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Chapter 7（第3页）

Inthissametinytimeperiod，aedpersoheRemovedcategory.HeheprobabilitythatIedsreduberwillbeproportionaltotheed，sotakestheformγ*I，whereγdependsonhowfastthoseiedrecover，areisolated，ordie.

Wehavefouiveincrease，oradetheed.Thebalawoprobabilities，β*S*Iaermiheranepidemicoccurs.Thereisagoodanalogywithgambling.Ifthegameisloadedagaiismorelikelytoreduceyourcapitaltha:yourcapitalfollowsarandomwalk，withaniowardszero.Butifthegamefavoursyou，providedthatbadluotbankruptyouearly，therandomwalkushersyhfromzerotooutrunanylosiowiisnecessary，butnotsuffietobeinyourfavour.

Intheepidemitext，thismeansthattheonlytimeanepidemic（=largefhtoccuriswhenaheedismorelikelytobeahanadesymbols，β*S*Imustexceedγ*I，whichisthesameasaskingthatS，thenumberofSusceptibles，exceedstheratioγβ，aquahethresholdforthispopulation.Thisisexactlyerelookihediseaseeoapopulation，

anepidemilyotheibleexceedsthisthreshold.

WilliamKermadersonMdritedthisresultin1927.Epidemibeavoidedbykeepingtheiblebelowthethreshold，whibeatwodistinctways.First，vaatetoreduumbersusceptible.Sedmeanstoihreshold.Beingaratio，athresholdiheorincreases–wespeeduprecoveryrates，orisolateier–orwheordecreases–wemaybeabletoredufeess，orwesurethatpeoplemixless，bygsporarily，orpostponingsportseventswherelargecrowdswouldgather.Wealsoassessthelikelysizeofthebehesedifferentresponses，andsojudgewhicharew.

Thesamepriepidemiimals.Thefirststeptake-and-mouthoutbreakincattleisusuallytorestrictents，whicreasesthethresholdsizebyredugthedenominator.Thisisoftenapaniedbymassslaughter（notavailablewithdiseasesinhumans!），whilyreducesthesizeofthesusceptiblepopulation，butalsoiheorihreshold.

Thisanalysisalsoexplainswhyweshouldexpectepidemicsofchildhooddiseasesatfairlyregularintervals.Afteranepidemicreducesthesizeofthesusceptiblepopulatiohreshold，hs，withiva，graduallytakethesizeabovethethreshthesforthebreak.Theloervalbetweehehigherthevulion，andthemoresevereidemicbe.

Knorobabilitiesmaynotcurediseases，butithelpmitigatetheireffects.

Batg

Thearmywishestoidentifywhichof1，000potesmaybevuloadisease，andthusunfittoserve.Abloodtestisavailable，butite.thejobbedohan￡50，000?

Providedthatonlyafairlysmallproportionwillprovevulheansweris‘yes’.berK，andpoolbloodsamplesfromKrecruits;thehispooledsample.Iftheresultishenallthosewhotributedbloodaredneedherwise，ositiveresult，atleastohegroupwouldtestpositive，soweuseKmoretests，ohem，tosettlethematter.Ifweareluetestwillsuffice，butwemighthavetomakeK+1tests.Wehopetomakefewerteststhanifeaistestedindividually.

&aopooldependsontheprobabilityofapositivetest.Supposethisis1%.Theehereositivesampleamo10%ofthetime，whiletheceallare90%.Sowewillneedo90%ofthetime，buteleves10%ofthetime.Thatleadstoabouttwotestse.Poolingtensamplesreducesthemeancostforthosetesfrom￡500to￡100.Ifwesplitthe1，000reto100groupsofsizeten，andpooltheirsamples，weexpecttosave80%oftheieof￡50，000!

Morerefinedcalsshowthat，whentheprobabilityofapositivetestisindeed1%，wewoulddoslightlybetterpoolinggroupsofeleveen，butthedifferenceisverymargihebestchoiceofthesizeofapoolinggroupisquiteseheprobabilityofapositivetest.Ifweexpect2%ofrecruitstotestpositive，inimizedifwepooleightbloodsamples;with5%thebestchoiceistopoolfivesamples，whileif10%willtestpositive，poolingfoursamplestur.（On，useofthebinomialdistributioheseanswers.）

InWorldWarII，thissimpleideasavedtheUSAabout80%ofitsiedcosts.

Airline

&houghtheymustpensatepassengersedofffiightsforwhichtheyhavepaid，airliiiaplane’scapacity.Simpleeicsisthereasootheairlineofmakihardlygeswiththenumberofpasseeachemptyseatislostrevehatnoteveryonewhohasbookedaparticularfiightdoesathebestamount?

Supposethattheplanehasos，atafareof￡100perseat，butapassengerispaid￡200iftheyhavetobeturnedawaybecausethefiightisfull.Theairlineneedsagoodestimateoftheprobabilitythatapassengerwhohasbookedwillactuallyturnup.Fhtstoholidaydestinations，thisprobabilitywillbecloseto100%，butitwillberatherlerswhohavemorefiexibilityiravelplaawillhelpairliheseces.

Perhapseagerwhobookshas，ily，an80%ingupfht.If120ticketsaresold，totalrevenueis￡12，000，and，althoughonly96passeurnupe，thereisad15%，thatmorethan100showup，aobeleftbehind.（Thesenumbersagainusingthebinomialdistributiotoberefundedbecauseinthissouttobe￡80.Selliicketsraisestherevenueby￡500，aalmeanrefundincreasesbyonly￡275，sothispolicyisexpeoreprofitable.Themostprofitablepoliaverage，selling128tiparedwithselling125，theextrarevenueof￡300justoutweighsthemeaof￡295，while129ticketswouldbeslightlyworsethanthis.

Itismorerealistictosupposethatsomepassengersaremorelikelytoturnupthahatgroupsofpassengerswhobooktogetherwilleitherallturnup，orhemwill.Butthesedetailscorporatedintothemodel，andairlineswilluetosellseatsuedpeheextrarevenue.

Queues

&hebest-developedapplisofprobabilityisthestudyofqueuesofvariouskiialimpetusptstouaionintelephoheworkoftheDanishtelephoneengineerAgnerErlangishoheuseofhisheunitofthevolumeoftelephoraffigtheorytributedtothesuccessoftheBerlinairliftof19489，aicstudyofqueuesflourishedduriyyears.

DavidKendalliion，withtheformatABn，notedasashorthandwayqueueswhereersarrivesipo，A，referstothedistributioweenerarrivals，whileBdescribesthedistributiotakestoserveaer，andnisthenumberofservers.

Forexample，intheexpressionDD3，‘D’isshortfordetermihatthereisnoraall.ersarriveofixedtimeintervals，allservicetimesalsohaveexactlythesamelength，ahreeservers.Thisqueuewouldbeoflittleihefieldofprobability，asthereisnovariability.Butsupposethereisahugeentialers，eahasatiurningupiimeinterval，sothatersarriveatsomeoverallaveragerate，butpletelyatrahesymbolMisused，tohonourAndreyMarkov，soMD2meansthatersarriveatrandom，aheroftwoservers，whoeaefixedtimetodotheirjob.

&oknoillbehave.Themaiionsarehowlongdoershavetowait，howfrequentlyareserversjustsittihumbs，andwhatprovematters?The‘servers’maybetheintensivecarebeds，while‘ers’arepatiecare.

Ifersarriveeveryfiveminutese，ahreeservers，thehemeaimeislessthaes，aqueueofihwillbuildup，andthewholeoperationisunsustainable.Sowemustassumethatthemeaakingaberofservers，islessthaimebetweenarrivals.Theratioofthesetwomeaermedthetraffisity，somefigurebetweenzeroandunity.

Inanidealworld，erswouldfalyshortqueuesandtheserverwouldbebusynearlyallthetime.Butthesetworequiremericallyopposed.Takeasimplecasewithoneserverandersarrivingatrandom.Ifthetraffisitywere0.9，calsshowthatwemightexpectaboutfivewaitingerse，ayqueueabout10%

ofthetime.Iftheiyroseto0.98，theserverwouldbeu2%ofthetime，butthemeahwouldshootupto25.Mosterswouldsiderthisawement.Unlessservershaveeime’，erswillgetahesystem，ordoboth.

Thequeuebehaviourdependsonmuchmoretharaffisity.sbeihemorevariabletheservicetime，thelongeryouexpectthequeuetobe.Withseveralservers，itmatterswhethertheyoperateasinmyrailwaystatioralqueuefeedsuptosixservers，orasi，whereIchoosewhichaisletojoin，andstaythere.Iions，e.g.ambulanceersmayhavehigherpriority.Mahe‘Firste，firstserved’rule，butwhennoodsarestoredoie，firstserved’mayapply.Somequeuesfeedotherqueues，theserversmayworkatdifferentspeeds，buomerscether.Eagle-eyedqueuishavefouhetralquestioaicmodelyoukof.