A stage-structured Bayesian hierarchical model for salmon lice populations at individual salmon farms – Estimated from multiple farm data sets

ContentslistsavailableatScienceDirect

Ecological Modelling

j o u r n al ho me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l

A stage-structured Bayesian hierarchical model for salmon lice populations at individual salmon farms – Estimated from multiple farm data sets

M. Aldrin

, R.B. Huseby

, A. Stien

, R.N. Grøntvedt

, H. Viljugrein

, P.A. Jansen

aNorwegianComputingCenter,P.O.Box114Blindern,NO-0314Oslo,Norway

bNorwegianInstituteforNatureResearch,P.O.Box6606Langnes,9296Tromsø,Norway

cINAQAS,P.O.Box1223Sluppen,NO-7462Trondheim,Norway

dNorwegianVeterinaryInstitute,P.O.Box750Sentrum,NO-0106Oslo,Norway

a r t i c l e i n f o

Articlehistory:

Received11January2017

Receivedinrevisedform18May2017 Accepted23May2017

Keywords:

Populationmodel Aquaculture Stochasticmodel Sealicecounts

a b s t r a c t

Salmonfarminghasbecomeaprosperousinternationalindustryoverthelastdecades.Alongwithgrowth intheproductionfarmedsalmon,however,anincreasingthreatbypathogenshasemerged.Ofspecial concernisthepropagationandspreadofthesalmonlouse,Lepeophtheirussalmonis.Togaininsightinto thisparasite’spopulationdynamicsinlargescalesalmonfarmingsystem,wepresentafullymechanistic stage-structuredpopulationmodelforthesalmonlouse,alsoallowingforcomplexitiesinvolvedinthe hierarchicalstructureoffullscalesalmonfarming.Themodelestimatesparameterscontrollingawide rangeofprocesses,includingtemperaturedependentdemographicrates,fishsizeandabundanceeffects onlousetransmissionrates,effectsizesofvarioussalmonlousecontrolmeasures,anddistancebased betweenfarmtransmissionrates.Modelparameterswereestimatedfromdataincluding32salmon farms,exceptthelastproductionmonthsforfivefarms,whichwereusedtoevaluatemodelpredictions.

WeusedaBayesianestimationapproach,combiningthepriordistributionsandthedatalikelihoodinto ajointposteriordistributionforallmodelparameters.Themodelgeneratedexpectedvaluesthatfit- tedtheobservedinfectionlevelsofthechalimus,adultfemaleandothermobilestagesofsalmonlice, reasonablywell.Predictionsfortheperiodsnotusedforfittingthemodelwerealsoconsistentwith theobservationaldata.Wearguethatthepresentmodelforthepopulationdynamicsofthesalmon louseinaquaculturefarmsystemsmaycontributetoresolvethecomplexityofprocessesthatdrivethis host-parasiterelationship,andhencemayimprovestrategiestocontroltheparasiteinthisproduction system.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Salmonfarminghasbecomealargeandeconomicallyprosper- ousinternationalindustryoverthelastdecades.Norwayholdsa leadingpositionasaproduceroffarmedsalmonidswithanannual productionofabout1.2milliontonnes,whichisroughlyhalfof theworldwideproduction(Anonymous,2015).Furthergrowthin theproductionofsalmonidsisindemand(Anonymous,2015),but thiswillcomeatthecostofincreasingrisksofpathogenpropaga- tionandtransmission.Large-scalehostdensitydependenceacting onpathogentransmissionhasbeendemonstratedinsalmonfarm- ingproduction systems, both formacro parasites (Aldrin etal.,

∗Correspondingauthor.

E-mailaddress:[email protected](M.Aldrin).

2013;Jansen etal.,2012;Kristoffersenetal.,2014)and viruses (Aldrinet al.,2011,2010; Kristoffersenet al., 2009).Of special concern,isthepropagationandspreadofthesalmonlouse,Lep- eophtheirussalmonis,whichisimplicitlyresponsibleforregulating thesalmonfarmingindustrythroughdensitydependenthostpara- siteinteractions(Frazeretal.,2012;Jansenetal.,2012;Groneretal., 2016b)Consequently,thisparasiteplaysadominantroleinthefor- mulationofmanagementpolicies(Anonymous,2015),dominates amongsalmonpathogensinthescientificliterature(Murrayetal., 2016)andisperceivedasamajorthreattowildsalmonpopulations (Tarangeretal.,2015;Vollsetetal.,2015;Forsethetal.,2017),all testifyingtothegravityofdetrimentaleffectsofthesalmonlouse onsalmonfarming.

Mathematicalandstatisticalmodelsareincreasinglybeingused toevaluateinfectionpathwaysandriskfactorsforpathogenprop- agationanddiseasedevelopment,bothinaquaticandterrestrial http://dx.doi.org/10.1016/j.ecolmodel.2017.05.019

0304-3800/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

animalfarming(Aldrinetal.,2013,2011,2010;SalamaandMurray, 2013;MurrayandSalama,2016;Jonkersetal.,2010;Diggle,2006;

Höhle,2009;Keelingetal.,2001;Scheeletal.,2007).Whensuch modelsarecapabletoreproducethemainpatternsinthehost- pathogenpopulationdynamics,includingthespreadwithinand betweenfarms,theycanbeusedtopredictfutureinfectionlevels aswellassimulatetheoutcomesofdiseasemitigationscenarios, examplesbeinginterventionstomitigatebovinetuberculosisin GreatBritain(Brooks-Pollocketal.,2014)andlongtermeffectsof infectioncontrolmeasurementstomitigatesalmonidalphavirus (SAV)incidencescausingpancreasdisease(PD)outbreaks(Aldrin etal.,2015).TheNorwegiansalmonidproductionsystemisexcep- tionallywellsuitedfordevelopingmodelsforsalmonliceinfection dynamicsbecauseofthewealthofsurveillancetime-seriesthat documentboththespatiallocationsandpopulationsizesofhost populationsatriskofinfection,aswellassalmonliceabundances inthesehostpopulations.Couplingthesehostandparasitepop- ulation data have provided insights into e.g. how salmon lice spreadbetweenfarmsdependingonbetween-farmdistances,and howtransmissionandparasiteabundancesdependonlocalhost biomasses(Jansen etal., 2012;Aldrinetal.,2013;Kristoffersen etal.,2014).However,previousmodelsthatdescribebothbetween andwithinfarmparasitepopulationdynamicshaveforsimplicity typicallybeenautoregressivestatisticalmodelsfocusingonsin- gleaggregatedmeasuresofparasiteinfectionlevels(Jansenetal., 2012;Aldrinetal.,2013;Kristoffersenetal.,2013,2014).Alterna- tively,modelshavebeendevelopedforsimulationpurposesonly (Groneretal.,2014)orfocusedonthepopulationdynamicsonsin- glefarmsoveralimitedperiod(Krkoˇseketal.,2010).Mostofthese approacheshavereliedheavilyonestimatesofdemographicrates obtainedinthelaboratory(Stienetal.,2005;Revieetal.,2005;

Gettinbyetal.,2011;Groneretal.,2013,2016a;Rittenhouseetal., 2016).

Theaimofthepresentpaperistoformulateafullymechanis- ticstage-structuredpopulationmodelforthesalmonlouse,that alsoallowsforthecomplexitiesinherentinfullscalesalmonfarm- ing.Furthermore,themodelaccountsforthehierarchicalstructure ofthedataobtainedfromtheproduction systemwheresalmon licearecountedonsubsamplesoffish,thefishbeingaggregated intoseparatecagesandthecagesbeingaggregatedtofarm.The modelestimatesparameterscontrollingawiderangeofprocesses, includingeffectsoftemperatureondemographicrates,fishsizeand abundanceeffectsontransmissionrates,thedifferenteffectsizes, temporalandstagespecificeffectsofawiderangeofsalmonlice controlmeasures,anddistance-basedtransmissionratesbetween farms.Theobjectivesfordevelopingsuchacomplexpopulation modelforthesalmonlousewastocoverseveralneedsinmodern salmonfarming:(1)Todevelopatoolthatkeepsaccountofthe salmonlousepopulationsattheproductionunitlevelinsalmon farms,basedonthesuccessivecountingofsalmonlouseinfections, andproducemorereliableestimatesofsalmonlousepopulation sizesatanypointintimethantheindividualsalmonlousecounts.

(2)Toproduceshorttermpredictionsoffuturesalmonlouseinfec- tionlevelstoenableproactiveuseofsalmonlousemanagement actions.(3)Toevaluatetheefficiencyofdifferentcontrolmeasures.

(4)Toevaluatewhetherestimatesofdemographicratesobtainedin thelaboratoryseemsapplicableinfullscaleproductionsettings.(5) Toexploreimportanceofdifferentsourcesofinfection(e.g.inter- nalversusexternalsources).(6)Finally,todevelopasufficiently realisticmodelthatcanbeusedforscenario-simulationsexploring theeffectsofvariousparasitecontrolstrategies.Inthispaper,we describethemodelindetailanddiscusstheresultsinrelationto thefirstfiveobjectives.Scenariosimulationsfromthemodelwill bethefocusinlaterwork.

2. Materialsandmethods

2.1. Data

Farm production of salmon comprises a freshwater juvenile phase,beingfollowedbyamarinegrowoutphase,thelatterwhich isthefocusofthisstudy.TheproductionofsalmononaNorwe- gianmarinefarmisinitiatedbystockingjuvenilesmoltstocages (ornet-pens)eitherinspringorinautumn.Salmonarekeptinthe marinefarmsforabout1.5yearsafterwhichtheyareslaughtered forfoodconsumption.InNorway,onlyfishofthesameyearclass ofagearekeptonagivenfarmandwetermthisacohortthrough- outthepresentpaper.Afterslaughtering,itismandatorytofallow thefarmforaperiodofatleasttwomonthsbeforestockinganew cohortofsalmon.

Themainbodyofdatainthepresentstudyconsistofcage-level datafrom32marinesalmonfarmsinNorway,ofwhich12farmsare aggregatedjustnorthoftheislandFrøyainMid-Norway(Fig.1).For eachfarm,thedatacoversafullproductioncycleforfarmedsalmon, fromstockingassmoltstoslaughteringasadultAtlanticsalmon (Salmo salarL.),includingfishproductiondata,licecounts,tem- peraturesandlousecontrolefforts.Salmonwerestockedbetween 2011and 2013andslaughteredabout11/2yearafterstocking, between2012and2014.Thenumberofproductionunits(cages) perfarmvariedfrom3to12,butwereusuallyaround8(mean7.7).

For9farms,thefishweremovedbetweencageswithinthefarm duringtheproductionperiod.

Seawater temperatureswere measured at 3m depthat the farms.Theaveragetemperaturewas9.1^◦C,and95%ofthetem- peratureswere between3.6 and 15.0^◦C. Data onsalinity were unavailableinsufficientdetailandhavethereforenotbeenused.

Theproductiondataconsistofdailynumbersandmeanweights ofsalmonpercageduringtheproductionperiod,informationon movementofsalmonbetweencageswithinfarmsandinformation onantiparasiticlicetreatmentusingchemotherapeuticmedicals (dayof applicationandtype ofmedical).Furthermore,thedata containinformationonstockingofcleanerfish(dayandnumber ofcleanerfishstocked),butwithlimitedinformationontheirmor- tality,andhenceonthenumberofcleanerfishpresentatagiven day.Thecleanerfishareusuallyeitherlumpsuckers(Cyclopterus lumpus), ballan wrasses (Labrus bergylta) or goldsinny wrasses (Ctenolabrusrupestris)oramixofthese,butwedonotdistinguish betweenvariousspeciesofcleanerfishinthemodel.

Theproductioncycleslastedonaverage16.5monthspercage, withonaverage140000fishpercage,typicallymoreinthebegin- ningofaproductioncycleandlesstowardstheend.Theaverage minimumandmaximumfishweightsduringaproductioncycle was 140g and 5.7kg,respectively. 89% of the cages contained cleanerfishinpartsoftheproductionperiod.

Asamainrule,licecountswereperformedonasampleofat least10fisheverysecondweekforeachcage.Thesalmonlicewere dividedintothreecategoriesaccordingtodevelopmentalstages, i.e.(i)chalimus(CH),(ii)othermobiles(OM),whichconsistofpre- adultsofbothsexesandadultmales,and(iii)adultfemales(AF).

Therewereonaverage41dateswithlicecountspercage,with averages(abundance)of0.23CH,0.76OMand0.18AFperfish.Six differenttypesofantiparasiticmedicalswereused(Table1),and therewereonaverage4.6eventsofmedicaltreatmentspercage.

Themedicalsemamectin benzoate anddiflubenzuron aregiven throughthefeed,typicallyoveraperiodofaroundtwoweeks.These treatmentshavearelativelylowdailyeffect,buteffectslastover aprolongedperiod.Theothermedicalsareappliedasbathtreat- mentsoveradurationofafewhours,withalargerdailyeffect,but lastingoverashorterperiod(seeSection2.2.3).

Inadditiontothedetailedcage-leveldataonthe32farms,we havemoreaggregateddataonallotherNorwegianmarinesalmon

Fig.1. Geographicalpositionsofthe32salmonfarmsontheWestcoastofNorway(greencircles).Thehighlightedareacontainsthe12farmsnorthoftheislandFrøya.(For interpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Table1

Overviewoftypesofmedicaltreatmentsused.Codes:Y=yes,N=no,NA=missinginformation.ThetablecontentisbasedoninformationinNygaard(2010)andOttesen etal.(2012).Thedelay^delandthedurationconstant^duraredefinedinSection2.2.3.Forhydrogenperoxide,theunitforı^durisdays.

Medical Productname Delay^del(days) Durationconstant ı^dur(^◦C·daysor days)

Temperature dependency

EffectonCH EffectonPA EffectonA Effectonegg

Hydrogen peroxide (H2O2)

0 7 N N Y Y NA

Deltamethrin Alphamax 2 84 Y Y Y Y N

Cypermethrin Betamax 2 84 Y Y Y Y N

Azamethiphos Salmosan 1 42 Y N Y Y N

Emamectin benzoate

Slice 5 210 Y Y Y Y NA

Diflubenzuron Releeze 10 126 Y Y Y N N

farms.Forafarmfatdayt,weknowthenumberofsalmon,denoted byN_tf^SAL .Wealsohaveanestimate ˆA^AF_tf oftheabundanceofAFlice atthefarm,basedonweeklylicecountsonasampleoffish,and thereforealsoanestimateofthetotalnumberofAFlice,givenby Nˆ^AF

tf =Aˆ^AF

tfN^SAL

tf .Finally,wehavetheseawaydistancesbetweenall farms,andweletd_ffdenotetheseawaydistancebetweenafarmf andanotherfarmf.Thesedatawereusedtocalculateanexternal infectionpressureindex(seeSection2.2.9).

Fig.2showsthemostrelevantdataforonecageatonefarm.The upperpanelshowstimeplotsoftheseawatertemperature(onthe

lefty-axis)andtheexternalinfectionpressureindex.Inaddition, thefirststockingofsalmoninthiscageisindicatedbythevertical pinklineandthevariousmedicaltreatmentsareshownasblue verticallines.Finally,thestockingofcleanerfishisalsoshownas verticallines.Theverticalextensionoftheselinesisproportionalto thestockedcleanerfishratio(ontherighty-axis),i.e.thenumber ofstockedcleanerfishdividedbythenumberofsalmon.Thelower threepanelsshowthecountedabundanceofliceintheCH,OMand AFcategories.

Fig.2.Countedliceabundanceandotherinformationforonecageatonefarm.Upperpanel:Seawatertemperature(greenline),externalinfectionpressureindex(reddotted curve),timeofstocking(pinkverticalline),treatments(blueverticallines)andstockedcleanerfishratio(blackverticallines).Threelowerpanels:Countsofchalimi,other mobilesoradultfemalesshownasgreencirclesconnectedbystraightlines.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

2.2. Modelframework

2.2.1. Modellingbackgroundandoverview

Manyauthors have previouslypresented modelsfor salmon louse populationdynamics. Mostof these modelsuseparame- ter values obtained in a laboratory (Stien et al., 2005; Groner etal.,2013,2016a)orfromsmallscaleexperimentalunitsinthe marineenvironment(Krkoˇseketal.,2009).Whenfullscalefarm- productiondatahavebeenusedforestimation,thesehavebeen aggregatedand only afew of themodelparameters wereesti- matedfromthesedata(Revieetal.,2005;Gettinbyetal.,2011).

Thepresentestimatingapproachisfundamentallydifferent.Allthe parametersareestimatedbyfittingourmodeltotheindividual licecountscollectedthroughtheproductionperiodincommercial fishfarms.However,toensurethatthefinalparametervaluesare withinbiologicalplausibleranges,weuselaboratorydata,mostly those summarisedin Stien etal. (2005), tospecify informative priordistributionsformanyoftheparameters.Thepriorsarethen updatedtoposteriordistributionsbythefullscalefarmdatausing Bayesianmethods.

Sincethemodelisestimatedonrealdatafrommanydifferent farmsundervariousconditions,itmustsimultaneouslyincorpo-

ratemany featurestohandleactivitiesoreventsthat affectlice abundance,includingvarioustypesoftreatments,externalinfec- tionfromneighbouringfarmsandthemovementoffish(andthen alsolice)betweencagesatthesamefarm.Ourmodelistherefore morecomplexthantheaforementionedmodels.

Biologically,thelifecycleofthesalmonlouseconsistsofeight developmentalstages(Hamreetal.,2013).Theseareaggregated intothefollowingfivestagesinourmodel:(i)recruits(R,eggsand naupliilarvae),(ii)copepodids(CO,infectiveplanktoniclarvae),(iii) chalimi(CH,sessileliceonfish),(iv)pre-adults(PA,mobileliceon fish)and(v)adults(A,alsomobileliceonfish).Theadultsarefurther dividedintoadultfemales(AF)andadultmales(AM).

Themainstagestructureandthefocusontemperaturedepen- denciesindevelopmentandrecruitmentareaspectsofthemodel thatareinspiredbythemodelinStienetal.(2005).Stienetal.

(2005)suggestadelayintegro-differentialequationmodel,amodel structurecommonlyusedforparasiticgastrointestinalnematodes offarmanimals(e.g.Grenfelletal.,1987)andfreelivinginsectspop- ulations(e.g.NisbetandGurney,1983;Nelsonetal.,2013).Wehave chosenadiscretetimeversionthatusesaonedaytimestep.This couldindicatethatmatrixmodelsandassociatedmethodsandthe- orycouldbeusedtoanalysethemodel(Caswell,2001).However,

Table2

Overviewofthemodelnotation.Whenrelevant,quantitiesmaybeusedwithsub- scriptsf,t,aandc,andwithsuperscriptsR,CO,CH,PA,AForAM.

f indexforfarm

t indexfortime(day)

a indexforstage-age

c indexforcage

N^R totalnumberoflicerecruits N^CO totalnumberofcopepodids

N^CH totalnumberofchalimuslarvaeonfishinagivencage N^PA totalnumberofpre-adultliceonfishinagivencage N^AF totalnumberofadultfemaleliceonfishinagivencage N^AM totalnumberofadultmaleliceonfishinagivencage s survivalrate(proportionperday)

m mortalityrate(m=1−s)

d developmentrate(proportionperday)

r reproductionrate(numbersperdayandperAFlice) e^Ext modifyingfactorforexternalrecruitment

N^SAL numberofsalmon

W averageweightofsalmon

M^SAL_cc numberofsalmonmovedfromcagectocagec wcc proportionofsalmonmoved,M_c^SAL_c/N^SAL_c

A=N^AF/N^SAL abundanceofadultfemaleliceonfishinagivencage N^CLF numberofcleanerfish

S^CLF numberofcleanerfishstocked

Y Numberoflicecountedonasampleofnfish parametersrelatedtomortality

ı parametersrelatedtodevelopment

expectedvalues

ˇ,,, variousparameters

² variances

z autoregressiveprocesses

autoregressivecoefficients

thestrongtemperaturedependenceindevelopmentandrecruit- mentratesmakesthisdifficult.Asthetemperaturevariesthrough theyear,thetemperatureliceexperiencewhileatagivenstage needstobekeptaccountof.Accordingly,wewillusetheexpres- sion“stage-age”throughoutthemanuscripttodenotethattheage ofacohortofsalmonlicewithinastageiskepttrackofthroughout, andusedtodeterminedevelopmentratestothenextstage.

Thegeneralideaisthatforstage-agea=0,thelicehavedevel- opedintothegivenstagefromthepreviousstage,andthatfora>0, thelicecandevelopintothesubsequentstage.Wefurtherassume thatwithinaday,inthefollowingorder;

(i)licemaybecountedonasampleoffish,

(ii)licemaydieduetonaturalmortalityortreatment,

(iii)thesurvivinglicemightdeveloptothenextstage,andfinally, (iv)fish,includingtheirsessileandmobilelice,canbemovedto

anothercagewithinthefarmorberemovedfromthefarm.

Removementoffishfromthefarmmaybeduetoslaughter, fishmortalityofotherreasonsor,occasionally,movementoffish toanotherfarm.Notethatwe havedataonstockingoffish,on movementoffishbetweencageswithinthefarmandonremoval offishfromthefarm,andthereisthereforenoneedtomodelthe populationdynamicsofthefarmedsalmonidsthemselves.When fisharestockedtomarinefarmsassmolt,theyarefreeoflice,and hencetheinitiallicetransmissioniscausedbyexternalinfections.

Notuntilsomeliceatthefarmhavedevelopedintoadults,canthe internalinfectionprocessstart.Thepopulationmodelforafarm withtwocagesisillustratedinFig.3.IntheRandCOstages,the liceareassociatedwiththefarm,butnotwithanyspecificcage.

FromtheCHstageandbeyond,however,thelicehaveinfectedthe fishandarethereforeassociatedwithspecificcages.Inthefollow- ingsubsectionswedescribethevariousaspectsofthepopulation modelandhowthemodelisrelatedtolicecountdata.Table2gives anoverviewofthemainnotationweuse.

2.2.2. Populationmodel

Below,wepresentthepopulationmodelforliceatagivenfarm (butforsimplicitywithoutthefarmindexf).Themodelconsists oftwoequationsperstage.Thefirstequationhandlesliceentering agivenstageatstage-age0,typicallyfromtheprecedingmodel stage.Thesecondper-stageequationhandlesliceageingintohigher stage-ageswithoutdevelopingintoanotherstage.

The submodels for survival, development and reproduction ratesarepresentedlaterinSections2.2.3–2.2.6.

Modelfortherecruitmentstage

AttheRstage,licearenotassociatedwithaspecificcage,but ratherseenasareservoirofrecruitswiththepotentialtoinfecta fishhostatthefarminquestioninthefuture.Themodelis:

N^R_t(a=0)=e^Ext_t N_t^AFExt₋₁ r_t^Ext₋₁+

c

a

[N_(t^AF_−1)acs^AF_(t−1)acr_(t−1)ac], (1)

N^R_t(a>0)=N^R_{(t−1)(a−1)}s^R_{(t−1)(a−1)}[1−d^R_{(t−1)(a−1)}]. (2) ThefirstterminEq.(1)representsrecruitment(intostage-age0) fromneighbouringfarms,alsocalledexternalrecruitment.Here, N^AFExt_t−1 isaweightedsumofadultfemalesatneighbouringfarms attimet−1(definedinSection2.2.9).Furthermore,weassume that thesereproducewitha rate r^Ext_t−1 (see Section2.2.7).Then, N^AFExt_t−1 r^Ext_(t−1)canbeinterpretedasapreliminaryestimateofthenum- berofexternalrecruitsreachingthefarm.However,thisaccounts forseaway distancestoneighbouringfarms,butnotforthesea currentsintheareathatmaybemoreorlessfavourableforagiven farm,andwhichalsomayvaryovertime.Therefore,wehaveintro- ducedthemodifyingfactore^Ext_t ,whichisafarm-dependentand timevaryingmodifyingfactor,seeSection2.2.8foranexactdef- inition.NotethatEq.(1)onlytakesintoaccountinfectionsfrom salmonliceonfarmedfish.Onecouldincludeanextratermto accountforinfectionsfromwildsalmonandtrout,buttheseare sofewcompared tothefarmedfishthattheycanbeneglected (Johansenetal.,2011).

Thesecondtermin(1)representsrecruitment(intostage-age0) fromadultfemaleliceatthesamefarm,alsocalledinternalrecruit- ment.TheproductN_(t^AF_−1)acs^AF_(t−1)acisthenumberofadultfemales atstage-ageaincagecthatsurvivesattimet−1andtheyrepro- ducewitharater_(t−1)ac (seeSection2.2.6).Thenewrecruitsare summedoverallpossiblestage-agesoftheadultfemalesandover allcages.

Eq.(2)keepstrackofthenumberofrecruitsofstage-agea>0 that(i)survivesfromtheprevioustimepointwithsurvivalrate s^R_{(t−1)(a−1)}(seeSection2.2.3)and(ii)donotdevelopintotheinfec- tiveCOstage,where d^R_{(t−1)(a−1)} isthedevelopmentrate,i.e. the proportionofrecruitsthatdevelopintotheCOstage(seeSection 2.2.4).

Theequationsforthenextstagesusesimilarnotationforsur- vivalrates,developmentratesandnumbersoflice.

Modelforthecopepodidstage

N^CO_t(a=0)=

a

N_(t^R_−1)as^R_(t−1)ad^R_(t−1)a, (3)

N^CO_t(a>0)=N^CO_{(t−1)(a−1)}s^CO_{(t−1)(a−1)}[1−

c

d^CO(t−1)(a−1)c]. (4)

Here, thedevelopment rate d^CO(t−1)(a−1)c (see Section 2.2.5)rep- resents the infection rate, i.e. the proportion of the available copepodidsthatduringadayinfectfishincagecandthusenterthe CHstage.Thesumovercages,d^CO_{(t−1)(a−1)}=

cd^CO_{(t−1)(a−1)c}isthen thetotalinfectionrateatthefarm.Thisismodelledasindependent ofstage-age.Wenotethatwhenaninfectivecopepodid(stageCO) attachestoafishhost,ittakesapproximately24hoursbeforeit

Fig.3. Overviewofthepopulationmodelforthesalmonlouse.Liceintheorange,redandgreenstagesarecounted,whereasliceinthebluestagesarenotcounted.Lice areassociatedwithacagefromthechalimusstage,hereillustratedbyafarmwithtwocages.Thed-s,m-sandr-ssymbolisedevelopment,mortalityandrecruitment, respectively.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

moultsintotheCHstage(Krkoˇseketal.,2009).However,inthe modelweignorethisshortperiodandassumethatacopepodid enterstheCHstageimmediatelyuponattachmenttoafishhost.

Modelforthechalimusstage

N_t(a=0)c^CH =

a

N_(t−1)a^CO s_(t−1)a^CO d^CO_(t−1)ac, (5)

N_t(a>0)c^CH =

c

N(t−1)(a−1)c^CH s^CH(t−1)(a−1)c[1−d^CH_{(t−1)(a−1)}]. (6)

FromtheCHstageon,theliceareattachedtoafish,andtherefore associatedwithaspecificcage.Weassumethattheattachedlice followthefishifthefisharemovedtoanothercageorifthefishare removedfromthefarm(includingslaughteringandotherfishmor- tality).Tohandlethis,theequationsgivenherefortheCH,PAandAF stagesareextendedslightly(seeSection1.2intheSupplementary material).

Modelforthepre-adultstage

N_t(a=0)c^PA =

a

c

N_(t−1)a^CH _cs^CH_(t−1)acd^CH_(t−1)a, (7)

N_t(a>0)c^PA =

c

N_(t^PA_{−1)(a−1)c}s^PA_{(t−1)(a−1)c}[1−d^PA_{(t−1)(a−1)}]. (8)

Modelfortheadultstages

Fortheadultstage,wedistinguishinprinciplebetweenmales andfemales.However,weassumethatmalesandfemaleshavethe samesurvivalanddevelopmentratesandthereforeeachconstitute 50%oftheadults.Themainreasonforthisisthatwedonothave dataonthenumberofadultmalesonthefish,andthereforedonot havetheinformationnecessaryforseparateestimationofadult maledemographicrates.Theequationsforadultfemalesarethen N_t(a=0)c^AF =0.5

a

N_(t−1)a^PA _cs^PA_(t−1)acd^PA_(t−1)a, (9)

N_t(a>0)c^AF =

c

N_(t^AF_{−1)(a−1)c}s^AF_{(t−1)(a−1)c}, (10)

whilethenumberofadultmalesisequaltothenumberofadult females:

N_tac^AM=N_tac^AF. (11)

2.2.3. Survivalrates

Weassumethatthesurvivalratesmaybefarm-specificforsome stages, and thereforeusetheindex fwhen convenient,but we sometimesdropthesuperscriptthatindicatesthestagename.We assumethatthetotalsurvivalrateistheproductofthreeterms;

s_tfac=s^nat_tfac·s^clf_tfac·s^cht_tfac=(1−m^nat_tfac)·(1−m^clf_tfac)·(1−m^cht_tfac), (12) wheres^nat_tfac issurvivalafternaturalmortality,s^clf_tfc issurvivalafter additionalmortalitydue tocleanerfishpredation(independent ofstage-age)ands^cht_tfc issurvivalafteradditionalmortalitydueto chemotherapeutictreatment(independentofstage-age).Them-s denotethecorrespondingmortalities.Thetwolattertermsarerel- evantonlyfortheCH,PAandAstages.Allthethreemortality(and survival)termsmustliebetween0and1,buttheyhavedifferent structures.

Naturalmortality

FortheRandCO stages,we simplyassumethatthenatural mortalityisaconstantthatiscommonforbothstages,i.e.

m^Rnat_tfac =m^Rnat=^RCOnat, (13)

m^COnat_tfac =m^COnat=^RCOnat (14)

ForeachoftheCH,PAandAstages,weassumethatthenatural mortalitiesarestochasticprocessesthatcanvaryover timeand betweenfarms,butarecommonforallcageswithinafarmand independentofstage-age.Thismayaccountforfactorsthatdiffer betweenfarmsandchangeovertime,forinstancesalinity,which isnotincludedinthemodel.Furthermore,includingthesemortal- itiesasfarm-specificandtime-varyingtermsimprovesthefitof themodeltodata.Foreachfarmandlousestage,themortalityis assumedtofollowanautoregressivemodeloforder1(AR(1))on thelogit-scaleas

m^nat_tfac=m^nat_tf =exp(z^nat_tf /(1+exp(z^nat_tf )), (15) (z^nat_tf −^nat₀ )= ^nat·(z^nat_(t−1)f−^nat₀ )+ε^nat_tf , (16)

Var(ε^nat_tf )=(^nat)². (17)

Herez_tf^nat=logit(m^nat_tf )=log(m^nat_tf /(1−m^nat_tf )).Furthermore,^nat₀ is theexpectedvalueonthelogit-scale, ^natanautoregressivecoef- ficientandε^nat_tf awhitenoiseprocesswithvariance(^nat)².These

parametershaveseparatevalues foreachstage.Inaddition,the time-varyingmortalitiesm^nat_tfacarerestrictedtoliewithinspecified intervals,whichare(0.0006–0.02)forCH,(0.002–0.21)forPAand (0.0003–0.70)forA.Theselimitsaremotivatedfromthevarious studiessummarisedinStienetal.(2005),andaresimplythemost extremelimitsoftheintervalsgivenintheirTable4.

Inaddition,weassumem=1fromstage-age80daysforadults andfromstage-age60daysfortheotherstages.Thisisanapprox- imationmadetosavecomputertime.

Mortalityduetocleanerfish

WeassumethatcleanerfishfeedonliceatthePAandAstages only(Leclercqetal.,2014),andthatthecorrespondingmortality forthesetwostagesareequal.Letx^clf_tfc=N_tfc^CLF/N_tfc^SALbetheratioof thenumberofcleanerfishtothenumberofsalmonincagec,at farmfandtimet.Thisratiodependsamongothersonthemortality ofcleanerfish,whichhastobeestimated,andthemodelforthis isdescribedlaterinSection2.2.10.LicemortalityinthePAandA stagesduetocleanerfishisthengivenby

m^clf_tfac=m^clf_tfc=1−exp(−^clfx^clf_tfc), (18) wheretheparameter^clfisnon-negative,suchthatthemortality alwaysisbetween0and1.Onereasonforassumingsuchasimple modelfortheeffectofcleanerfish(asopposedtotheeffectofmed- icaltreatmentsdiscussedbelow)isthatwealsomustestimatethe cleanerfishratio,whichismultipliedbythecleanerfisheffect.

Mortalityduetochemotherapeutictreatment

We assumethatthechemotherapeutictreatments introduce extramortalityofliceinsomeorallthestagesCH,PAandA,depend- ingonthetypeoftreatment.However,forsimplicityweassume thatliceareonlyaffectedaslongastheystayinthestagetheywere atthetimeoftreatment.Iftheymanagetodeveloptothenextstage, theyareclearofthetreatmenteffect.Letthesetofsubscriptsfcbi denotetheithapplicationofachemotherapeuticoftypebincage catfarmf.Assumingthatthistreatmentwasgivenattimet⁰_fcbi,we defineanindicatorvariablex^cht_tfacbithatis1whenthetreatmentis active(aperiodafterthetreatmentisgiven)forliceatstage-agea, i.e.when

t ∈[t_fcbi⁰ +^del_b ,t⁰_fcbi+^del_b +^dur_fcbi)−1]anda≥t−t_fcbi⁰ . (19) Here^delisatimedelay(indays)fromapplicationuntilatreatment givesavisibleeffect,whichvariesbetweentreatments(Table1).

Furthermore,^duristhedurationoftheeffect,whichdependson theseawatertemperatureaccordingto

^dur=ı^dur/T_t0, (20)

whereı^durisaconstant(withunitdegreedays,i.e.^◦C·days)given inTable1andT_t0 istheseawatertemperaturewhenthemedical isapplied.Oneexceptioniswhenhydrogenperoxidewasapplied, forwhich^duristemperatureindependentandgivenby^dur=ı^dur (indays).

Themortalityduetochemotherapeutictreatmentisgivenby m^cht_tfac=1−exp(

b

−u^cht_fcbix^cht_tfacbi), (21)

whereu^cht_fcbiisaregressioncoefficientexpressingtheeffectofthe specificapplicationofthetreatment.Theseregressioncoefficients varysystematicallybetweentreatmenttypes,accountingforvary- ingefficiencyofdifferenttypesoftreatments.Inaddition,theyvary randomlybetweendifferentapplicationsofthesametreatment type,whichforinstancemaybeduetoavaryingdegreeofresis- tanceinthelicepopulations(Jansenetal.,2016).Thisishandled bythefollowingformulation

u^cht_fcbi=log(1+exp(u^cht∗_fcbi)), (22)

u^cht_fcbi^∗∼N(^cht,(^cht)²). (23) Ingeneral,theparameters^chtand^chtdifferbetweenvarioustypes oftreatment,but aresetequalfor sometreatmenttypes.These parametersareequalforthestagesforwhichatreatmenthaseffect, buttheeffectofaspecifictreatmentfcbi,representedbytheran- domcoefficientu^cht_fcbi,mayvarybetweenstages(thisisforsimplicity omittedfromthenotationabove).

Weassumethattheeffectsofdeltamethrinandcypermethrin areequal,sincethesearesimilarcompounds.Furthermore,when azamethiphosisusedincombinationwithdeltamethrinorcyper- methrin,weassumeithasthesameeffectasusingdeltamethrinor cypermethrinalone,sincethiscombinationisusedwhenreduced treatmenteffectisexpectedduetoresistancetowardsthemedicals.

2.2.4. Developmentrates

Weconsiderherethedevelopmentratefromonestagetothe next,forthestagesR,CHandPA.ThedevelopmentrateforCOcan beinterpretedasaninfectionrate,andistreatedinthenextsub- section.Inallthepopulationmodelsforlicethatwementionedin theintroduction,thedevelopmentrateis0untilsome,minimum stage-ageandafterwardspositiveandconstant.Inouropinion,the conceptofastrictandabsoluteminimumdevelopmenttimecan bequestionedinapopulationwithmillionsofindividuals,andthe assumptionofaconstantdevelopmentthereaftermaybeunrealis- tic.Wehavechosentoconsiderthedevelopmenttothenextstage asatime-to-eventprocess,andmodelitasadiscretisedversion ofaWeibulldistribution.TheWeibulldistributioniswidelyused instatisticalmodelsfortime-to-eventorsurvivalanalysis(Aalen etal.,2008).Italsogaveabetterfittoourdatathantheminimum developmenttimemodelmentionedabove(datanotshown).

When thetime to an event is continuous and Weibull dis- tributed,theeventrate(oftencalledhazardinsurvivalanalysis)is (ı^sc)^−ısı^sa^ıs−1,whereaisthestage-ageortime,ı^sisashapeparam- eterandı^scisascaleparameter(sometimes(ı^sc)^−ısistermedthe scaleparameter).Inourcase,itisconvenienttore-parameterise thisasafunctionofthemediantimetoevent,ı^m,andtheshape parameter.Theeventratethenbecomeslog(2)(ı^m)^−ısı^sa^ıs−1,since themedianintheWeibulldistributionisı^sc(log(2))^(1/ıs).

Weuseadiscretisedversionofthis,i.e.ourdevelopmentrate istheprobabilitytodeveloptothenextstagewithinaday,and itmustthereforealsoberestrictedtobeatmostone.Weassume thatthemediantimetodevelopmayvaryovertimeandbetween farms,andintroducethereforethesubscriptstfaonit.Themodel forthedevelopmentrateisthen

d_tfa=min(log(2)(ı^m_tfa)^−ısı^sa^ıs−1,1)fora=0,1,.... (24) Wefurtherassumethatthemediandevelopmenttimedepends onthetemperaturehistoryası^m_tfa=c/( ¯T_tfa)^ıp,where ¯T_tfaistheaver- agetemperaturethatliceatstage-ageaatfarmfhaveexperienced, i.e.theaveragetemperaturefromtimet−atotimet,cisaconstant andı^pisanotherconstantthatperformsapowertransformationof T¯_tfa.Togetamoreclearinterpretationoftheconstantc,weparam- eteriseitasafunctionofthemediandevelopmenttimeat10^◦C, denotedbyı^m10.Thefinalmodelforthemediandevelopmenttime thenbecomes

ı^m_tfa=(10^ıpı^m10)/( ¯T_tfa)^ı

p=ı^m10(10/T¯_tfa)^ı

p

. (25)

ThedevelopmentratedefinedbyEqs.(24)and(25)alsodependson thestageinthewaythattheparametersı^m10,ı^sandı^parestage- specific.Onemotivationforintroducingı^m10asabasicparameteris thatweuseresultsondevelopmenttimesaround10^◦Cfromother studiesaspriorinformation,toensurethatourestimatesarewithin biologicalplausibleranges.FortheRstage,whichconsistsofeggs

andnauplii,thispriorinformationisgivenseparateforeggsand nauplii.Therefore,fortheRstage,ı^Rm10isthesumofoneparameter ı^Em10foreggsandanotherquantityı^Nm10fornauplii,i.e.

ı^Rm10=ı^Em10+ı^Nm10, (26) andı^Em10isalsocontainedinthereproductionrateintroducedlater inSection2.2.6.

Intheestimation,werestrictı^stobelargerthan1,andthedevel- opmentratewillthenbe0 atstage-age0and thenincreaseby increasingstage-age.Furthermore,thelargerı^sis,themoresteep willthedevelopmentrateincreasefrom0to1aroundthemedian.

Whenı^s>2,thedifferencebetweenthemeanandthemedianwill belessthan7 %.Itshouldfurtherbenotedthattheparameter ı^m10isonlyapproximatelythemediandevelopmenttime,since weconsideratime-discreteversionoftheWeibulldistribution.

AssumingaconstanttemperatureT,Stienetal.(2005)modelled theminimumdevelopmenttimeasc₁/(T+c₂)^c3,wherec₁,c₂ and c₃areconstants.Theyfurtherassumedthatc₃=2andestimatedc₁ andc2.Weuseasimilarformulationforthemediandevelopment time,butassumec2=0andestimatec1andc3.Inpractice,these twoformulationsarequitesimilarfortherelevanttemperatures andfortheestimatedvaluesofc₃=ı^p(between0.4^◦Cand1.3^◦C, seeTable4).

2.2.5. Infectionrate

Theinfectionrateistheproportionofthecopepodidsthatinfect fishduringadayandthusdevelopintotheCHstage.Itisfarm-and cage-dependent,butdoesnotdependonstage-agea,exceptthat weassumethatdevelopmentmayonlyhappenfora≥1.Thisis modelledas

d^CO_tfc=exp(^CO_tfc)/(1+

c

exp(^CO_tfc)), (27)

where

^CO_tfc =ı^CO_0fc+log(N^SAL_tfc )+ı^CO₁ (log(W_tfc)−0.55). (28) HereN_tfc^SALandW_tfc arethenumber(inmillions)andtheaverage weight(inkg),respectively,offishincagecatfarmfandtimet, and0.55isroughlythemeanofthenaturallogarithmoftheweight offish.Withthisformulation,d^CO_tf =

cd^CO_tfc willbetheproportion ofcopepodidsthatinfectfishinanycageduringdayt,andthiswill alwaysbebetween0and1.Furthermore,whentheproportionsor ratesaresmall,therated_tfac^CO foreachcagewillapproximatelybe proportionaltothenumberoffishN_tfc^SALinthecageandtoW^ı

CO 1 tfc . Theparameterı^CO_0fccontrolsthemagnitudeoftheinfectionrate conditionedonthenumberandweightoffishwithinagivencage.

Inourmodelı^CO_0fcdependsoncageandfarm,reflectingthatsome farmsorcagesmaybemoreexposedtoinfectionthanothersdueto forinstanceseacurrentconditions.Thisishandledbythefollowing hierarchicalstructure:

ı^CO_0fc∼N(ı^CO_0f,(^COdf)²), (29) ı^CO_0f∼N(ı^CO₀ ,(^COd)²), (30) whereı^CO_0f isafarm-specificmeanandı^CO₀ anoverallmean.Fur- thermore,^COdfreflectsthevariabilitybetweencagesatthesame farm,whereas^COdreflectsthevariabilitybetweenfarms.

2.2.6. Reproductionrate

Therecruitmentmodel,Eq.(1),includestheinternalreproduc- tionrater_tac,whichismodelledtakingintoaccountthefollowing factors:Femaleadultsextrudepairsofeggstrings.Theycanextrude anewsetofeggstringswithin24houraftertheprevioussetwas

hatched,buthatchingcantakeseveraldays(Stienetal.,2005).The numberofeggsperstringmayincreaseforeachconsecutiveextru- sion,whichweapproximatewithstage-age.Finally,notalleggsare viable.Inaddition,weallowfordensitydependenceinrecruitment, duetopotentiallyreducedprobabilityofmatefindingatlowlice abundance,assuggestedbyStormoenetal.(2013),Krkoˇseketal.

(2012)andGroneretal.(2014).

Thereproductionratertac forinternal recruitmentattime t, stage-ageaandcagecisthusmodelledas

rtac=ˇ₀^r·(a+1)^ˇr1·1/(ı^Em_t +1)·(1−exp(−^r·Atc)). (31) ThefirstterminEq.(31),ˇ^r₀,representsthenumberofviableeggs forthefirstextrusion.Thenextterm,(a+1)^ˇr1 modelshowthe numberofviableeggsperextrusionincreasesbystage-age.The thirdterm,1/(ı^Em_t +1),representstherateofpairsofeggstrings producedperday,whichistheinverseofaveragetimebetween each eggextrusion, whichfurtheris approximatelythemedian hatchingtimeplusonedayfordevelopingneweggstrings.The medianhatchingtimeisgivenby

ı^Em_t =ı^Em10(10/Tt)^ıRp, (32)

whereTtistheseawatertemperatureandı^Em10andı^Rpareparam- etersdefinedinSection2.2.4.Finally,theterm(1−exp(−^r·Atc)) allowsfordensitydependentrecruitment.Here,Atc=N_tc^AF/N_tc^SALis theabundanceofadultfemalesincagecattimet.Averylargevalue of^rcorrespondstoamodelwithoutdensitydependentrecruit- ment.OftheparametersinvolvedinEq.(31),weestimateı^m10E,ı^pR and^randfixˇ^r₀andˇ^r₁to172.5and0.2,respectively(seeSection 2.5intheSupplementarymaterialforamotivationofthesevalues).

2.2.7. Reproductionrateforexternalrecruitment

Thereproductionrater^Ext_t forexternalrecruitmentinEq.(1)is similartotheinternalone,butthefemaleliceabundanceAtc in Eq.(31)isreplacedbyaweightedaverageofthecountedabun- danceatneighbouringfarms,A^AFExt_t (Section2.2.9).Furthermore, weassumethatallthesefemaleliceatneighbouringfarmsareat stage-agea=10.Theassumedstage-ageof10isratherarbitrary, buttheresultsareinsensitivetothischoice.

2.2.8. Modifyingfactorintheexternalrecruitment

The modifying factor e^Ext_t for external recruitment is farm- specific,soweincludethefarmindexfaswell.Atthelog-scale, itvariesovertimearoundafarm-specificlevelaccordingtothe followingAR(1)model:

e^Ext_tf =exp(z^Ext_tf ), (33)

(z^Ext_tf −^Ext_f )= ^Ext·(z^Ext_(t−1)f−^Ext_f )+ε^Ext_tf , (34)

(ε^Ext_tf )∼N(0,(^Extar)²), (35)

^Ext_f ∼N(^Ext,(^Ext)²). (36) Here,^Ext_f isthefarm-specificexpectedvalueonthelog-scale, ^Ext theautoregressivecoefficientand(^Extar)²theresidualvariance.

Furthermore,^Ext is theoverall expected valueand (^Ext)² the between-farmvarianceof^Ext_f .

2.2.9. DefinitionsofN_tf^AFExtandA^AFExt_tf

Theweightedsumofadultfemalesatneighbouringfarmsused inEq.(2),denotedbyN_tf^AFExtforfarmfattimet,isgivenby N^AFExt_tf =

f=/f

g(d_ff) ˆN^AF_tf, (37)

whereg(·)isafunctiondecreasingbyincreasingdistancegivenby

g(d)=exp(−0.618d^0.568). (38)

ThisdistancefunctionistakenfromAldrinetal.(2013),andisbased onadata-drivenmodelforliceabundanceestimatedfrommore thaneightyearsofdataonall1400Norwegiansalmonfarmsthat wereactiveinthedataperiod.

Wehavealsocalculatedacorrespondingweightedaverageof thecountedabundanceofadultfemalesat neighbouringfarms, A^AFExt_tf ,whichreplacesA_tcinEq.(31)whenthereproductionrate r^Ext_t forexternalrecruitmentiscomputed(Section2.2.7).Itisgiven by

A^AFExt_tf =

f=/f

g(d_ff) ˆA^AF_tf/

f=/f

g(d_ff). (39)

2.2.10. Cleanerfishmodel

LetS^clf_tc denotethenumberofcleanerfishstockedandN_tc^clfthe totalnumberofcleanerfishincagecattimet.S_tc^clf isobserved, whereasN_tc^clfisunknownandmodelledas

N^clf_tc =N_(t−1)c^clf (1−^clf)+S^clf_tc, (40)

where^clfisthedailyconstantmortalityrateofcleanerfish,com- monforallfarms.

2.2.11. Datamodelandmodelfitting

Inthis subsection,wedescribehowthepopulationmodelis relatedtothelicecountdata.LetY_tc^CGbethenumberofliceincount groupCGfoundonntccountedfishattimetandcagec,wherethe countgroupsareeitherchalimus(CH),adultfemales(AF)orother mobiles(OM,i.e,pre-adultsandadultmales).Weassumethatthese followanegativebinomialdistributionwithmean^CG_tc =E(Y_tc^CG) andaheterogeneityoraggregationparameterntc^CG,suchthatthe varianceofY_tc^CGis^CG_tc +(^CG_tc)²/(n_tc^CG).Deletingthesuperscript CGandsubscripttcforamoment,theprobabilitydistributionofY is

P(Y=y)=(y+n) y!(n)

n+

n+

. (41)

Wegetthetotallikelihoodforeach countgroupbymultiplying overallcounts,cagesandfarms.Wefurtherassumeindependence betweencountgroupsandgetthetotallikelihoodbymultiplying thecontributionfromeachcountgroup.

TheexpectednumbersofthevariousY_tc^CG’saregivenfromthe populationmodelas

E(Y_tc^CH)=ntc·p^CHcount_tc ·N_tc^CH/N^SAL_tc , (42) E(Y_tc^AF)=ntc·N_tc^AF/N_tc^SAL, (43) E(Y_tc^OM)=ntc·(N^PA_tc +N_tc^AM)/N_tc^SAL, (44) wheretheroleofthefactorp^CHcount_tc istoadjustforunder-reporting ofCHlice,sincetheyareverysmallanddifficulttocount,espe- ciallyonlargefish.Weassumethatthisfactorisfarm-specific(for instance,thestaffatsomefarmsmaybemoretrainedormotivated thanstaffatotherfarms),andweintroducefromnowontheindex fforfarm.Then,themodelforp^CHcount_tfc is

p^CHcount_tfc =exp(^CHcount_tfc )/(1+exp(^CHcount_tfc )), (45) where

^CHcount_tfc =ˇ^CHcount_0f +ˇ^CHcount₁ (W_ftc−0.1), (46) whereW_ftcasbeforeisthemeanweightoffishincagecatfarmfat timet.Theconstant0.1ischosentomakeiteasiertospecifyprior distributionsforˇ^CHcount_0f andˇ^CHcount₁ .Here,ˇ^CHcount₁ is common

forallfarms,butˇ^CHcount_0f variesbetweenfarmsaccordingtothe followinghierarchicalmodel:

ˇ^CHcount_0f ∼N(ˇ^CHcount₀ ,(^CHcount)²). (47)

Themodelwasestimatedfromthedataincluding32farms,except thelastmonths(3–11)ofdataforfiveofthefarmsthatwereused forevaluatingconditionalpredictions.WeusedaBayesianesti- mationapproach,combiningthepriordistributionsandthedata likelihoodintoajointposteriordistributionforallmodelparame- ters.Manyofthepriordistributionsusedwereinformative,based onresultsfromlaboratoryexperiments,e.g.thosereportedinStien etal.(2005).Forsomepriors,however,weusemorevagueset- tings(seeSection2intheSupplementarymaterialfordetails).The modelwasfittedtodatausingMarkovChainMonteCarlo(MCMC) simulations(Gilksetal.,1996).First,severalinitialchainswererun toidentifyaroughrangeforplausibleparametervalues.Thenfour independentchainswerestartedfromslightlydifferentstarting valueswithinthisrange.Thefirst25000iterationswereusedas burn-intoestablishconvergence,andtheposteriordistributions werecalculatedbycombining100thinnedsamplesfromthelast 14000iterationsfromeachofthechains.SeeSection4intheSup- plementaryMaterialformoredetailsontheMCMCalgorithm.

3. Resultsanddiscussion

3.1. Fittedandpredictedvalues

Themodelgeneratedexpectedvaluesthatfittedtheobserved infection levels of chalimus (CH), adult female (AF) and other mobilestages(OM)well(Table3,Fig.4,andSection3intheSupple- mentarymaterialwithresultsforsevenotherfarms).Predictions fortheperiodsnotusedforfittingthemodelwerealsoconsistent withthedatawithrespecttothetimingofpopulationgrowthof adultfemale(AF)andothermobilestages(OM)(Fig.4,andFig- ure1–5inSection3intheSupplementarymaterial),eventhough thepredictionerrorsnaturallytendtobelargerintheprediction periodsthanintheestimationperiods(Table3).Theseresultssup- portthenotionthatthereisasubstantialdeterministiccomponent inthetransmissionpathwaysandpopulationdynamicsofsalmon liceinfishfarms.Thisemphasisesapotentialforutilisingthemas- sivebodyofdatagatheredbythesalmonfarmingindustrytogain controlofsalmonlouseinfectionsinfarms,whichisaprerequisite forsustainablegrowthinNorwegiansalmonfarming(Anonymous, 2015).

Forperiodswithelevatedpredictedpopulationsizes,however, abundancesofsalmonliceweresometimesover-estimated(e.g.

AFabundanceinAugust2013,Figure3,Section3inSupplemen- tarymaterial)andsometimesunder-estimated(e.g.AFandOMin firstpartofSeptember2013,Fig.4).Theselargedeviationsinsome predictionsarelikelytoreflect(1)thattherearepredictorvariables thathavenotbeenincludedinthepresentmodel(e.g.salinity),(2) substantialuncertaintyinsomepredictorvariablesliketheabun- danceof cleanerfishinthecages, and (3)that stochasticity,in particularwithrespecttotheinfectionprocess,limitsourability tomakeprecisepredictions.Accordingly,alsothecredibleinter- valsforthepredictionswerewidewhenelevatedabundancesof infectionwerepredicted(e.g.Fig.4).Thecredibleintervalswere wellcalibratedintheestimationperiodsforallthreecountingcat- egoriesoflice.ThiswasalsothecaseforAFandOMinthefirst monthofeachpredictionperiod,inthattheactualcoveragewas closetothenominal95%(Table3).However,forpredictionsmore thanonemonthahead,theactualcoveragewasslightlytoolow, indicatingthatthecredibleintervalsforlongtermforecastswere slightlytoonarrow.