AcellularautomatonmodelforthetransmissiondynamicsofschistosomiasisYunLiu,KaiChu,XiaoliXu,HaiweiWu*DepartmentofPathogenBiologyNanjingMedicalUniversityNanjing,ChinaChengWanDepartmentofPublicServiceManagementNanjingMedicalUniversityNanjing,ChinaAbstractInthispaper,anewstochasticmodelbasedoncellularautomataisestablishedtosimulatetheoccurrenceanddevelopmentofSchistosomajaponicum(S.japonicum)infectioninanendemicpopulation.Weincludedtheprocessofthepathogeninvasionfromexposuretowormdevelopmentandtillwormdeathwhentheinfectionisclearedinthemodel.WefurtherutilizedthemodeltopredicttheprevalenceastheoutcomesoftheselectedchemotherapycarriedoutinJiahuvillage.Comparingmodelpredictedprevalenceandintensitieswiththeobservedparameters,itisanticipatedthatourcellularautomatontransmissionmodelcanserveasatoolforstudyingschistosomiasistransmissiondynamicsinendemicareas.Keywords-S.japonicum；cellularautomata；SjCAmodel；transmissiondynamics；prevalence；intensityofinfectionI.INTRODUCTIONSchistosomiasisisamajorparasiticdiseasethatrankssecondonlytomalariaintermsofhumansufferinginthetropics.Amongthethreespeciesofschistosomes,onlyS.japonicumisendemicinChinaandmainlydistributedintheareasofintermediateandlowerreachesoftheYangtzeRiver,especiallythePoyangLakeandDongtingLake1,2.Thesimulationofdiseasedynamicsisvitallyimportantasitenablesonetobetterunderstandtheepidemiologyandcontrolmeasuresofschistosomiasis.Inthepastfewdecades,manyresearchers3-5devotedtothemathematicalmodelforthetransmissiondynamicsofschistosomiasisbyequation-basedmodeling(EBM),themainformofwhichwereaseriesofdetermineddifferentialequations.Aftermathematicalmodelswereestablished,theshort-termandlong-termoutcomesofdiseasescanbepredictedinaparticularparameterssetting.Itisnotedthatthosemodelswereperformedincontinuoussystemsimulation.Translationofthepopulationdynamicsandstochasticsintodifferential/differenceequationformwasnottrivial.Besides,thosemodelsusuallytrackedtheaveragewormburdenwithouttakingthevariationsamongindividualsintoaccount.Itisimportanttonotethatomittingthestochasticityorthediscretenatureoftheindividualsoftenleadstoabsurdresults.Mostrecently,computerstochasticsimulatingmethodshavebeenwidelyusedtomimicthetransmissionandimmunedynamicsofdiseases6,7,includingcellularautomata(CA).CAisadynamicalsysteminwhichspace,time,andthestatesarediscrete.Unlikeequation-basedmodels,CAmodelsfocusontheindividualandcanthereforehandlebothhomogeneousandheterogeneouspopulations.Thetranslationoftheconceptualmodelintoanumericalsettingisusuallytechnicallysimpleandstraight-forward.Hence,CAmodelshavebeendevelopedtosimulatemanybiologicalsystems.Inthispaper,astochasticmodelbasedonCA(wenameitasSjCA)isestablishedtosimulatethetransmissiondynamicsofschistosomasisjaponicaintheendemicarea.TheSjCAmodelisusedtomimiceachindividualsstateandeffectsoftheselectedchemotherapycarriedoutinJiahuvillage.Andtheoutcomesofshistosomiasisinthisareacanbepredicted.Bycomparingtheobservedprevalenceandintensitieswiththepredicteddataof2006inJiahuvillage,theeffectsofSjCAmodelareshownsatisfactory.II.MATERIALSANDMETHODSA.StudypopulationandsamplecollectionInDecember2005,peoplelivinginJiahuvillagewererecruitedfora2-yearcross-sectionalinvestigation8.LocatedonthesoutheasternshoreofPoyangLakeinJiangxiProvince,thisvillagewasaS.japonicumendemicarea.Aquestionnairewassubmittedtocollectthewatercontactofeachresident.Variablesrecordedinthequestionnaireincludedthetypeofactivity,andthefrequencyofwatercontactandsoon.Thescoresofwatercontactrangedfrom0to80.Ateachsurvey,twostoolsampleswerecollectedfromparticipantsatof3-5daysintervals.Kato-Katztechniquewasusedtodetecteggsinthestoolsandresultswererecordedaseggspergram(EPG)9.Allresidentsfoundtobeegg-positivereceivedasingleoraldoseofpraziquantelof40mg/kgbodyweight.TheselectedchemotherapywascarriedoutinJanuaryof2006,onemonthlaterthanthefirstsurvey.ThestudypopulationinSjCAmodelismadeupofparticipantswithEPGresultsinthe2consecutiveyears.Thereare706subjectsinthestudypopulationaged5-74years,outofwhich,134egg-positiveindividualsandtheprevalencewas18.98%in2005while94egg-positiveindividualsand13.31%ofprevalencein2006.ThegeometricmeanofEPGwas0.9851.548and0.4721.103,respectively.Figure1showstheresultsofthegeometricmeanofEPGby8agecategoriesof10yearseachinthe2-yearsurveys.B.FrameworkofSjCAmodelThedefiningcharacteristicsofCAmodelsarecell,itsneighbourcells,rulesandthespatialenvironment-lattice.Each978-1-4244-4713-8/10/$25.002010IEEEFigure1.ResultsofthegeometricmeanofEPGby8agecategoriesof10yeaseachin2005(greybars)and2006(whitebars).Figure2.ModellingflowchartforthesimulationofthranmissiondynamicsoftheinfectionwithS.japonicumcell,definedbyapointinaregularspatiallattice,canhaveanyoneofafinitenumberofstatesthatareupdatedaccordingtoasetoflogicalrules.Thestateofeachcellatasubsequenttimestepisdependentonthecellsownstateand/orthestatesofitsneighboursatthecurrenttimestep.Inourmodel,eachgridrepresentscertaingeographicdistrictinJiahuvillageofJiangxiProvince.TheelementsofeachindividualaredescribedbyPERSON.Wetakeonedayasatimestep.Ateachtimestep,thechangesofwormburdenineachindividualarerelatedtowaterexposure,numberofschistosomesacquiredperwatercontact,wormestablishmentfunction(i.e.fractionofacquiredcercariaesurviving)andthedeathofadultworm.Thepredictionsofprevalenceandintensitiescanbeobtainedaftercertaintimesteps.TheflowchartofsimulationisshowninFigure2.Foreachindividuali,theprincipalvariabletrackedinthemodeliswormburdeniw.Twodifferentparametertypesareused(TableI):(1)fixedparametersthathavethesamevalueinallsimulations,(2)uncertainparametersforwhichweperformanuncertainanalysis.Besides,inthemodel,thehouselocationisdefinedbythelongitude,latitudeandaltitudeingeographicinformationsystem(GIS).C.RulesinSjCAmodel1)Initiation:a)Productionofbasicelements:Thenumberofsimulatedindividualsis706,thesizeofstudypopulation.Accordingtotheobserveddata,elementsofeachindividualincludingageia,sexis,longitude,latitudeandaltitudebyGISofeachresidentcanbedetermined.b)Wormburdeniw:EPGofeachindividualiEPGcanbeobtainedfromthe2-yearsurveys.Butdataonwormburdenofeachindividualiwarenotavailable.Itisassumedthatthereisaconstantaveragenumberofeggsperworm(matingprobabilitiesarenottakenintoaccount)givenbyie.iwisrelatedtoiEPGby/iiiwEPGe=(1)c)Averagewormestablishmentfunctionf:Itisadensitydependentwormestablishmentfunctionwhichdescribesaprocessinwhichthelikehoodofdevelopingintoanadultwormisassumedtobereducedwhenthecurrentwormburdenishighduetoacrowdingeffect,toconcomitantimmunity,orboth.fcanbegivenas10()1)(1(1)()kMMfekM+=+(2)ThedistributionofwormsamongstudypopulationinJiahuvillagereflectsanegativebinomialdistributionwiththemeanintensityofinfectionM=4.64andtheaggregationparameterk=0.0149.With=0.0015,f=0.7595d)Watercontacti:Consideringsomepeoplewiththescoreof0onwatercontact,i.e.i=0,couldcontactwithcontaminantedwater,wereviseiwithRandscaledtotherequiredvalueofi.e)Numberofcercariaeacquiredperpersonperwatercontact:isassumedtobepropotionaltothedensityofOncomelaniasnails.Accordingtothepastfielddata11,isanalysisedtorangein0.003798-0.0456.2)simulationateachtimestepa)ti():Foreachindividuali,theti()isheredefinedastheexpectednumberofcercariaeacquiredperpersonperwatercontactateachtimestepwithoutacquiredimmunity.ti()isassumedtobearandomnumberwithintherangeof.b)Seasonalfluctuation:UnlikeschistosomiasisinPhilippines,severalprocessesofhumanschistosomeinfectioninChinacanbedisturbedbyclimateanddisplayseasonalluctuation14.Furthermore,winteristoocoldandOncomelaniasnailscouldnotshedcercariaelowerthan115。Hence,2=spring&summerautumn(3)intwer=0(4)c)Establishmentfunctionforeachindividualif:TheestablishmentfunctionamongindividualsinCAmodelwasassumedtoobeyanegativebinomialdistribution.ThenaTABLEI.DEFAULTPARAMETERSETseriesofrandomnumbersobeyingthisnegativebinomialdistributioncanbegeneratedby,(,706,1)iffnbinrandAAnbinrandAAnbinrandrp=NbinrandANbinrandA(5)nbinrandAAisarandomnumber,whichbelongstothesetofNbinrandA.Alltheelementsinthesetobeyanegativebinomialdistributionwiththemeanpandaggregationparameterr.Themeanofalltheelementsarestandardedto1.AndthesizeofNbinrandAis706.d)Theforceofinfectioni:Thenumbeofnewsuccessfuladdedschistosomesisrelatedtoiiiif=(6)e)i:Notonlythosefactorsdescribedabove,butalsothetypeandlocationofcontaminantedwatercaninfluencetheprobabilityandseverityofhumanschistosomeinfection.ItisassumedthatthenearertheresidentsliveawayfromthePoyangLake,thehigherprobabilitiesandthemoresevereofschistosomeinfectionsare.ThedistancethateachindividualliveawayfromthePoyangLakeisdefinedasid.iisrelatedtoidby,()iiiiiiiitdceiltdround=(7)f)Themortalityrateforworm:ispercapitamortalityrateforschistosomes.g)Thewormburdenofeachindividualatasubsequenttimestepisdependentonthenewaddedwormsandthemortalityrateforwormatthecurrenttimestep,givenby(1)()()iiiiwtwtwt+=+(8)3)TheeffectsofchemotherapyInourmodel,weassumethatonaverage98%oftheinfectedresidentstreatedwithpraziquantalarecured,andthata60%reductioninwormintensityoccursinhumanswhoareinfected,treated,butnotcured.Thesimulationofeffectsofchemotherapystartatthe31sttimestep.III.RESULTSA.ExperimentaltimesItisshownthatinthesameparameterssetting,theeffectsofsimulationsfor100timesarenotsignificantlydifferentfromthatofsimulationsfor1000times.Hence,weadopt100timessimulationineachparametersetting.B.EffectsofSjCAmodelThesimulationssuggestthebestfitbetweentheobservedandthesimulatedresultswasobtainedusingR=20.Thedesignofotherparametershavebeendescribedabove.1)Prevalence:AsshowninFigure3,theobservedprevalencein2006is13.31%.Allpredictedprevalencesfluctuatearoundthisvalue.Althoughtherearesignificantdifferences(P0.05)betweenthesimulatedandobservedprevalence,theaverageprevalenceof100predictedresults(13.59%)isproximatelyidenticaltotheobservedprevalence.2)Intensityofinfection:Figure4showsthepredictedandobservedresultsofthegeometricmeanofEPGby8agecategoriesof10yearseachin2006.Thereareslightdifferencesbetweenthesimulatedandobservedage-specificintensityofinfections.C.ModelingeffectsafterparametersregulationFigure5and6showthepredictedprevalencesof2006inJiahuvillagewithdifferentRanddifferentf.Itisindicatedthatthepredictedprevalencesof2006aredirectlyproportionaltoRandtheestablishmentfunctionf,andthebestfitbetweentheobservedandthesimulatedresultswereobtainedusingtheparameterssettingweadopt,withtheotherparametersunchanged.IV.DISCUSSIONWefirstlyestablishanewapproachbasedoncellularautomatatomodelingthetransmissiondynamicsofschistsosomiasisjaponica.ComparedwiththetraditionalsimulationmethodofEBM,CAmodelingischaracterizedbymimickingtheoccurrenceanddevelopmentofschistosomesineachindividualusingastochasticapproach.Althoughtherearesignificantdifferencesbetweentheobservationsandpredictionsofprevalence(P0.05)andintensities,consideringtherandomnessofeachsimulationbyCAmodel,westillthinkthatCAmodelcansatisfactorilydescribethedevelopmentofS.japonicuminfectioninJiahuvillageofJiangxiProvincebetween2005-2006.Thispapershowsthattheone-yeareffectsofthismodelaresatisfactory,butitstillneedsfurtherextensionandrefining.ParameterSymbolValueandunitsReferenceFixedTimestept1Minimundurationofsimulation1(day)AgeiayearsSexis1or2LongitudeLatitudeAltitudeDistancethateachindividuallivesawayfromthePoyangLakeidEPGperwormie1014ScoresofwatercontactiRevisedvaluesforwatercontactR20WatercontactiAveragewormestablishmentfunctionf0.75955,10Wormlifespan1/4(years)15Efficacyofchemotherapy98%Uncertain(forindividuali)wormestablishmentfunctionifvariesExpectednumberofcercariaeacquiredperpersonperwatercontact0.003798-0.045611Expectednumberofcercariaeacquiredperwatercontactateachtimestep()itvariesThetranmissioncyclecanbeclosedbyincludingthedynamicsoftheintermediatehostOncomelaniasnailsandthereserviorhostespeciallythebuffaloeswhichplayanimportantroleinFigure3.Resultsofprevalenceincludingthepredictedprevalenceineachsimulation(squarespot)andtheobservedprevalence(blackline).Figure4.Resultsoftheage-specificgeometricmeansofEPGinafunctionof8agecategoriesof10yearseach,includingtheobservations(whitebars)andpredictions(greybars)byagecategory.Figure5.Resultsofpredictedprevalencesof2006inJiahuvillagewithdifferentR.Figure6.Resultsofthepredictedprevalencesof2006inJiahuvillagewithdifferentestablishmentfunctionf.thetranmissionandasinfectionsourcesinendemicareas.Furthermore,parametersinthemodelarebasedonfielddataandestimatedequationinexpertopinion,andsomeofthemarenotavailableinfieldinvestigation.InthehopethatCAmodelwilleventuallycontributetotheepidemiologyandtransmissiondynamicsofschistosomiasisjaponica,effortstofurtherdevelopthemodelwillbecontinued.ACKNOWLEDGMENTThisresearchissupportedbytheNationalNaturalScienceFoundationofChina(30671836)andNaturalScienceFoundationofJiangsuEducationalCommittee(08KJD310006).REFERENCES1Ross,A.G.,etal.,FaecaleggaggregationinhumansinfectedwithSchistosomajaponicuminChina.ActaTrop,1998.70(2):p.205-10.2YangJZ,Z.Z.,AnalysisontheHouseholdsClusterofPositiveSerumofSchistosomaisis.ChinseJOURNALofChangzhiMedicalCollege,2008.22(1):p.29-303Chan,M.S.,etal.,ThedevelopmentofanagestructuredmodelforschistosomiasistransmissiondynamicsandcontrolanditsvalidationforSchistosomamansoni.EpidemiolInfect,1995.115(2):p.325-44.4Chan,M.S.,etal.,Dynamicaspectsofmorbidityandacquiredimmunityinschistosomiasiscontrol.ActaTrop,1996.62(2):p.105-17.5Liang,S.,D.Maszle,andR.C.Spear,Aquantitativeframeworkforamulti-groupmodelofSchistosomiasisjaponicumtransmissiondynamicsandcontrolinSichuan,China.ActaTrop,2002.82(2):p.263-77