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Help with Homework UBC Math 102
HelpwithHomework
UBCMath102
Assignment10:Problem4
DeadleavesaccumulateonthegroundinaforestatarateofAgramspersquarecentimeterperyear.Atthesametime,theseleavesdecomposeatacontinuousrateofkpercentperyear.WriteadifferentialequationforthetotalmassQofdeadleaves(persquarecentimeter).
UBCMath102
Balanceequationformass/area(Q(t))
=-
UBCMath102
Rateofchangeofamtleaves(gm/cm2/yr)
Rateofaccumulation(gm/cm2/yr)
Rateofdecay
(gm/cm2/yr)
Balanceequationformass/area(Q(t))
=-
UBCMath102
dQ/dt(gm/cm2/yr)
A(gm/cm2/yr)
bQ(gm/cm2/yr)
Assignment10:Problem8Youhaveapondwithdecorativefishinyourbackyard.Thepondholds800gallonsofwater.Onceaweekyoupourfreshwaterintothepondattherateof100gallonsperhour.Thepondisfilledtothebrim,andsoasyoupourwaterintothetankwaterflowsoutatthesamerate.Thereisapumpinthepondthatkeepsthewaterperfectlymixed.Yourgoalistoaddwateruntilanypollutantsinthepondarereducedbyafactor1/2.YoukeepthefreshwaterflowingforHowmanyhours?
Hint:Writedownamassbalanceequation
Massbalance
NopollutantenteringSomepolutantmassleavingwithoutflowdm/dt=0–ratemassoutcheckyourunits!
Assignment10:Problem11
• Givendatafornewton’sLawofCooling• Knownexactsolution:
• Wanted:findvaluesofconstantsTinitandk• Fitbestlinetotransformeddata• NOTE:YouhavetosubtractambienttempEfromthedatabeforetakingln!
Modelforthespreadofadisease
UBCMath102
DynamicsofadiseaseGoal:tousemathematicstounderstandpandemic/spreadofdiseaseoutbreak1995https://www.youtube.com/watch?v=Mj9SUJdpJS4Contagion(2011)(staringMattDamon)https://www.youtube.com/watch?v=4sYSyuuLk5g
UBCMath102
DynamicsofadiseaseGiven:1infectedindividualinahealthypopul.• Howmanypeoplewillbecomeinfected?• Willthediseasepersistornot?
Goal:Deriveandanalyzeadifferentialequationmodeltoaddressthisquestion.
UBCMath102
Definitions
t=timeS(t)=numofhealthy(susceptible)peopleI(t)=numberofinfectedpeopleN(t)=totalpopulationsize=S(t)+I(t)
UBCMath102
Assumptions
• Diseasetransmittedbycontactbetweeninfectedandsusceptibleindividuals
• Infectedindividualsrecover(withrateμ),andarethensusceptibleagain(noimmunity).
• Ignorebirths,deaths,migration.• Randommixing,identicalindividualsineachgroup.
UBCMath102
Spreadofinfection
UBCMath102
??
Transmittingthedisease
Anewinfectioncanoccurwhenaninfectedindividualcomesintocontactwithahealthysusceptibleindividual
UBCMath102
Lawofmassaction:
Therateofcontactbetweentwo(idealgas)moleculesisproportionaltotheproductoftheirdensities.
UBCMath102
(1)Diseasetransmissionrate
AccordingtotheLawofMassAction,therateofdiseasetransmissionwouldbe
(A)(D)(B)(E)(C)
UBCMath102
(1)Diseasetransmissionrate
AccordingtotheLawofMassAction,therateofdiseasetransmissionwouldbe
(A)(D)(B)(E)(C)
UBCMath102
Contactrate• DiseasetransmissionrequirescontactbetweenSandI.
• TherateofdiseasetransmissionisproportionaltotheproductofSandI,soisoftheform
UBCMath102
Keepingtrackoftransitions
Writedownadifferentialequationfortherateofchangeofthenumberofinfectedindividuals.
UBCMath102
(2)Myequationlookslike
(A)
(B)(C)(D)
(2)Myequationlookslike
(A)
(B)(C)(D)
Keepingtrackoftransitions
Writedownadifferentialequationfortherateofchangeofthenumberofsusceptibleindividuals.
UBCMath102
(3)Myequationlookslike
(A)
(B)(C)(D)
(3)Myequationlookslike
(A)
(B)(C)(D)
Keepingtrackoftransitions
UBCMath102
(4)Units
Whataretheunitsofβandμ?(A) Bothhaveunitsof1/time.(B) Bothhaveunitsofperpersonperunittime.(C) βhasunitsofperpersonperunittimeand
μhasunitsof1/time(D) Bothhaveunitsofpeople/time(E) Noneoftheabove.
UBCMath102
(4)Units
Whataretheunitsofβandμ?(A) Bothhaveunitsof1/time.(B) Bothhaveunitsofperpersonperunittime.(C) βhasunitsofperpersonperunittimeand
μhasunitsof1/time(D) Bothhaveunitsofpeople/time(E) Noneoftheabove.
UBCMath102
(5)Totalpopulation:N(t)=S(t)+I(t)
Basedonthemodel,thetotalpopulationwill(A) Growexponentiallywithrate(B) Decreaseexponentiallyatrate(C) Growatrate(D) Decreaseaspeoplediefromthedisease,at
rate(E) Stayatsomeconstantvalue.
UBCMath102
(5)Totalpopulation:N(t)=S(t)+I(t)
Basedonthemodel,thetotalpopulationwill(A) Growexponentiallywithrate(B) Decreaseexponentiallyatrate(C) Growatrate(D) Decreaseaspeoplediefromthedisease,at
rate(E) Stayatsomeconstantvalue.
UBCMath102
Totalpopulation:N=S+I=constant
Proof:Addeqs:ThenSoconclude:
UBCMath102
Simplifyingthemodel
EliminatethevariableS(t)usingWriteadifferentialequationthatcontainsonlythevariableI(t)andtheconstantsN,βandμ.
UBCMath102
(6)Myequationlookslike
WheretheconstantKis:
(A)(B)(C)(D)(E)
UBCMath102
(6)Myequationlookslike
WheretheconstantKis:
(A)(B)(C)(D)(E)
UBCMath102
Solution:
Analyzethemodel
Drawastatespacediagramand/oraslopefielddiagramforthedifferentialequation.(Assumeβ>0).AddsolutioncurvesforI(t).Interprettheresults.Considertwocases:K>0andK<0.
UBCMath102
IfKispositive(K>0)
•
UBCMath102
PredictedbehaviourforI(t)
UBCMath102
I(t)K0
IfI(0)>0thenI(t)àK
(7)IfKispositive(K>0)
ThenumberofinfectedindividualsI(t)alwaysapproachesK,andthenumberofsusceptibleindividualsS(t)(A) AlsoapproachesK.(B) ApproachesN.(C) Approachesμ/β.(D) ApproachesμN/β.(E) Approacheszero.
(7)IfKispositive(K>0)
ThenumberofinfectedindividualsI(t)alwaysapproachesK,andthenumberofsusceptibleindividualsS(t)(A) AlsoapproachesK.(B) ApproachesN.(C) Approachesμ/β.(D) ApproachesμN/β.(E) Approacheszero.
IfKisnegative(K<0)
UBCMath102
Solution,cont’d
(8)IfKisnegative(K<0)
Inthiscasethepredictionisthat(A)Ià0andSàK(B) IàNandSà0(C) Ià0andSàμ/β(D) Ià0andSàN(E) Noneoftheabove
UBCMath102
(8)IfKisnegative(K<0)
Inthiscasethepredictionisthat(A)Ià0andSàK(B) IàNandSà0(C) Ià0andSàμ/β(D) Ià0andSàN(E) Noneoftheabove
UBCMath102
Prediction“itdependsonR0”:
WerefertothisratioasR0
UBCMath102
R0• TheratioR0=βN/μisthebasicreproductivenumberofthedisease.
• Itrepresentsthenumberofnewinfectionscausedbyasingleinfectedindividualduringthecourseoftheirillness.
• ThediseasewillbecomeendemicifR0>1.
UBCMath102
Example:
Supposeaninfectiousdiseasehasapproximatedurationof10daysandtransmissionrate0.001perpersonperday.WhatisthesmallestsizeNofapopulationinwhichthisdiseasecouldbecomeendemic?
R0=(βN/μ)>1
(9)Example:Supposeaninfectiousdiseasehasapproximatedurationof10daysandtransmissionrate0.001perpersonperday.WhatisthesmallestsizeNofapopulationinwhichthisdiseasecouldbecomeendemic?
(A) 10(B)100(C)1000(A) 10000(B) Notenoughinformationtosay.
(9)Example:Supposeaninfectiousdiseasehasapproximatedurationof10daysandtransmissionrate0.001perpersonperday.WhatisthesmallestsizeNofapopulationinwhichthisdiseasecouldbecomeendemic?
(A) 10(B)100(C)1000(A) 10000(B) Notenoughinformationtosay.
Example:Supposeaninfectiousdiseasehasapproximatedurationof10daysandtransmissionrate0.001perpersonperday.WhatisthesmallestsizeNofapopulationinwhichthisdiseasecouldbecomeendemic?Diseaseduration=10daysàμ=1/10=0.1R0=βN/μ>1soneedpopulationsizeN>μ/β=(1/10)(1/0.001)=1/0.01=100Needapopulationofatleast100forthediseasetobecomeendemic!
CurrentdiseasemodelsThisisanACTIVEAREAOFRESEARCHincludingourownDeptofMathematics,aswellastheBCCDC(CenterforDiseaseControl)andmanyactiveresearchgroupsworld-wide.Researchersstudytheeffectsof:socialnetworks(notrandomlymixingpopulations),vaccinationstrategies,publichealthmeasures,birthsanddeaths,aswellasothermodesofdiseasespread.Askmeifyouwanttoknowmore.
Extrastuff:WhereR0comesfromandwhatitmeans
Note:thisisforyourowninterest
andwillnotbetestedonafinalexam
UBCMath102
Units
Eachtermhasunitsofpeople/timeβhasunitsofperpersonperunittimeμhasunitsof1/time
UBCMath102
Unitsandmeaning
Theconstantμhasunitsof1/time.So1/μhasunitsoftime.1/μisthetypicaltimethatapersonstaysintheinfectedclass(i.e.typicaldurationoftheinfectiousperiod.)
UBCMath102
Theconstantβhasunitsofperpersonperunittime.(person)-1(time)-1
ThusβNhasunitsof(time)-1ThisistheratethatnewinfectionsoccurperinfectiousindividualinapopulationofsizeN.
UBCMath102
Unitsandmeaning
WhatdoestheratioβN(1/μ)mean?
(A) Thenumberofpeoplewhogetsickafteralongtime
(B) Theratioofdiseasetransmissionratetodiseasedurationtime.
(C) Thenumberofnewinfectionscausedbyonesickindividualinahealthypopulation.
(D) Noneoftheabove
Unitsandmeaning
ThusβNistheratethatnewinfectionsoccurperinfectiousindividualinapopulationofsizeN.
R0=βN(1/μ)=βN/μisthenumberofnewinfectionsthatarestartedby1sickindividualduringthecourseoftheirillness.
UBCMath102
Unitsandmeaning
Fromdifferentialequationstotrigonometricfunctions
Introducingsineandcosine
ThemathematicsofLOVE
UBCMath102
ThemathematicsofStormyLOVE
https://www.youtube.com/watch?v=dBgn8wLOElU
RomeoandJuliet
Bythefamousmathematician,physicist,andauthoroffavouritessuchas:
UBCMath102
StevenStrogatz
RomeoandJuliet
Juliet:MypassionforRomeodecreasesinproportiontohislove.Romeo:MypassionforJulietincreasesinproportiontoherlove.
UBCMath102
RomeoandJuliet
Juliet:MypassionforRomeodecreasesinproportiontohislove.(ThemoreRomeolovesme,themoreIrunawayfromhim..Butwhenhehatesme,Istarttolovehim.)Romeo:MypassionforJulietincreasesinproportiontoherlove.(ThemoreJulietlovesme,themoreIloveher!Butwhenshehatesme,myloveforherdecreases.)
UBCMath102
Lovemeter
• Ihateyou Iloveyou
-101
UBCMath102
RomeoandJuliet
Letx(t)=Juliet’sloveforRomeo,y(t)=Romeo’sloveforJuliet
Bothx(t)andy(t)willchangewithtime,asthestar-crossedloverschaseeachotheracrossthelovemeter.
UBCMath102
(1)Juliet’slove,x(t)
MypassionforRomeodecreasesatarateproportionaltohislove.(Assumek1>0).(A) dx/dt=k1x (B)dx/dt=-k1x(C)dy/dt=-k1y (D)dx/dt=-k1y (E)Notsure
UBCMath102
(1)Juliet’slove,x(t)
MypassionforRomeodecreasesatarateproportionaltohislove.(Assumek1>0).(A) dx/dt=k1x (B)dx/dt=-k1x(C)dy/dt=-k1y (D)dx/dt=-k1y (E)Notsure
UBCMath102
(2)Romeo’slove,y(t)
MypassionforJulietincreasesatarateproportionaltoherlove(Assumek2>0).(A) dx/dt=k2x (B)dy/dt=-k2x(C)dy/dt=k2y (D)dy/dt=-k2y (E)dy/dt=k2x
UBCMath102
(2)Romeo’slove,y(t)
MypassionforJulietincreasesatarateproportionaltoherlove(Assumek2>0).(A) dx/dt=k2x (B)dy/dt=-k2x(C)dy/dt=k2y (D)dy/dt=-k2y (E)dy/dt=k2x
UBCMath102
Love-haterelationship
• Juliet: dx/dt=-k1y• Romeo: dy/dt=k2x
Thisisoncemoreapairofcoupleddifferentialequationsfortwofunctionsoftime,(x(t),y(t)).What’sgonnahappen?
UBCMath102
Letk1=k2=1
Thexyplanedx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
Directionfield
dx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
(3)Whatdoyouthinkwillhappen?
dx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
(3)Whatdoyouthinkwillhappen?
dx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
Solutioncurve
dx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
Solutioncurve
dx/dt=-k1ydy/dt=k2x
Juliet’slovex(t)
Romeo
’slovey(t)
RomeoandJulietchaseeachotherinanendlesslove-circle!
Shownin3Dwithtimeaxis:
UBCMath102
Timetà
Juliet’slovex(t)
Romeo’slovey(t)
RomeoandJuliet
UBCMath102 Timetà
(4)Doyourecognizethesefunctions?
Yes!Theseare(A) Polynomials(B) Exponentials(C) Powerfunctions(D) Sineandcosine(E) Notsure
UBCMath102
(4)Doyourecognizethesefunctions?
Yes!Theseare(A) Polynomials(B) Exponentials(C) Powerfunctions(D) Sineandcosine(E) Notsure
UBCMath102
RomeoandJuliet
Thesecurvesarex(t)=cos(t),y(t)=sin(t).Next,afterashortbreak,wewilldiscussthesetrigonometricfunctions!
UBCMath102
Timetà
Introducing:thetrigonometricfunctionssin(t),cos(t)
What’sspecialaboutthesefunctions?- Classic“periodicfunctions”- Describeoscillatingsystems-Specially“nice”derivs!-Closerelatives..
UBCMath102
(x(t),y(t))
Derivativesofcosineandsine
Cosine:Sine:(SeecoursenotesSection15.1wherethisisshownusingthedefinitionofthederivative.)
UBCMath102
Findthesecondderivativeofthefunctiony(t)=sin(t)
Whatdifferentialequationdoesthisfunctionsatisfy?
UBCMath102
(5)Sin(t)satisfies
(A)(B)(C)(D)(E)Notsure
UBCMath102
(5)Sin(t)satisfies
(A)(B)(C)(D)(E)Notsure
UBCMath102
Solution:
2ndderivativeoffunction=-(originalfn)
Easytoshowthat:
Buttheseare“thesameequation”intwonotations:bothsaythat 2ndderivativeoffunction=-(originalfn)
Bothsineandcosinesatisfythesamekindofdifferentialequation
2ndderivativeoffunction=-(originalfn)
Confused???
“Ithoughttrigfunctionshavetodowithanglesandtriangles”Yestheydo!Providedweinterpretanglesinaspecialway:
UBCMath102
Anglesandtrigfunctions
AnglesinradiansWedefineanewmeasureforangles:1revolutionaroundacircle2πradiansAngleßàlengthofanarcsubtended
UBCMath102
Convention
• Anglesincreasecounterclockwise
(6)Convertfromdegreestoradians:
Intermsofradians,theangles30,45,60,90oare:
(A) π/6,π/4,π/3,π/2(B) π/3,π/2,π/6,π(C) π/30,π/45,π/60,π/90
UBCMath102
(6)Convertfromdegreestoradians:
Intermsofradians,theangles30,45,60,90oare:
(A) π/6,π/4,π/3,π/2(B) π/3,π/2,π/6,π(C) π/30,π/45,π/60,π/90
UBCMath102
Specialangles
UBCMath102(SeeTrigreview,AppendixofM102CourseNotes)
Connectionwithangle()
Nowletthetadependontime,sothepointwillmovearoundthecircle
