Nine challenges for deterministic epidemic models

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Original citation:

Roberts, Mick, Andreasen, Viggo, Lloyd, Alun and Pellis, Lorenzo. (2015) Nine

challenges for deterministic epidemic models. Epidemics, 10 . pp. 49-53.

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ContentslistsavailableatScienceDirect

Epidemics

jo u r n al h om ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / e p i d e m i c s

Nine

challenges

for

deterministic

epidemic

models

Mick

Roberts

a,∗

_,

_Viggo

_Andreasen

b

_,

_Alun

_Lloyd

c,d

_,

_Lorenzo

_Pellis

e

a_Infectious_Disease_Research_Centre,_Institute_of_Natural_and_Mathematical_Sciences,_and_New_Zealand_Institute_for_Advanced_Study,_Massey_University, PrivateBag102904,NorthShoreMailCentre,1311Auckland,NewZealand

b_Department_of_Science,_Roskilde_University,₄₀₀₀_Roskilde,_Denmark

c_Department_of_Mathematics_and_{Biomathematics}_Graduate_Program,_North_Carolina_State_University,_Raleigh,_NC_27695,_USA d_Fogarty_{International}_Center,_National_Institutes_of_Health,_Bethesda,_MD_20892,_USA

e_Warwick_Infectious_Disease_Epidemiology_Research_Centre_(WIDER)_and_Warwick_Mathematics_Institute,_University_of_Warwick,_Coventry,_CV4_7AL,_UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received26February2014

Receivedinrevisedform4September2014 Accepted16September2014

Availableonline27September2014

Keywords:

Deterministicmodels Endemicequilibrium Multi-strainsystems Spatialmodels

Non-communicablediseases

a

b

s

t

r

a

c

t

Deterministicmodelshavealonghistoryofbeingappliedtothestudyofinfectiousdisease epidemiol-ogy.Wehighlightanddiscussninechallengesinthisarea.Theﬁrsttwoconcerntheendemicequilibrium anditsstability.Weindicatetheneedformodelsthatdescribemulti-straininfections,infectionswith time-varyinginfectivity,andthosewheresuperinfectionispossible.Wethenconsidertheneedfor advancesinspatialepidemicmodels,anddrawattentiontothelackofmodelsthatexplorethe rela-tionshipbetweencommunicableandnon-communicablediseases.Theﬁnaltwochallengesconcernthe usesandlimitationsofdeterministicmodelsasapproximationstostochasticsystems.

Introduction

Deterministicmodelshavealonghistoryofbeingappliedtothe

studyofinfectiousdiseaseepidemiology.Manyearlierstudieswere

conﬁnedtoestablishingcriteriaforthestabilityofthe

infection-freesteadystateandexistenceofanendemicsteadystate,perhaps

insimplecaseswithexplicitexpressionsfortheproportion

sus-ceptible,prevalenceofinfectionandherdimmunity.Studiesofthe

endemicstateinvolvedemographicprocessesthatoccurata

dif-ferent(andlonger)timescale,aswellasepidemiologicalprocesses.

Importantconcepts forstructuredpopulations suchas

vaccine-inducedage-shiftandcoregroupsarefundamentalinsightsthat

arisefromthis analysis,soeven thoughdiseasetransmissionis

inprincipleadiscretestochasticprocess,deterministicmodelling

offersafruitfulavenuetostudyproblemsofendemicity.Thisgives

risetoourﬁrsttwochallenges.

Thetransmission dynamicsof genetically varying pathogens

have received considerable interest in recent years, driven by

advances in molecular biology, theimpact of multivalent

vac-cinesandtheemergenceofdrug-resistance.Importantchallenges

remainwithregardtothemulti-strainmodelsthatarise.Theseare

addressedasChallenge3.Indevelopingmulti-scalemodelsthat

∗Correspondingauthor.Tel.:+6494140800.

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

linkwithin-andbetween-hostdynamics,forexampletostudythe

long-termevolutionofpathogens,oneoftenfacestheproblemof

howtomodelsuperinfection.TheseareaddressedasChallenges4

and5.

Spatially explicit models are usually treated in a stochastic

framework,althoughthiswasnotalwaysthecase(Andersonand

May,1991;Diekmannetal.,2013).Diffusion modelshave been

proposedthat giverisetotravellingepidemicwaves througha

homogeneouspopulation.However,in realitycontactsbetween

individualsaredifferentduetoavarietyoffactors,andnotjust

spa-tiallydetermined,henceaheterogeneousdescriptionisrequired.

TakingaccountofthisisChallenge6.

Itiswell-knownthatnon-communicablediseases(NCDs)such

asasthma,somecancersandcardio-vasculardiseaseshaverisk

fac-torsincommonwithinfectiousdiseases:thepredominantonesare

lowsocio-economicstatus,poornutritionandpoorhousing.While

changesinthesefactorscouldleadtochangesininfectious

dis-easesandNCDs,therehasbeenrelativelylittleinvestigationofthe

interactionbetweenthem.ThispresentsChallenge7.

Deterministicmodelsaregenerallyregardedassimplerto

han-dlethanstochasticmodels.Hence,theyareoftentheﬁrsttooltried

whenanewproblempresentsitself(Diekmannetal.,2013).Their

limitationsarefrequentlyalludedto,butoftenignored.Challenge

8istodeﬁnetheselimitations.Manyinfectiousdiseasesystemsare

fundamentallyindividual-basedstochasticprocesses,andaremore

naturallydescribedbystochasticmodels.Analysisofanequivalent

http://dx.doi.org/10.1016/j.epidem.2014.09.006

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50 M.Robertsetal./Epidemics10(2015)49–53

(insomesense)deterministicmodelmaythenyieldinformation

aboutthesolutionofthestochasticsystem.Ourﬁnalchallengeisto

understandtherelationshipbetweenso-calledequivalent

stochas-ticanddeterministicrepresentationsofthesamesystem.

1.Understandingtheendemicequilibrium

Theendemicequilibriumarisesasabalancebetween

transmis-sionofinfectionandreplenishmentofthesusceptiblepool,either

throughloss ofimmunity or demographic turnover.The

math-ematicalstarting point for thecharacterization of the endemic

equilibrium is typically a renewal equation which, under

suit-ableassumptionsaboutseparabilityofthemixingfunction,may

beexpressedinterms ofa scalarequationintheforceof

infec-tion.Introducingthe effectivereproductionnumber _Reff asthe

number of new infections that a typical infectious individual

produces,onemayinterprettheendemicequilibriumasthe

con-ditionReff=1(Diekmannetal.,2013).Itwouldbeinterestingto

determinetheunderlyingstructureofthisequation,inparticular

forthenon-separablecase,whichwouldaddresstheimplications

ofempiricallyobservedmixingpatternssuchasthosereportedin

Mossongetal.(2008).

Forendemicinfections,parameterestimationisnaturallybased

oninformationabouttheendemicequilibrium.Insimplesettings

onemaydetermineR0,forexample,fromtheobservedaverage

ageatinfectionorfromthefractionofthepopulationthatremains

susceptible.ThustheestimationofR0isindirectinthesensethat

itreliesheavilyonourunderstandingoftheendemicequilibrium.

ItisstrikingthatwhereR0isestimatedinthisway(e.g.for

child-hooddiseases,seeTable4.1inAndersonandMay,1991),thevalues

obtainedaretypicallyhigherthanthosefordiseaseswhereR0is

determinedfromdirectlystudyingtheonsetoftheepidemic(e.g.

inﬂuenza,SARSorHIV).

Whilehost renewalthrough demographic processesis fairly

well understood, the renewal process associated with waning

immunityis considerablyless clearalthoughthere are

applica-tionstoimportantdiseasessuchaspertussis(Rohanietal.,2010)

andmalaria(Bailey,1982).Thereisaneedtostudythisprocess,

bothintermsoftheunderlyingbiologyandintermsofitsdynamic

consequences(Bredaetal.,2012).

Itisknownthatregularperiodicepidemicsofchildhood

dis-easesdependontheseasonalityofthetransmissioncoefﬁcientin

combinationwiththepopulationbirthprocess.Oscillationsaround

theendemicequilibriumareobservedforawiderangeof

infec-tiousdiseases(GrasslyandFraser,2006),butseveralaspectsofthis

processarenotwellunderstood.Itisclearthatthevariation in

transmissibility(forexample duetoschoolholidays)affectsthe

qualitativepatternofepidemics,inawaythatcould(atleastin

principle)bestudiedbyFloquettheory.However,wedonothave

acomprehensivetheoryfortheinteraction,oranunderstanding

ofwhetherstochasticvariationinthetroughsbetweenepidemics

maybeneglected(BillingsandSchwartz,2002,seealsoChallenge

9),orknowledgeofhowthesepatternsrelatetoso-calledskipping

dynamics(Stoneetal.,2007).Achallengeistoexaminetherenewal

equation,anddevelopadeeperunderstandingoftherelationship

betweenReffandR0thatmightclarifytheseissues.

2.Deﬁningthestabilityoftheendemicequilibrium

Althoughitisusuallystraightforwardtodeterminethesmall

amplitudelinearperturbationsabouttheequilibriumandderive

theassociatedcharacteristicequation,thisequationistypicallytoo

complextoprovidegeneralstabilityresults.Forexample,itremains

anopenquestionunderwhichconditionstheinternalequilibrium

oftheage-structuredSIR modelwithdemographic turn-over is

stable,andstudieshaveshownthatstabilityaswellas

instabil-ity(throughaHopfbifurcation)mayoccurforspeciﬁcconditions

(Andreasen, 1993).Singular perturbations utilisingthemultiple

timescalesthatareinherentinendemicmodelsmayofferan

alter-nativeapproach.ConsideranSIRmodelwheretimeismeasured

inunitsofhostlife-span,andthesizesofeachepidemic

compart-mentaremeasuredasfractionsofthetotalpopulationsize.The

underlyingdynamicsfollow

˙

S =B−S−ˇ

IS

˙

I = ˇ

IS−1

I−I

˙

R = 1

I−R

where

1denotestheratiooftheinfectiousperiodtothehost

life-span,the transmissioncoefﬁcientˇ=R0(1+

)∼1 andthe

birthrateB∼1.Onewouldthenlookforafasttimescaleofthe

epidemic,whereI_∼1,andaslowtimescaleofthedemographics,

whereI∼

.Forthedisease-freestatethisispossible(seefor

exam-pleOwuoretal.,2013),butitisnotclearhowasimilarseparation

wouldworkfortheendemicstate.Inparticular,thelinear

anal-ysissuggeststhattheremayinadditionbeanintermediatetime

scaleoforder√

.Tobeuseful,onewouldthenhavetoextendthe

analysistostructuredpopulations.Findingageneralparadigmfor

thestabilityoftheendemicequilibriumremainsa challengefor

theoreticians.

3.Modellingmulti-strainsystems

Thenatureofdiversityisaspoorlyunderstoodinepidemiology

asinmanyotherbranchesofpopulationbiology.Wehaveonlytwo

generalmodelsavailable:thequasi-speciesmodel of

mutation-selectionbalanceand thecompetitiveexclusion principle.Most

modelsofstraindynamicsmaybeseenasspecialcasesofthesetwo

basicmodels,withthecomplicationthatcompetitioncaneitherbe

directlybetweenstrainswithinthehost(asmaybethecasefor

bac-terialcolonization),orindirectcompetitionforasharedresource

(asmaybethecaseforimmunizingpathogens,theresourcebeing

susceptiblehosts).Super-infectionandcross-immunityarespecial

casesofthesetwomodesofinteractionthathavereceivedsome

attention(seeChallenges5and6),butweneedtounderstand

bet-terthenatureofthenichesthatariseduetothedynamicalaspects

oftransmission.Examplesarethepathogenstrainswithsuperior

survivalduringtroughsoflowdiseaseactivity(Gogetal.,2003),and

themechanismsbywhichlongtermhostimmunitymayinteract

withstraindynamics(KucharskiandGog,2012).Asthenumberof

co-existingspeciesislimitedbythenumberofsharedresources,

thesemodelswillingeneralonlyallowforarestricteddiversity.

Thechallengeistoextendepidemicmodelsofstraindynamicsto

allowforgreaterdiversity,assuggestedbyLipsitchetal.(2009)for

thecaseofbacterialcolonization.

4.Modellingtime-varyinginfectivity

Themajorityofdeterministicmodels,andespeciallythoseused

for applications in veterinary and public health, are

compart-mentalmodels.Theseinvolveconstant transitionrates between

compartments,and hence sojourntimes that are exponentially

distributed(orErlangdistributedinthecaseofmultipleidentical

sequentialcompartments).Theadvantageofthesemodelsisthat

onecanuseordinarydifferentialequationsand,without

special-istknowledge,canbeneﬁtfromthetheoryofdynamicalsystems

andwell-developedandreadilyavailablenumericalmethods.The

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generalityandremovesthepossibilityofembeddingmorerealistic

infectivityproﬁles.

Modelsinwhichindividualsareassumedtohaveatime-varying

infectivityproﬁle,oftenreferredtoastime-since-infectionmodels,

representavalidalternative.Thisformalismprovidesa

compre-hensivetheoryforinvasion, andusefultools forcalculatingthe

ﬁnalsizeofSIRepidemicsandtheequilibriaofSISmodels(orSIR

modelswithdemography,etc.).However,ageneraltheoryofsuch

modelsisstill lacking,and trackingtheirdynamics numerically

isnon-trivial. Many challengesliehere.First, developing better

methodsfor handlingintegral equationsin inﬁnite-dimensional

spaces,boththeoreticallyandnumerically.Second,developingthe

frameworktoincludewaningimmunity,forexampleby

provid-inggeneraltoolsforsolving therenewalequationfor theforce

ofinfectionattheendemicequilibrium,whichiscomplicatedby

thepresenceofre-infection. Third,extendingsucha framework

tomulti-strainsystems,withalltheattendantchallengesalready

highlighted(Challenge3),inparticularkeepingtrackofhow

indi-viduals’pastinfectionhistoriescombinetoshape theimmunity

proﬁleinthepopulation,whichinturnregulatesnewinfections

and contributes to updating theinfection histories themselves.

Finally, a majorchallengewould beto extendtheentire

time-since-infectionframeworktostructuredpopulationsofincreasing

complexity,forexampleonnetworks(seeChallenge6).

5.Modellingsuperinfection

Superinfectionoccurswhen,followinganinfectionthathasnot

yetbeencleared(andmayormaynothavetriggeredanimmune

response),thehostisinfectedbyaheterologousstrainofthesame

pathogen(Smithetal.,2005).Mostmicroparasitemodelsignore

thepossibilityofahostbeinginfectedasecondtimebefore

recov-ering(Diekmannetal.,2013).Suchanassumptioniskeybecause

itallowstheinfectiontobetreatedasaprocessthatevolves

inde-pendentlywithinthehost.Amodelingframeworkbasedonthe

conceptoftime-since-infectioncanthereforebeformulated,where

aclockattachedtoeach infectedhoststartstickingatthetime

ofinfection,andanythingthatfollowsinthathostdependsonly

onthis relative time.Thisapproach leadstothestandard

deﬁ-nition of R0 and the well-developedtheory of next-generation

operators(Diekmannetal.,2013).Relaxingtheassumptionofno

superinfectionis non-trivial, astheentire modeling framework

collapses.However,superinfectionisknowntooccurandmightbe

importantforinfectionssuchasHIVorTB.Someattemptsat

mod-ellingsuperinfectionhavebeenreported,althoughalmostalways

intheMarkoviancase ofconstant infectionand recoveryrates.

Theseassumptionsleadtoexponentiallydistributeddurationsin

eachcompartment,aconditiontoounrealisticforinfectionslike

HIVwithacomplexinfectivityproﬁle.Thefew exceptions(e.g.,

MartchevaandThieme,2003)assumeanad-hocimpactof

super-infection,whichseemsdifﬁculttogeneralise.Theﬁrststepwill

betheconstructionof R0 (oranextensionof theconcept),and

thesecondthedevelopmentofthenext-generationoperator

for-malism.Thechallengeistodevelopageneraltheoryformodelling

superinfection(seealsoGogetal.,inthisissue).

6.Constructingrealisticspatiallyexplicitmodels

Networkmodelsdeﬁnecontactsinasocialspace.Networkshave

anumberofmeasurableproperties,forexampledegree

distribu-tion,transitivityandclusteringcoefﬁcient(seePellisetal.,inthis

issue).Thesepropertiesarenotsufﬁcienttodeterminehowgood

anapproximationonenetworkmaybetoanother,andasyetno

metricthatfulﬁllsthisfunctionhasbeendeﬁned.Itshouldbe

possi-bletodeﬁneacorrespondencebetweenlargenetworkmodelsand

modelsoncontinuousspaceinsuchawaythatsomeanalytical

resultscan bederived fromthe continuousrepresentation. We

couldthen havea correspondence betweennetworkproperties

andthresholdquantitiesforinvasionorpersistenceinthespatial

model,andameasurethatdetermineshowclosetwonetworks

appeartobewithregardtotheﬁnalsizeofanepidemic.

Aspatiallyexplicitmodelhighlightstheuseoftheword

typi-calinthedeﬁnitionofR0.Forexample,ifmodellinganinfectious

diseasepreviouslynotpresentinageographicregion,thenthe

pri-marycaseismore likelytooccurataportofentry.Indeﬁning

athresholdquantityforinvasion,theconnectivityoftheprimary

caseofteninﬂuenceswhetheranepidemictakesoff,andforsmall

networksthechoiceofprimarycaseclearlyinﬂuencestheﬁnalsize

oftheoutbreak.Henceamethodologyisrequiredthatassigns

dif-ferentthresholdsforinvasiontodifferentnodesofanetwork,orto

differentlocationsforspatiallycontinuousmodels.

Analternativespatialmodellingparadigmisbasedon

metapop-ulation structures, witha gravity model deﬁning contact rates

betweenthenodesrepresentingcommunities(seeRileyetal.,in

thisissue).Thecontactratebetweennodesiandjisproportionalto

(forexample)ninj/r_ij2,whereniisthepopulationdensityatnodei

andrijisthedistancebetweenthenodes.Morecomplexvariations

havebeenproposed, butwheneverthecontactrateisastrictly

increasingfunctionofniitispossibleforittoattainunrealistically

highvalues.Smallvaluesofrijhaveasimilareffect,sosaturating

functional formsarerequired. Aradiationmodel hasbeen

pro-posedfortravelbetweencitiesinanattempttoaddresstheseissues

(Siminietal.,2012).Itisnotclearhowthiscouldbeusedtomodel

epidemics,andﬁndingatractableandrealisticdeterministicmodel

withanexplicitspatialstructureremainsachallenge.

7.Exploringtheinteractionwithnon-communicable diseases

Cancersare usuallyregarded as non-communicablediseases

(NCDs),thenotableexceptionsbeingcaninetransmissible

vene-realtumourandTasmaniandevilfacialtumourdisease(McCallum,

2008).However,someinfectiousdiseasesareknowntoberisk

fac-torsforNCDs.Forexample,Jaagsiektesheepretroviruscauseslung

tumoursinsheepandgoats(Woottonetal.,2005),andthelink

betweenhumanpapillomavirusinfectionandcervicalcancerhas

beenestablished(Walboomersetal.,1999)andinvestigatedina

deterministicframework(Baussanoetal.,2010).Theinteraction

betweena transmissibleagent and an NCDmay bemore

sub-tle.Antibioticusehasbeenassociatedwithbreastcancer(Velicer

et al., 2004), and theexposureto microbesduring early

child-hoodhasbeenassociatedwithincreasedriskofconditionssuch

asinﬂammatoryboweldiseaseandasthma(Olszaketal.,2012).

Increasingevidenceindicatesakeyroleforthebacterialmicrobiota

incarcinogenesis(SchwabeandJobin,2013).Therearecomplex

interactionsbetweenahostanditsmicrobiota,andanalteration

tothedynamicsofthisinteractionthroughinfectionmaypromote

avarietyofdiseases,manyofwhichmayusuallyberegardedas

non-communicable.Inordertoevaluatethepotentialroleof

trans-missibleagentsinthedevelopmentoftheseconditions,newtypes

ofmathematicalmodelofthehost–pathogeninteractionwillbe

required.Thesemodelswillneedtosuggesttestablehypotheses

thatconnectthepathogen,thehost’simmuneresponse,andthe

variousregulatorypathwaysofthehostthatmayplayapart.

8.Deﬁningthelimitationsofdeterministicmodels

Allepidemicmodelsareinherentlystochasticatthe

individ-uallevel. However,Kurtz (1970, 1971) provedthat, for afairly

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52 M.Robertsetal./Epidemics10(2015)49–53

ofthestochasticsystemsatisﬁesasuitablydeﬁneddeterministic

model.Thisresultisnotlimitedtohomogeneouslymixing

mod-els:forexamplehouseholdmodelsfeaturelocalmixinginsmall

groups,andKurtz’sresultcanbeusedtounderpintheapproach

ofHouseandKeeling(2008)for theiranalysis.Analogously, for

SIRepidemicmodelsontree-like(unclustered)networks,the

vari-ousdeterministicmodelsproposedintheliterature(BallandNeal,

2008;Lindquistetal.,2011;Volz,2008),togetherwiththestandard

pairwiseapproximation,havebeenshowntobeexactintheinﬁnite

populationlimit.However,allofthesemodelsarerestrictedbythe

assumptionofconstantrecoveryrates.Oneimportantchallengeis

toextendKurtz’sresultstoothermorecomplicatedsituationsof

practicalinterest,forexamplethetime-since-infectionframework

describedinChallenge5.

Arelatedproblemisthewidespreaduseofdeterministic

mod-elsinepidemiologywithoutfullrecognitionoftheirlimitations,

andwithpotentiallymisleadingconclusions.Forexample,

multi-annualpredictionsmightinvolvelownumbersofinfectedhostsin

thebetween-epidemictroughs,andstochasticeffectsmaynotbe

negligible(seeBrittonetal,inthisissue).Analogously,

determinis-ticmetapopulationmodelshavealongsuccessfulhistory(Hanski

andGaggiotti,2004), butwhenthecouplingbetween

subpopu-lationsisweaktheystruggletorepresentthesystemdynamics.

Theyareunabletocapturetypicallystochasticphenomena,like

fade-out,extinction,andlackofsynchronyduetorandomdelays

(Rocketal.,2014).Whenmetapopulationmodelsexhibitamore

complexstructure,deterministicanalysismaydescribean

attrac-torasstable,butstochasticﬂuctuationsmayinterferewithmodel

componentsresultingincomplexorbits.Asdeterministicmodels

cannotcapturethesephenomena,weneedtounderstandwhen

theyarelikelytooccur.

9.Developingrobustdeterministicapproximationsof stochasticmodels

Theimportanceof stochasticeffects can bequantiﬁedusing

deterministic moment equations. The nonlinearity in the

pro-cessleadstoacouplingbetweenmomentequationsofdifferent

orders.For instancethe nonlinear transmission term of an SIR

modelleadstoequationsforexpectationsthatincludesecondorder

moments:E(SI)=E(S)E(I)+cov(S,I).Thelowerorderapproximation

inwhich allvariancesand covariancesareassumed tobezero,

i.e.stochasticeffectsareignored,leadstothesimplest

determin-isticapproximation–theso-called meanﬁeld model.Whenthe

populationunderconsiderationislargebutﬁnite,thesolutionof

themeanﬁeldmodel maynotbea goodapproximationtothe

meanbehaviourofthecorrespondingstochasticmodel.Toobtain

a higher order approximation, an alternative assumption must

beemployed.Themultivariatenormal(MVN)closure

approxima-tion(Isham,1991)isoftenusedwhenmodellingmicroparasites.

NumericalresultsshowthattheMVNapproximationoftenfails

insituationsofinterest,e.g.forrecurrentepidemicsnearthe

crit-icalcommunitysize,whereextinction isnotuncommon(Lloyd,

2004).TheMVNandmultivariatenegativebinomial

approxima-tionshavebeenusedtomodelmacroparasites(HerbertandIsham,

2000).

InsituationswhereresultssimilartothoseofKurtz’s(see

Chal-lenge8)arenotavailable,orwhenintuitiondoesnotsuggest a

naturalchoiceforasuitabledeterministicmodel,other

approx-imation schemes must be adopted. Network moment-closure

approximations(seee.g.HouseandKeeling,2008;Keeling,2000;

Pellisetal.,inthisissue) areessentiallytheonlyviablemethod

toavoid computationallyexpensivefully stochasticsimulations

forSIRmodelsonclusterednetworksorSISmodels(onany

net-work).

Conclusion

Wehavediscussedaselectionofninechallengesinapplying

deterministicmodelstoepidemiology.Clearly,ourselectionisnot

exhaustiveand numerous otherimportant challenges involving

deterministicmodelsarediscussedelsewhereinthisspecialissue.

Furthermore,manychallengesdescribedheredonotexclusively

relatetodeterministicmodels.Inselectingourchallenges,wehave

focussedonthosewherepreliminaryapproacheshaveemployed

deterministicmodels,andthosewhereitislikelythatdeterministic

modelswillmakeasigniﬁcantcontribution.

Acknowledgments

Thispaperwasconceivedwhiletheauthorstookpartinthe

IsaacNewtonInstituteforMathematicalSciencesprogrammeon

InfectiousDiseaseDynamics.Theauthorsaregratefultothe

Insti-tute, and the programme organisers, for their support.MGR is

supportedbytheMarsdenFundundercontractMAU1106.ALLis

supportedbytheResearchandPolicyforInfectiousDisease

Dynam-ics(RAPIDD)program of theScienceandTechnologyDirectory,

DepartmentofHomelandSecurity,andFogartyInternational

Cen-ter,NationalInstitutesofHealth,andbygrantsfromtheNational

InstitutesofHealth(R01-AI091980)andtheNationalScience

Foun-dation(RTG/DMS-1246991).LP issupportedbytheEngineering

andPhysicalSciencesResearchCouncil.

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