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AcellularautomatonmodelforthetransmissiondynamicsofschistosomiasisYunLiu,KaiChu,XiaoliXu,HaiweiWu*DepartmentofPathogenBiologyNanjingMedicalUniversityNanjing,ChinaChengWanDepartmentofPublicServiceManagementNanjingMedicalUniversityNanjing,ChinaAbstractInthispaper,anewstochasticmodelbasedoncellularautomataisestablishedtosimulatetheoccurrenceanddevelopmentofSchistosomajaponicumS.japonicuminfectioninanendemicpopulation.Weincludedtheprocessofthepathogeninvasionfromexposuretowormdevelopmentandtillwormdeathwhentheinfectionisclearedinthemodel.WefurtherutilizedthemodeltopredicttheprevalenceastheoutcomesoftheselectedchemotherapycarriedoutinJiahuvillage.Comparingmodelpredictedprevalenceandintensitieswiththeobservedparameters,itisanticipatedthatourcellularautomatontransmissionmodelcanserveasatoolforstudyingschistosomiasistransmissiondynamicsinendemicareas.Keywords-S.japonicum；cellularautomata；SjCAmodel；transmissiondynamics；prevalence；intensityofinfectionI.INTRODUCTIONSchistosomiasisisamajorparasiticdiseasethatrankssecondonlytomalariaintermsofhumansufferinginthetropics.Amongthethreespeciesofschistosomes,onlyS.japonicumisendemicinChinaandmainlydistributedintheareasofintermediateandlowerreachesoftheYangtzeRiver,especiallythePoyangLakeandDongtingLake[1,2].Thesimulationofdiseasedynamicsisvitallyimportantasitenablesonetobetterunderstandtheepidemiologyandcontrolmeasuresofschistosomiasis.Inthepastfewdecades,manyresearchers[3-5]devotedtothemathematicalmodelforthetransmissiondynamicsofschistosomiasisbyequation-basedmodelingEBM,themainformofwhichwereaseriesofdetermineddifferentialequations.Aftermathematicalmodelswereestablished,theshort-termandlong-termoutcomesofdiseasescanbepredictedinaparticularparameterssetting.Itisnotedthatthosemodelswereperformedincontinuoussystemsimulation.Translationofthepopulationdynamicsandstochasticsintodifferential/differenceequationformwasnottrivial.Besides,thosemodelsusuallytrackedtheaveragewormburdenwithouttakingthevariationsamongindividualsintoaccount.Itisimportanttonotethatomittingthestochasticityorthediscretenatureoftheindividualsoftenleadstoabsurdresults.Mostrecently,computerstochasticsimulatingmethodshavebeenwidelyusedtomimicthetransmissionandimmunedynamicsofdiseases[6,7],includingcellularautomataCA.CAisadynamicalsysteminwhichspace,time,andthestatesarediscrete.Unlikeequation-basedmodels,CAmodelsfocusontheindividualandcanthereforehandlebothhomogeneousandheterogeneouspopulations.Thetranslationoftheconceptualmodelintoanumericalsettingisusuallytechnicallysimpleandstraight-forward.Hence,CAmodelshavebeendevelopedtosimulatemanybiologicalsystems.Inthispaper,astochasticmodelbasedonCAwenameitasSjCAisestablishedtosimulatethetransmissiondynamicsofschistosomasisjaponicaintheendemicarea.TheSjCAmodelisusedtomimiceachindividual’sstateandeffectsoftheselectedchemotherapycarriedoutinJiahuvillage.Andtheoutcomesofshistosomiasisinthisareacanbepredicted.Bycomparingtheobservedprevalenceandintensitieswiththepredicteddataof2006inJiahuvillage,theeffectsofSjCAmodelareshownsatisfactory.II.MATERIALSANDMETHODSA.StudypopulationandsamplecollectionInDecember2005,peoplelivinginJiahuvillagewererecruitedfora2-yearcross-sectionalinvestigation[8].LocatedonthesoutheasternshoreofPoyangLakeinJiangxiProvince,thisvillagewasaS.japonicumendemicarea.Aquestionnairewassubmittedtocollectthewatercontactofeachresident.Variablesrecordedinthequestionnaireincludedthetypeofactivity,andthefrequencyofwatercontactandsoon.Thescoresofwatercontactrangedfrom0to80.Ateachsurvey,twostoolsampleswerecollectedfromparticipantsatof3-5daysintervals.Kato-KatztechniquewasusedtodetecteggsinthestoolsandresultswererecordedaseggspergramEPG[9].Allresidentsfoundtobeegg-positivereceivedasingleoraldoseofpraziquantelof40mg/kgbodyweight.TheselectedchemotherapywascarriedoutinJanuaryof2006,onemonthlaterthanthefirstsurvey.ThestudypopulationinSjCAmodelismadeupofparticipantswithEPGresultsinthe2consecutiveyears.Thereare706subjectsinthestudypopulationaged5-74years,outofwhich,134egg-positiveindividualsandtheprevalencewas18.98in2005while94egg-positiveindividualsand13.31ofprevalencein2006.ThegeometricmeanofEPGwas0.9851.548and0.4721.103,respectively.Figure1showstheresultsofthegeometricmeanofEPGby8agecategoriesof10yearseachinthe2-yearsurveys.B.FrameworkofSjCAmodelThedefiningcharacteristicsofCAmodelsarecell,itsneighbourcells,rulesandthespatialenvironment-lattice.Each978-1-4244-4713-8/10/25.002010IEEEFigure1.ResultsofthegeometricmeanofEPGby8agecategoriesof10yeaseachin2005greybarsand2006whitebars.Figure2.ModellingflowchartforthesimulationofthranmissiondynamicsoftheinfectionwithS.japonicumcell,definedbyapointinaregularspatiallattice,canhaveanyoneofafinitenumberofstatesthatareupdatedaccordingtoasetoflogicalrules.Thestateofeachcellatasubsequenttimestepisdependentonthecell’sownstateand/orthestatesofitsneighboursatthecurrenttimestep.Inourmodel,eachgridrepresentscertaingeographicdistrictinJiahuvillageofJiangxiProvince.TheelementsofeachindividualaredescribedbyPERSON.Wetakeonedayasatimestep.Ateachtimestep,thechangesofwormburdenineachindividualarerelatedtowaterexposure,numberofschistosomesacquiredperwatercontact,wormestablishmentfunctioni.e.fractionofacquiredcercariaesurvivingandthedeathofadultworm.Thepredictionsofprevalenceandintensitiescanbeobtainedaftercertaintimesteps.TheflowchartofsimulationisshowninFigure2.Foreachindividuali,theprincipalvariabletrackedinthemodeliswormburdeniw.TwodifferentparametertypesareusedTableI1fixedparametersthathavethesamevalueinallsimulations,2uncertainparametersforwhichweperformanuncertainanalysis.Besides,inthemodel,thehouselocationisdefinedbythelongitude,latitudeandaltitudeingeographicinformationsystemGIS.C.RulesinSjCAmodel1InitiationaProductionofbasicelementsThenumberofsimulatedindividualsis706,thesizeofstudypopulation.Accordingtotheobserveddata,elementsofeachindividualincludingageia,sexis,longitude,latitudeandaltitudebyGISofeachresidentcanbedetermined.bWormburdeniwEPGofeachindividualiEPGcanbeobtainedfromthe2-yearsurveys.Butdataonwormburdenofeachindividualiwarenotavailable.Itisassumedthatthereisaconstantaveragenumberofeggsperwormmatingprobabilitiesarenottakenintoaccountgivenbyie.iwisrelatedtoiEPGby/iiiwEPGe1cAveragewormestablishmentfunctionfItisadensitydependentwormestablishmentfunctionwhichdescribesaprocessinwhichthelikehoodofdevelopingintoanadultwormisassumedtobereducedwhenthecurrentwormburdenishighduetoa‘crowdingeffect’,toconcomitantimmunity,orboth.fcanbegivenas[10]111kMMfekMγ−−−2ThedistributionofwormsamongstudypopulationinJiahuvillagereflectsanegativebinomialdistributionwiththemeanintensityofinfectionM4.64andtheaggregationparameterk0.0149.Withγ0.001[5],f0.7595dWatercontactiρConsideringsomepeoplewiththescoreof0onwatercontact,i.e.iθ0,couldcontactwithcontaminantedwater,wereviseiθwithRandscaledtotherequiredvalueofiρ.eNumberofcercariaeacquiredperpersonperwatercontactεεisassumedtobepropotionaltothedensityofOncomelaniasnails.Accordingtothepastfielddata[11],εisanalysisedtorangein0.003798-0.0456.2simulationateachtimestepatiεForeachindividuali,thetiεisheredefinedastheexpectednumberofcercariaeacquiredperpersonperwatercontactateachtimestepwithoutacquiredimmunity.tiεisassumedtobearandomnumberwithintherangeofε.bSeasonalfluctuationUnlikeschistosomiasisinPhilippines,severalprocessesofhumanschistosomeinfectioninChinacanbedisturbedbyclimateanddisplayseasonalluctuation[14].Furthermore,winteristoocoldandOncomelaniasnailscouldnotshedcercariaelowerthan1℃[15]。Hence,2ρρspringsummerautumn3intwerε04cEstablishmentfunctionforeachindividualifTheestablishmentfunctionamongindividualsinCAmodelwasassumedtoobeyanegativebinomialdistribution.ThenaTABLEI.DEFAULTPARAMETERSETseriesofrandomnumbersobeyingthisnegativebinomialdistributioncanbegeneratedby,,,,706,1iffnbinrandAAnbinrandAAnbinrandrp∗∈NbinrandANbinrandA5nbinrandAAisarandomnumber,whichbelongstothesetofNbinrandA.Alltheelementsinthesetobeyanegativebinomialdistributionwiththemeanpandaggregationparameterr.Themeanofalltheelementsarestandardedto1.AndthesizeofNbinrandAis706.dTheforceofinfectioniΛThenumbeofnewsuccessfuladdedschistosomesisrelatedtoiiiifερΛ6e iΛNotonlythosefactorsdescribedabove,butalsothetypeandlocationofcontaminantedwatercaninfluencetheprobabilityandseverityofhumanschistosomeinfection.ItisassumedthatthenearertheresidentsliveawayfromthePoyangLake,thehigherprobabilitiesandthemoresevereofschistosomeinfectionsare.ThedistancethateachindividualliveawayfromthePoyangLakeisdefinedasid. iΛisrelatedtoidby ,iiiiiiiitdceiltdroundρρ≥ΛΛΛΛ7fThemortalityrateforwormispercapitamortalityrateforschistosomes.gThewormburdenofeachindividualatasubsequenttimestepisdependentonthenewaddedwormsandthemortalityrateforwormatthecurrenttimestep,givenby 1iiiiwtwtwtΛ−83TheeffectsofchemotherapyInourmodel,weassumethatonaverage98oftheinfectedresidentstreatedwithpraziquantalarecured,andthata60reductioninwormintensityoccursinhumanswhoareinfected,treated,butnotcured.Thesimulationofeffectsofchemotherapystartatthe31sttimestep.III.RESULTSA.ExperimentaltimesItisshownthatinthesameparameterssetting,theeffectsofsimulationsfor100timesarenotsignificantlydifferentfromthatofsimulationsfor1000times.Hence,weadopt100timessimulationineachparametersetting.B.EffectsofSjCAmodelThesimulationssuggestthebestfitbetweentheobservedandthesimulatedresultswasobtainedusingR20.Thedesignofotherparametershavebeendescribedabove.1PrevalenceAsshowninFigure3,theobservedprevalencein2006is13.31.Allpredictedprevalencesfluctuatearoundthisvalue.AlthoughtherearesignificantdifferencesP0.05betweenthesimulatedandobservedprevalence,theaverageprevalenceof100predictedresults13.59isproximatelyidenticaltotheobservedprevalence.2IntensityofinfectionFigure4showsthepredictedandobservedresultsofthegeometricmeanofEPGby8agecategoriesof10yearseachin2006.Thereareslightdifferencesbetweenthesimulatedandobservedage-specificintensityofinfections.C.ModelingeffectsafterparametersregulationFigure5and6showthepredictedprevalencesof2006inJiahuvillagewithdifferentRanddifferentf.Itisindicatedthatthepredictedprevalencesof2006aredirectlyproportionaltoRandtheestablishmentfunctionf,andthebestfitbetweentheobservedandthesimulatedresultswereobtainedusingtheparameterssettingweadopt,withtheotherparametersunchanged.IV.DISCUSSIONWefirstlyestablishanewapproachbasedoncellularautomatatomodelingthetransmissiondynamicsofschistsosomiasisjaponica.ComparedwiththetraditionalsimulationmethodofEBM,CAmodelingischaracterizedbymimickingtheoccurrenceanddevelopmentofschistosomesineachindividualusingastochasticapproach.AlthoughtherearesignificantdifferencesbetweentheobservationsandpredictionsofprevalenceP0.05andintensities,consideringtherandomnessofeachsimulationbyCAmodel,westillthinkthatCAmodelcansatisfactorilydescribethedevelopmentofS.japonicuminfectioninJiahuvillageofJiangxiProvincebetween2005-2006.Thispapershowsthattheone-yeareffectsofthismodelaresatisfactory,butitstillneedsfurtherextensionandrefining.ParameterSymbolValueandunitsReferenceFixedTimestept1Minimundurationofsimulation1dayAgeiayearsSexis1or2LongitudeLatitudeAltitudeDistancethateachindividuallivesawayfromthePoyangLakeidEPGperwormie10[14]ScoresofwatercontactiθRevisedvaluesforwatercontactR20WatercontactiρAveragewormestablishmentfunctionf0.7595[5,10]Wormlifespan1/4years[15]Efficacyofchemotherapy98UncertainforindividualiwormestablishmentfunctionifvariesExpectednumberofcercariaeacquiredperpersonperwatercontactε0.003798-0.0456[11]ExpectednumberofcercariaeacquiredperwatercontactateachtimestepitεvariesThetranmissioncyclecanbeclosedbyincludingthedynamicsoftheintermediatehostOncomelaniasnailsandthereserviorhostespeciallythebuffaloeswhichplayanimportantroleinFigure3.Resultsofprevalenceincludingthepredictedprevalenceineachsimulationsquarespotandtheobservedprevalenceblackline.Figure4.Resultsoftheage-specificgeometricmeansofEPGinafunctionof8agecategoriesof10yearseach,includingtheobservationswhitebarsandpredictionsgreybarsbyagecategory.Figure5.Resultsofpredictedprevalencesof2006inJiahuvillagewithdifferentR.Figure6.Resultsofthepredictedprevalencesof2006inJiahuvillagewithdifferentestablishmentfunctionf.thetranmissionandasinfectionsourcesinendemicareas.Furthermore,parametersinthemodelarebasedonfielddataandestimatedequationinexpertopinion,andsomeofthemarenotavailableinfieldinvestigation.InthehopethatCAmodelwilleventuallycontributetotheepidemiologyandtransmissiondynamicsofschistosomiasisjaponica,effortstofurtherdevelopthemodelwillbecontinued.ACKNOWLEDGMENTThisresearchissupportedbytheNationalNaturalScienceFoundationofChina30671836andNaturalScienceFoundationofJiangsuEducationalCommittee08KJD310006.REFERENCES[1]Ross,A.G.,etal.,FaecaleggaggregationinhumansinfectedwithSchistosomajaponicuminChina.ActaTrop,1998.702p.205-10.[2]YangJZ,Z.Z.,AnalysisontheHouseholdsClusterofPositiveSerumofSchistosomaisis.ChinseJOURNALofChangzhiMedicalCollege,2008.221p.29-30[3]Chan,M.S.,etal.,ThedevelopmentofanagestructuredmodelforschistosomiasistransmissiondynamicsandcontrolanditsvalidationforSchistosomamansoni.EpidemiolInfect,1995.1152p.325-44.[4]Chan,M.S.,etal.,Dynamicaspectsofmorbidityandacquiredimmunityinschistosomiasiscontrol.ActaTrop,1996.622p.105-17.[5]Liang,S.,D.Maszle,andR.C.Spear,Aquantitativeframeworkforamulti-groupmodelofSchistosomiasisjaponicumtransmissiondynamicsandcontrolinSichuan,China.ActaTrop,2002.822p.263-77.[6]Vlas,S.J.,etal.,SCHISTOSIMamicrosimulationmodelfortheepidemiologyandcontrolofschistosomiasis.AmJTropMedHyg,1996.555Supplp.170-5.[7]Chavali,A.K.,etal.,Characterizingemergentpropertiesofimmunologicalsystemswithmulti-cellularrule-basedcomputationalmodeling.TrendsImmunol,2008.2912p.589-99.[8]Lin,D.D.,etal.,EvaluationofIgG-ELISAforthediagnosisofSchistosomajaponicuminahighprevalence,lowintensityendemicareaofChina.ActaTrop,2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