,isassumedtobesmall,insuchawaythatthetotalnumberofindividualscanbethoughttobeconstantovertheentireobservationperiod.Inthesecondpartofthepresentpapertheresultingnon-lineardelay-differentialequations[11]fortheincrementinthenumberofinfectedindividualsaresolvedbystandardnumericalroutines.Thedurationofthediseaseasafunctionofarescaledinteractionparameterπˆisplottedfordifferentinitialnumbersofinfectiousindividuals0~µandfordifferentpopulationsizes.Endemic-likeandepidemicregimesfortheinfectiousdisease,asalsofoundbymeansoftraditionalapproaches[6],aredefinedbyintroducingthecross-overvaluecπˆoftheeffectiveinteractionparameter.Conclusionsaredrawninthelastsection,wherefurtherdevelopmentsofthemodifiedmodelarealsodiscussed.IITHEMODELLetusconsiderafixedsizepopulationofindividuals,subdividedinthreedistinctspecies:Susceptible(S-species);Infectious(I-species);Recovered(R-species).Assumethatthemobilityofeachindividualissuchtocauseacompleteinteractionamongallmembersofthecommunitywithinonemonitoringtimeinterval.Equivalently,onemightalsoassumethatthemembersofeachspeciesareuniformlydistributed,atanytimet,overtheobservationlandscape.Underthishypothesiswecantakethenumberofnewinfectionst∆Aµ∆,duetoanon-lethalvirus,tobegivenbythefollowingrelation[1]:()()tttA∆=∆σπµµ,(1)where()tµand()tσarethenumberofinfectiousandsusceptibleindividualsattimet,respectively,andwhereπistheeffectiveinfectionrate.Aslightdifferentmeaningistobeattributedtothestatisticalexchangeparameterπ,dependingwhetherweareobservinghighlymobileindividualsor4uniformlydistributespeciesovertheentirelandscape.Inthepresentmodel,indeed,weshallintroducearecoverytimeinterval(orinfectiousperiod)τ,takentobeequalforallindividuals.Thisquantityactsasareferencetimescale,sothat,inthecaseofhighlymobileindividuals,πisrepresentativeoftheeffectivenessoftheinteractionbetweentheS-andtheI-speciesinatimeτ,whileitaccountsalsoforthenumberoftheseinteractionsinthesameintervaloftimeinthecaseofuniformdistributionofthethreespecies.Bynowdefining()tρasthetotalnumberofrecoveredindividualsattimet,wemaywrite:)[]τn,)()t−)()−+at)(−+−t10(n()()(tttNρσµ++=,(2)whereNisthetotalconstantnumberofmember
本文标题:Time Evolution of Non-Lethal Infectious Diseases A
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