Nonlinear Dynamics of Infectious Diseases Transfer

1NONLINEARDYNAMICSOFINFECTIOUSDISEASESTRANSFERWITHPOSSIBLEAPPLICATIONSFORTUBERCULARINFECTIONV.D.Krevchik1,5,T.V.Novikova2,Yu.I.Dahnovsky3,M.B.Semenov1,5,E.V.Shcherbakova1,KenjiYamamoto41PhysicsDepartment,PenzaStateUniversity,Penza440017,[e-mail]2Penzaregionaltubercularprophylacticcenter,Russia3DepartmentofPhysics&Astronomy/3905,1000E.UniversityAve.,UniversityofWyoming,Laramie,WY,82071,USA4ResearchInstituteofInternationalMedicalCenter,[e-mail]5InstituteforBasicResearch,P.O.Box1577,PalmHarbor,FL34682,[e-mail]10,2007Inthispaper,wemodelanonlineardynamicsofinfectiousdiseasestransfer.Particularly,westudypossibleapplicationstotubercularinfectioninmodelswithdifferentprofiles(peakvalues)ofthepopulationdensitydepend-enceonspatialcoordinates.Ourapproachisbasedonthewellknownmethodofinstantonswhichhasbeenusedbytheauthorstodescribekineticsofadiabaticchemicalreactionsasafunctionoftheheat-bathtemperatureandothersystemparameters.Inourapproach,weuse“socialtemperature”T∗asoneofthecon-trollingparameters.IncreaseofT∗leadstoaccelerationoftheinfectiousdis-easestransfer.The“blockage”effectfortheinfectiousdiseasestransferhasbeendemonstratedinthecasewhenpeakvalues(inthepopulationdensity)areequaltooneandunderconditionthatthe“socialtemperature”islow.Existenceofsucheffectessentiallydependsfromenvironment“activity”(socialandpro-phylactic).ResultsofourmodelingqualitativelymeetthetuberculosisdynamicspreaddatainPenzaregionofRussia.Keywords:infectiousdiseases,transfermodeling.21.IntroductionTheproblemofexactmodelingofdifferenttypesofinfectiousdiseasestransferisofgreatimportance[1,2,3,4].Inmostcasesexistingmodelsareof“macro-scopic”character(thescaleiscountrywide).Anumberof“macroscopic”mod-elscanbedescribedas“network”models,percolation-,reactionary-diffusion-,oscillating,stochastic-,multivariableMarkovianmodels[1],etc.Anothertypeofmodelsareofamicroscopiccharacterwhichareusedatthemolecularlevel:mutationmodels;models,whichrevealinfectiousstabilitytodifferentmedicineinfluence,etc.Uptodate,modelsofanintermediate(mesoscopic)levelpracti-callyarenotdeveloped.Suchmodelscouldgiveimportant(integrating)infor-mationaboutinfectious“carrier”(likeincubationperiodorlatency).Also,suchmodelswouldallowinvestigationoffeaturesofcontrollableinfectioustransferatthescaleofconnectingsettlements,e.g.,towns.Existingspectrumoftheinfectioustransfermodelshasbeendiscussedatthe“InternationalSymposiumonTransmissionModelsforInfectiousDiseases”[2,3,4]attheTokyoInternationalMedicalCenterofJapan,andvitalnecessityofthemesoscopicmodelsforinfectioustransferhasbeenconvincinglydemon-strated.AlthoughthemostreportsatthisConferencehasbeendevotedtomod-elingofthesupposedglobalpandemicinfluenza(withaccountforhighrisksandfortheinfectioustransferhighrates);themesoscopicmodelforthetubercu-losistransferhasbeenrecognizedasimportantoneaswell.ImportanceoftheresearchatthislevelcanbealsoconfirmedbytheintegratedstatisticsforthetuberculosisdiseasesgrowthinRussiasince1991(seeFig.1).ThistendencyiskepttodayinRussia,andproblemiscomplicatedbecauseofhighpopulationdecrease(seeFig.2).3(a)(b)(c)Fig.1.(a)TendenciesforthetuberculosisdiseasesgrowthinRussiasince1991;(b)PopulationdecreaseinPenzaregion,Russia;(c)TuberculosisdiseasesgrowthinPenzaregion,Russia.2.ThemodeldescriptionPriortomathematicalformulationofthemodel,wewilldiscussonsomeassumptions.Forexample,withanaccountforthetuberculosisdiseasesfeaturesPenzaregion14200001440000146000014800001500000152000015400001560000199819992000200120022003200420056.49%Penzaregion0102030405060708090199819992000200120022003200420054(associaldiseases),preferential(orprobable)directionsofinfectioustransferaredirectionsfromsettlements(towns)withlowerpopulationdensity(Fig.3)tosettlementswithhigherpopulationdensity.Theinfectiontransferrateessen-tiallydependsfromsocial-economicconditions,whichweindirectlydescribebythe“socialtemperature”parameter.Withhigher“socialtemperature”,morecomplicatedsocial-economicconditionsarerealized(unemploymentlevelin-creases,inflationincreases,medicalservicefortheessentialpopulationpartisgettingworse,etc.);astheresult,infectioustransferratebecomeshigher.Envi-ronment(whereinfectiontransfers)essentiallyinfluenceontheinfectiousdis-easetransferdynamics.Particularly,activeprophylacticscanincreaseresistibil-ity(or“dissipation”)ofenvironmenttotheinfectiousdiseasetransfer.Ontheotherhand,destabilizingorexcitation(oscillating)influenceontheenvironmentcanleadtotheinfectioustransferrategrowth.Besides,“inertia”,whichisre-latedtoincubationperiodofdisease,canbeincludedinthemodelthroughtheparametersimilarto“mass”.Alltheabovementionedfactorsareconsideredinsuggestedmesoscopicmodel.Toconstructthemodel,wedevelopandapplywellapprovedinstantonmethod,whichhasbeenusedearlierindissipativetunnelchemicalkineticsre-search(withaccountoftheheatbathinfluence)[5,6].Forthispurpose,wewillspecifytheformulationoftransfer(kinetic)problemandwillshowhowHamil-tonianoftheinitialreactionarysystemcanbetransformedtothetraditionalone