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MCMA题
TheLeavesofaTree
Nowadaystheheavymetalpollutionissocommonthatpeoplepaymoreandmoreattentiontoit.Theaimofthispaperistocalculatethemaximumofmethylmercuryinhumanbodyduringtheirlifetimeandthemaximumnumberoffishtheaverageadultcansafelyeatpermonth.FromCityOfficialsresearch[1],wegetinformationthatthemeanvalueofmethylmercuryinbasssamplesoftheNeversinkReservoiris1300ug/kgandtheaverageweightofbasspeopleconsumepermonthis0.7kg.Accordingtothedifferentconsumingtimeineverymonth,weconstructadiscretedynamicalsystemmodelfortheamountofmethylmercurythatwillbebioaccumulatedintheaverageadultbody.Inidealconditions,weassumepeopleconsumebassatfixedtermpermonth.Basedonit,weconstructfixed-ingestionmodelandwereachtheconclusionthatthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3505ug.Asmethylmercuryingestedisnotonlycomingfrombassbutalsofromotherfood,hence,wemakefurtherrevisetoourmodelsothatthemodelisclosertotheactualsituation.Asaresult,wefigureoutthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3679ug.Asamatteroffact,althoughweassumepeopleconsumeonefishpermonth,theconsumingtimehasgreatrandomness.Takingtherandomnessintoconsideration,weconstructarandom-ingestionmodelatthebasisofthefirstmodel.Throughcomputersimulations,weobtainthemaximumofmethylmercuryinhumanbodyis4261ug.Wealsocalculatethemaximumamountis4420ugafterrandom-ingestionmodelisrevised.Asitisknowntous,differentcountriesanddistrictshavedifferentcriterionsformercurytoxicity.Inourcase,weadoptLD50asthetoxiccriterions(LD50isthedosageatwhich50%ofthehumansexposedtoaparticularchemicalwilldie.TheLD50formethylmercuryis50mg/kg.).Wespeculatemercurytoxicityhaseffectontheabilityofeliminatingmercury,therefore,wesetupvariable-eliminationmodelatthebasisofthefirstmodel.Accordingtothefirstmodel,theamountofmethylmercuryinhumanbodyis50ug/kg,farlessthan50mg/kg,sowereachtheconclusionthatthefishconsumptionrestrictionsputforwardbythereservoiradvisoriescanprotecttheaverageadult.Iftheamountofmethylmercuryingestedincreases,theamountofbioaccumulationwillgoupcorrespondingly.If50mg/kgisthemaximumamountofmethylmercuryinhumanbody,wecanobtainthemaximumnumberoffishthatpeopleconsumesafelypermonthis997.
Keywords:
methylmercurydiscretedynamicalsystemmodelvariable-eliminationmodel
discreteuniformrandomdistributionmodelrandom-ingestionmodel
Introduction
Withthedevelopmentofindustry,thedegreeofenvironmentalpollutionisalsoincreasing.Humanactivitiesareresponsibleformostofthemercuryemittedintotheenvironment.Mercury,abyproductofcoal,comesfromacidrainfromthesmokestackemissionsofold,coal-firedpowerplantsintheMidwestandSouth.Itsparticlesriseonthesmokestackplumesandhitcharideonprevailingwinds,whichoftenblownortheast.AftercollidingwiththeCatskillmountainrange,theparticlesdroptotheearth.Onceintheecosystem,micro-organismsinthesoilandreservoirsedimentbreakdownthemercuryandproduceaverytoxicchemicalformknownasmethylmercury.Ithasgreateffectonhumanhealth.
PublicofficialsareworriedabouttheelevatedlevelsoftoxicmercurypollutioninreservoirsprovidingdrinkingwatertotheNewYorkCity.Theyhaveaskedforourassistanceinanalyzingtheseverityoftheproblem.Asaresultofthebioaccumulationofmethylmercury,ifthereservoirispolluted,wecanmakesurethattheamountofmethylmercuryinfishisalsoincreasing.IfeachpersonadherestothefishconsumptionrestrictionsaspublishedintheNeversinkReservoiradvisoryandconsumesnomorethanonefishpermonth,throughanalyzing,weconstructadiscretedynamicalsystemmodeloftimefortheamountofmethylmercurythatwillbioaccumulateintheaverageadultperson.Thenwecanobtainthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetime.Atthesametime,wecanalsogetthetimethatpeoplehavetakentoachievethemaximumamountofmethylmercury.Asweknow,differentcountriesanddistrictshavedifferentcriterionsforthemercurytoxicity.Inourcase,weadoptthecriterionofKellerArmyCommunityHospital.Ifthemaximumamountofmethylmercuryinhumanbodyisfarlessthanthesafecriterion,wecanreachtheconclusionthatthereservoirisnotpollutedbymercuryorthepolluteddegreeisverylow,otherwisewecansaythereservoirisgreatpollutedbymercury.Finally,thedegreeofpollutionisdeterminedbytheamountofmethylmercuryinhumanbody.
ProblemOne
discretedynamicalsystemmodel
ThemeanvalueofmethylmercuryinbasssamplesoftheNeversinkReservoiris1300ug/kgandtheaverageweightofbassis0.7kg.Accordingtothesubject,peopleconsumenomorethanonefishpermonth.Forthesafetyofpeople,wemustconsiderthebioaccumulationofmethylmercuryundertheworstconditionthatpeopleabsorbthemaximumamountofmethylmercury.Therefore,weassumethatpeopleconsumeonefishpermonth.
Assumptions
●Theamountofmethylmercuryinfishisabsorbedcompletelyandinstantlybypeople.
●Theeliminationofmercuryisproportionaltotheamountremaining.
●Peopleabsorbfixedamountofmethylmercuryatfixedtermpermonth.
●Weassumethehalf-lifeofmethylmercuryinhumanbodyis69.3days.
Solutions
Let
denotetheproportionofeliminatingmethylmercurypermonth,
denotetheaccumulationproportion.Asweknow,methylmercurydecaysabout50percentevery65to75days,ifnofurthermethylmercuryisingestedduringthattime.Consequently,
Throughcalculating,weget
Let’sdefinethefollowingvariables:
denotestheamountofmethylmercuryatinitialtime,
denotesthenumberofmonth,
denotestheamountofmethylmercuryinhumanbodyatthemomentpeoplehavejustingestedthemethylmercuryinthemonth
denotestheamountofmethylmercurythatpeopleingestpermonthand
.
Moreover,weassume
Though,
weget
Withtheremainingamountofmethylmercuryincreasing,theeliminationofmethylmercuryisalsogoingup.Weknowtheamountofingestedmethylmercurypermouthisaconstant.Therefore,withtimegoingby,therewillbeabalancebetweenabsorptionandelimination.Wecanobtainthesteady-statevalueofremainingmethylmercuryas
approachesinfinity.
Thevalueof
isshownbyfigure1.
Figure1.merthylmercurycompletelycomingfromfishandingestedatfixedtermpermonth
Ifthedifferenceoftheremainingmethylmercurybetweenthemonth
and
islessthanfivepercentoftheamountofmethylmercurythatpeopleingestpermonth,thatis,
Thenwecanget
Atthesametime,wecanworkoutthetimethatpeoplehavetakentoachieve3380ugis11months.
Fromourmodel,wereachtheconclusionthatthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulateintheirlifetimeis3505ug.
Ifpeopleingestmethylmercuryeveryhalfofamonth,however,thesumofmethylmercuryingestedpermonthisconstant,consequently,
Asaresult,weobtainthemaximunamountofmethylmercuryinhumanbodyis3270ug.Whenthedifferenceiswithin5%,wegetthetimepeoplehavetakentoachieveitis11months.
Similarly,ifpeopleingestmethylmercuryperday,wegetthemaximumamountis3050ug,andthetimeis10months.
RevisingModel
Asamatteroffact,theamountofmethylmercuryinhumanbodyisnotcompletelycomingfromfish.AccordingtotheresearchofHongKongSARFoodandEnvironmentalHygieneDepartment[1],undernormalcondition,about76percentofmethylmercurycomesfromfishand24percentcomesfromotherseafood.Inordertomakeourmodelmoreandmoreinlinewiththeactualsituation,itisnecessaryforustoreviseit.TheU.S.environmentalProtectionAgency(USEPA)setthesafemonthlydoseformethylmercuryat3microgramperkilogram(ug/kg)ofbodyweight.IfweadoptUSEPAcriterion,wecancalculatetheamountofmethylmercurythattheaverageadultingestfromseafoodis50.4ugpermonth.
Assumptions
●Theamountofmethylmercuryintheseafoodisabsorbedcompletelyandinstantlybypeople.
●Theeliminationofmethylmercuryisproportionaltotheamountremaining.
●Peopleingestfixedamountofmethylmercuryfromotherseafoodeveryday.
●Weassumethehalf-lifeofmethylmercuryinhumanbodyis69.3days.
Solutions
Let
denotetheamountofmethylmercuryatinitialtime,
denotethenumberofdays,
denotetheremainingamountontheday
and
denotetheamountofmethylmercurythatpeopleingestperday.Moreover,weassume
Inaddition,weworkout
Theproportionofremainingmethylmercuryeachdayis
then
Throughcalculating,weget
Becauseof
weobtainsteady-statevalueofmethylmercury
Ifthedifferenceofremainingmethylmercurybetweentheday
and
islessthanfivepercentoftheamountofmethylmercurythatpeopleingesteveryday,thatis,
Wehave
Sowecanreachtheconclusionthatthemaximumamountofmethylmercurytheaverageadulthumanwillbioaccumulatefromseafoodis160ugandthetimethatpeopletaketoachievethemaximumis301days.
Let
denotetheamountofmethylmercurypeopleingestthroughbassatfixedtermpermonth,sotheamountofmethylmercuryanaverageadultaccumulateontheday
is
Thevalueof
isshownbyfigure2.
Figure2.merthylmercurycomingfromfishandotherseafoodandingest