Age and bite structured models for vector borne diseases

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warwick.ac.uk/lib-publications

Original citation:

Rock, Kat S., Wood, D. A. (David A.) and Keeling, Matthew James. (2015) Age- and

bite-structured models for vector-borne diseases. Epidemics, 12 . pp. 20-29.

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ContentslistsavailableatScienceDirect

Epidemics

jo u rn al h om ep age : 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

Age-

and

bite-structured

models

for

vector-borne

diseases

K.S.

Rock

a,b,∗

_,

_D.A.

_Wood

a

_,

_M.J.

_Keeling

a,b

a_Warwick_Mathematics_Institute,_Zeeman_Building,_University_of_Warwick,_Coventry_CV4_7AL,_United_Kingdom b_WIDER_Centre,_University_of_Warwick,_Coventry_CV4_7AL,_United_Kingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received10October2014

Receivedinrevisedform23February2015 Accepted24February2015

Availableonline5March2015

Keywords:

Vector-bornedisease PDEmodel Feedingpatterns Vectorbehaviour

Structuredpopulationmodel

a

b

s

t

r

a

c

t

Thebiologyandbehaviourofbitinginsectsisavitallyimportantaspectinthespreadofvector-borne dis-eases.Thispaperaimstodetermine,throughtheuseofmathematicalmodels,whateffectincorporating vectorsenescenceandrealisticfeedingpatternshasondisease.Anovelmodelisdevelopedtoenablethe effectsofage-andbite-structuretobeexaminedindetail.ThisoriginalPDEframeworkextendsprevious age-structuredmodelsintoafurtherdimensiontogiveanewinsightintotheroleofvectorbitingandits interactionwithvectormortalityandspreadofdisease.ThroughthePDEmodel,therolesofthevector deathandbiteratesareexaminedinawaywhichisimpossibleunderthetraditionalODEformulation. Itisdemonstratedthatincorporatingmorerealisticfunctionsforvectorbitingandmortalityinamodel maygiverisetodifferentdynamicsthanthoseseenunderamoresimpleODEformulation.Thenumerical resultsindicatethattheefﬁcacyofcontrolmethodsthatincreasevectormortalitymaynotbeasgreatas predictedunderastandardhost–vectormodel,whereasothercontrolsincludingtreatmentofhumans maybemoreeffectivethanpreviouslythought.

1. Introduction

Theroleofbitinginsects isoftheutmostimportanceinthe transmissiondynamics of vector-borne diseases;without them manydiseases simplycouldnot spread.Vectorbiology suchas longevityandbitingratehaslongbeenknowntodeterminenot onlythepersistenceofsuchdiseasesbutalsotoaffectthesizeand speedofepidemicsandtheequilibriumprevalenceofendemics. Indeed,intheearlymathematicalmodelsofmalaria, Ross indi-catesthatvectordeathrateandbiterateareimportantwithboth featuringinhisthresholdtheoremformalaria(Ross,1916).

The Ross–Macdonald ordinary differential equation (ODE) model(Macdonald,1957;Ross,1911)anditsmanyvariations dom-inatetheliteratureinvector-bornediseasemodelling.However, keyassumptionsregardinginsectbehaviourandbiologyareoften disregardedoroverlooked.Takingabasicmodelofvector-borne disease,onecanuseamechanisticapproachdrivenbyobservation ofthebiologyoftransmissionandintroducemoreoftheinherent complexity.Itisimportantthatthisisintroducedinsuchawaythat thedirecteffectsofthenewelementscanbeascertained.Here,the biologyandcorrespondingbehaviourofthevectorisscrutinised.

∗ Correspondingauthorat:WarwickMathematicsInstitute,Universityof War-wick,CoventryCV47AL,UnitedKingdom.Tel.:+442476150774.

E-mailaddress:[email protected](K.S.Rock).

Itwillbeassumedthatthebasicunderlyingvector-borne dis-easemodeltakestheform:

Hosts

⎧

⎪

⎨

⎪

⎩

dSH

dt = bHNH−dHSH+HIH−HSH

dEH

dt = −dHEH−HEH+HSH

dIH

dt = −(dH+DH)IH+HEH−HIH

Vectors

⎧

⎪

⎨

⎪

⎩

dSV

dt = bVNV−VSV−dVSV

dEV

dt = VSV−VEV−dVEV

dIV

dt = VEV−dVIV

(1.1)

wherei=˛piIj/(NH+m)istheforceof infectionofspecies jon

species i(j=/ i).This termis a standard “criss-cross” transmis-siontermassociated withpurelydisassortativemixing.Itarises throughavectorbitingatarate˛,pickingasinglehostfromall otherhosts(NH)andotheranimals(m)andtheprobabilityof

trans-missionfrominfectedhost/vectortosusceptiblevector/hostbeing successful(pV/pH).OtherparameternotationisgiveninTable1.

Thissusceptible-exposed-infected(SEI)host–vectormodelhas recovery(atarateH)forhosts,butnotvectorsandadditionally

http://dx.doi.org/10.1016/j.epidem.2015.02.006

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Table1

ParametersfortheSEIRoss–Macdonaldmodel(1.1).

Parametersandvariables Description

bH Percapitahostbirthrate dH Hostdeathrate

H Forceofinfectionuponhost

H Hostrecoveryrate

H Inverseofhostlatentperiod DH Disease-inducedhostdeathrate

pH Probabilityofhostbecominginfectedfroma

singleinfectedbite

bV Percapitavectorbirthrate dV Vectordeathrate

V Forceofinfectionuponvector

V Inverseofvectorlatentperiod

˛ Averagebiterate

pV Probabilityofvectorbecominginfectedfroma

singlebiteonaninfectedhost

m Numberofotheranimalsavailableforblood feeding(assumingnofeedingpreference betweenhosts)ornumberofotheranimals scaledbythevector’srelativepreferenceof theseanimalsovertheprimaryhosts(seeRock etal.,inpressformoredetailsonvector preference).

Table2

Newparametersfortheageandbite-structuredmodel(otherparametersremain thesameasthestandardODEmodel(1.1).)

Parametersand variables

Description Note

t Chronologicaltime Timesincelastbite(TSLB)

a Ageofvector Sincebitingmaturity ˛() Percapitabiterate ˛()=ˇr() ˇ Maximumpercapitabiterate Constant

r() “Desiretobite”probabilitythat avectorwilltakeablood-meal ifitﬁndsahost

ı Kroneckerdelta ı(x)=

1 ifx=0 0 otherwise

disease-inducedmortality(DH)forhosts. Next amore complex

modelisderivedfrom(1.1),howeverthefollowingmethodology

couldbeappliedtoalmostanyODEvectormodel.

2. Methodology

2.1. Agestructure(vectorsenescence)

Theageatwhichavectorbecomesinfectiousaffectsthe num-berofsecondaryinfectionsthatcanresultfromthisoneindividual. Ifinfectionoccursnearthestartofthevector’slife,itwillinflict a highernumberof bites(Styeret al.,2007;Bailey,1982).This notionisthat onaveragethevector whichis infected ata low agewillspendlongerinfectiousthanitscounterpartwhichwas infectednearertotheendofitslife;morebitesoccur(onaverage) whilstitisinfectedandconsequentlyitspreadsdiseasemoreto thehostpopulation.Therelationshipbetweenvectorsurvivorship anditsimportanteffectsonbothvectorialcapacityandthebasic reproductiveratiowasfirstdiscussedbyMacdonaldinthe1950s (Macdonald,1956,1952,1961),howeveritwasnotuntilmuchlater thatdifferenttypeofdistributionsforvectormortalitywereused ratherthansimplyalteringthefixeddailysurvivorship.

TraditionalODEmodelssuchastheRoss–Macdonaldmodel, makeuseofthesimpleMarkovianformulationbyassumingthat the(instantaneous)deathrateisconstantregardlessofage;this leadstoexponentiallydistributedlifeexpectancies.Insomecases thismaybeareasonableand/orjustiﬁableassumption,however

morerecentworkonvectorssuchasthemosquito(Styeretal., 2007; Bellan, 2010) and tsetse (Hargrove et al., 2011)indicate thatnotmodellingrealisticdeathratesmayleadtoinaccuracies whenestimatingthetransmissionandprevalenceofvector-borne disease.Thiscertainlywarrantsfurtherinvestigationandiscited as one of themost overlooked aspectsof vector-borne disease modelling; Styeret al.(2007)and Bellan (2010)emphasise the importanceofvectorsenescenceaspartofthemodelling proce-dure.

Othershavealsoattemptedtoresolvethisneglectedinsightinto vector-bornediseasemodellingbymeansofLumped-AgeClasses

wherebythevectorpopulationispartitionedintoclassesinwhich parameters(inparticularthedeathrate)areassumedtobeconstant (HancockandGodfray,2007).Thismethodiscommonlyfoundin singlepopulationage-structuredmodels;insteadofmodelling age-ingbysomerateoflossandgainbetweenclasses,thetechnique utilisesadelaydifferentialequation(DDE)frameworkwhere indi-vidualseffectivelyspendﬁxedtimesineachstage.DDEsaregeneral morecomplextoworkwiththanODEs,particularlyduring numer-icalsimulation.

Anaturalwaytointroduceagestructurewithinthevector pop-ulationisviaapartialdifferentialequation(PDE)typemodelina similarmannertocreatinganagestructureinsinglespeciesdisease models(describedbyvariousauthorsKeelingandRohani,2008; Murray,2002;Britton,2003),wherebyamorerealisticdeathrate whichisafunctionofageischosen.

ImposingaPDE-typeagestructureontheSEIhost–vectormodel necessitates:

•Dependenceofbothchronologicaltimeandageforvectors(but notforhosts,althoughhostscouldbetreatedsimilarly):

SH(t),EH(t),IH(t),SV(a,t),EV(a,t),IV(a,t)

•Forcedbirthsforvectors(birthsmustoccuratagezero,a=0):

bVı(a)

•Agedependentdeathsforvectors:

−dV(a)

•Inclusionoftheageingprocessforvectors:

−

∂

NV

∂

a

•A new infection term within the hostpopulation (the infec-tiontermforthevectorpopulationremainsunchangedandit isassumedthatinfectiousnessdoesnotvarywithagehencethe probabilityoftransmissionisindependentofage):

H=˛pH

1 (NH+m)

∞

0

IV(u,t)du

2.2. Bitestructure(vectorfeedingbehaviour)

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Oncesatedfromfeedingthevectoris unlikelytofeedagainfor sometime,duringwhichthedesiretobitewillriseonceagainuntil thefeedingcyclerepeats.Todealwiththemechanicsofbitingthe modelmaybefurtheradapted,inacomparablewaytoaddingage structure,howeverwiththeadditionalpropertyofresetting“time

sincelastbite”(TSLB)afterthevectorhasfed.

Inadditiontotheagestructure,TSLB structuregivesfurther additionstothePDEmodel:

•Dependenceonage(a)andTSLB()aswellaschronologicaltime (t),forvectors(butnotforhosts):

SH(t),EH(t),IH(t),SV(a,,t),EV(a,,t),IV(a,,t)

•Forcedbirthsforvectors.BirthsmustoccuratagezeroandTSLB zero(a=0,=0):

bVı(a)ı()

•TSLB-dependentbitingrate:

−˛()

and so vectors are upon biting vectors are transferred into thesame agebut =0TSLBcategory (i.e.a non-infectivebite moves SV(a, ,t)toSV(a,0, t)). In generalvectorswill either

remainsusceptibleand somovetoSV(a,0,t)withprobability

1−(pVIH)/(NH+m)or becomeexposedand enter EV(a,0,t)in

asimilar mannertotheRoss–Macdonaldmodel.Vectors that arealreadyexposedorinfectiousdonotchangeinfectionstatus throughbiting,onlyTSLBcategory.

•Inclusionofthe“notbiting”processforvectors:

−

∂

NV

∂

•Aslightchangetothehostinfectionterm:

H=pH

1 (NH+m)

∞

0

u

0

˛(q)IV(u,q,t)dqdu

notingthatvectorscannothavegonelongerwithoutbitingthan theirage(≤a).Herethefunction˛()appearsalongwiththe infected vectors under theintegral as now thebiting rate is dependentnotonlyonthetotalnumberofvectorsofallages, butalsotherespectivebitingratesofalltheindividualvectors fromthosethathavejust fed(TSLBequalszero)tothosethat haveneverfed(TSLBequalsage).

2.3. Novelmodel

Anewsystem(2.1)canbederivedusingtheconceptsof age-dependenceandbiteratestructure.Fornotationalease,SH,EHand

IHwillbewrittenassuchdespitebeingfunctionsoftime(t).

Like-wiseSV,EVandIVarefunctionsofage,TSLB,andtime(a,,t)(see

Table2):

Hosts

⎧

⎪

⎨

⎪

⎩

dSH

dt =bHNH−dHSH+HIH−pH SH

(NH+m)

∞

0

u

0

˛(q)IV(u,q,t)dqdu

dEH

dt =−dHEH−HEH+pH SH

(NH+m)

∞

0

u

0

˛(q)IV(u,q,t)dqdu

dIH

dt =−(dH+DH)IH+HEH−HIH

Vectors

⎧

⎪

⎨

⎪

⎩

∂

SV

∂

t =ı()ı(a)bV

∞

0

u

0

NV(u,q,t)dqdu+ı()

1−pV₍ IH

NH+m)

a

0

˛(q)SV(a,q,t)dq−dV(a)SV−˛()SV−

∂

SV

∂

a−

∂

SV

∂

EV

∂

t =ı()

a

0

˛(q)

pV₍ IH

NH+m)SV

(a,q,t)+EV(a,q,t) dq−dV(a)EV−˛()EV−VEV−

∂

EV

∂

a −

∂

EV

∂

IV

∂

t =ı()

a

0

˛(q)IV(a,q,t)dq−dV(a)IV−˛()IV+VEV−

∂

IV

∂

a −

∂

IV

∂

(2.1)

InordertogeneratesolutionsfromthissystemofPDEsarange ofanalyticaland numericalmethodsareoutlined.Firstthe sys-temisconsideredinthedisease-freecasewhichallowsanalytical solutionstobeobtained.Withtheadditionofdiseasethesystem becomesunsolvableusingthesetechniques,howeverthe station-arydisease-free solutionsyieldplausibleinitialconditionsfrom which toinitialise numericalsimulations. Suitable methods for solvingPDEsareoutlinedanddiscussedbeforeresultscanﬁnally begeneratedandconclusionsdrawn.

ItisnotedthattheRoss–Macdonaldmodel(1.1)isthelimiting caseofthenewsystem(2.1)wherethedeathrateandbiterateare ofindependentofageandTSLBrespectively.

3. Disease-freesolutions

In the absence of disease, the underlying dynamics (births, deathsandbiting)ofthevectorpopulationdonotchange. There-forebysolvingthePDEforthevectorpopulationforIH,IV=0the

ageandbiteratestructureddistributionofthevectorpopulation canbefound.

3.1. Age-structuredPDEs

First,theagestructurealoneisconsidered:thiscanbesolved usingtheMcKendrickapproachtoagestructure(Britton,2003).

ALexisdiagram(Fig.1)isausefulwaytovisualisethevector population.Eachlinerepresentsanindividualageingwithtime. BirthsaredenotedwithcirclesandoccurattheratebVNV(t)(theper

capitabirthratemultipliedbythetotalpopulationsizeattimet). Deathsaremodelledaccordingtotherelevantdistribution,dV(a),

andareshownascrosses.Thenumberofvectorsagedawillbe denotedby

v

(a,t)orsimply

v

inthefollowing.

Thecorrespondingequationsforthisage-structuredPDEare:

∂

v

∂

a+

∂

v

∂

t =−dV(a)

v

(0,t)=bV(t)

_∞

0

v

(a,t)da

(3.1)

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0 5 10 15 0

2 4 6 8 10 12

time

age

individual 1 individual 2 individual 3 individual 4 death new birth

Fig.1. 2DLexisdiagramforage-structuredpopulations.Individualsarerepresented bylinespassingthroughtimeandageatthesamerate.Hereoneindividualsisalive attimezeroandbirthsofothers,whicharedenotedbycircles,occurattimes4,6 and8.Deathsaremarkedbycrosses;asingleindividualisaliveattime15.

ItcanbeenseenfromtheLexisdiagramthatcharacteristicsof thisPDEaregivenbya=t+cwherecisconstant.Usingthemethod of characteristicsthis system canbesolved and it is seen that

v

0(a)=Aexp(−

a

0 dV(u)du)isastationarydistribution(S.1gives

moredetails).

ToﬁndA,itisnecessarytoselectthefunction,dV(a);biologically

motivatedchoicesaredescribedandusedintheSection5.1.

3.2. Ageandbite-structuredPDEs

Returningtothemainage-andbite-structuredPDEmodelof theform:

∂

v

∂

a+

∂

v

∂

+

∂

v

∂

t =−(dV(a)+˛())

v

(0,0,t)=bV

∞

0

a

0

v

(a,,t)dda

v

(a,0,t)=

a−

0

v

(a,q,t)˛(q)dq for a=/0

(3.2)

addsone furtherdimensiontotheproblem. The previous two-dimensional Lexis diagram is now extended into this third dimension,ascanbeseeninFig.2.Hereitisdemonstratedhow individualsmovethroughtime,ageandTSLBclasseswiththesame rate,howeverwhenabiteoccurs(atatotalrateof−˛()

v

),thereisa discontinuityinthegraphunlikethatofthe2Dversion.Tohandle thisdiscontinuitymathematically,itiseasiertoformulate equa-tionsfortheresettingofthebiteclassasanewindividualentering attheboundary=0whilethebitingvectorexitsthepopulation.

Themethodofcharacteristicsmaybeutilisedagain,thistimeto yieldageneralsolutionof:

v

(a,,t)=

v

0(a−t,−t)exp

−

a

a−t

dV(u)du−

−t

˛(q)dq

(3.3)

where

v

0istheinitialdistributionofvectorsacrossagesandTSLBs

(furtherdetailsandcomputationaregiveninS.2).

Toﬁndastationarysolution,theboundaryconditionsareused. If

v

0(a,):=B(a−)exp

−

a

a−

dV(u)du−

0

˛(q)dq

(3.4)

Fig.2.3DLexisdiagramforage-andbite-structuredpopulations.Theseindividuals areidenticaltothoseinFig.1,aswouldbeseenbyatop-downview,howeverthe linesnowpassthrough3dimensions.Hereindividualstravelthroughtime,ageand TSLBwiththesamerateandbirthsanddeathsoccurasbefore.Additionally,biting eventsoccurandaredenotedbyasquare.Uponbitinganindividualmovesdirectly backontothezeroTSLBplaneleadingtothesaw-toothpatternseen.

inthedomain0≤≤a,a≥0wherethefunctionBisdeﬁnedbythe non-localboundaryconditionsofeitherbirths(inthecasea=)or bybiting(otherwise):

B(a−)=

⎧

⎪

⎨

⎪

⎩

a−

0

v

(a,q,t)˛(q)dq if a> bites

bV

∞

0

a

0

v

(a,,t)dda if a= births

(3.5)

thenthis

v

0 (givenby(3.4)and(3.5))isastationarydistribution

deﬁnedimplicitly.

Toclarify,thetotalinﬂuxofbirthsintoapopulationenterat (a,)=(0,0).Birthshereareassumedtoarisefromeachvector producingoffspringataratebV.

Onceborn(ormoreaccurately,uponreachingbiting matura-tion)thevectorwillageandmovethroughTSLBclassesuntilit eitherdiesorbites;beforeeitheroftheseeventsoccurthe indi-vidualisclassiﬁedsuchthata=.Upondyingorbiting,individuals moveoffthecharacteristiclinea=andsothenumberofvectors decaysaccordingtothefunctionalformsofthedeathandbiterates. Deceasedvectorsareremovedfromthetotalpopulation, how-ever upon biting a vector is assigned zero TSLB; this can be visualisedmathematicallyasanewindividualenteringthe pop-ulationontheboundary(a,)=(a,0).Thenewlyfedindividualsat ageaarealltheindividualswhichwerepreviouslyalsoageabut ofanyTSLB.

4. Numericalmethods

Theaboveworkenableddisease-freeanalyticsolutionsofthe vector PDEto beobtained, however introducing theadditional vector and host classes to capture disease spread necessitates numericalschemestobeutilised.Thissectionoutlinesonesuch methodbeforetheresultsaregiveninSection5.

TherearemanynumericalmethodstosolvePDEs,howeverin

thiscasethemethodoflines(MOL)isasensiblechoiceforthistype

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Table3

Additionalparametersfornumericalanalysisoftheage-andbite-structuredvector model.

Parameters Description Note

h Step-size Thisisthewidthofthe gridinbothageand TSLBdirections

A Maximum

life expectancy

T Maximum

TSLB

N1 Numberof

ageintervals

N1=A_h

M Numberof

agegrid points

M=N1+1

N2 Numberof

TSLB intervals

N2=T_h

Q Numberof

TSLBgrid points

Q=N2+1

togenerateasurface.TheMOLisacommontechniqueinother

disciplinessuchasphysicsandmaybeusedhereinsteadofother

approaches(suchaspartitioningthevectorpopulationintoageand

TSLBclasses).Itischosenhereasitisknownthattherearepreferred

directionsinthissystemcorrespondingwiththecharacteristicsof

thePDE.TheMOL,whichareusedinconjunctionwithﬁnite

differ-ences,arestandardchoicesformanynumericalanalysesofPDEs

(Ekolin,1991;Dehghan,2003).

Auniform2DgridistakenwherethemaximumageisA, maxi-mumTSLBisTandthespacingbetweenlinesineachdirection(age andTSLB)issettoh1andh2respectively.AcrossthisgridthePDEs

canbediscretisedbycomputingﬁnitedifferences.

UnderthismethodNV(a,)isthenumberofindividualvectorsof

thespeciﬁcageaandTSLB.Byinterpolatingbetweengridpoints intheMOLaquasi-smoothsurfacerepresentingthesolutionmay beprojectedabovethedomain.

Fromhereonwardsitwillbeassumedthath1=h2=h;thisallows

forfastercalculationduringsimulationduetosimplerformulation. ToﬁndthenumberofvectorsbetweenanytwoagesandTSLBsthe volumebelowthesurfacemustbecalculated.Duetointerpolation isitpossibletogenerateanreasonableapproximationforthis num-berofindividualseveniftheagesandTSLBsdonotcorrespondto gridlines.

Therearetwotypesofgridpointtoconsider:theboundary=0 wherenewbitesorbirthsoccurandallotherpointsinthedomain. Eachtypeiscomputeddifferently.Pointsnotonthe=0boundary canbecalculatedbeusingthetechniqueoffinitedifferences(see S.3);atthesepointsthereisaninfluxofageingandnon-biting vec-torsandanoutfluxofdeathsandbiting.Ontheboundary,newbites andbirthsarecalculatedfromtheintegralequations(3.5)using

thecompositetrapezoidalrule.Thissystemhasnon-localDirichlet

boundaryconditionsalong =0andso,whilstitisnecessaryto computethemateverytimestepusingthecompositetrapezoidal rule,thederivativeatu(a,0,t)doesnothavetobecomputed.

Thevectorpopulationwillberepresentedbythree(M-by-Q) matrices,SV,EVandIV(seeTable3).Itwillbeassumedthat

bit-inginstantaneouslymovesanindividualtothe=0categorybut tothesameagecategory(i.e.SV(a,)→SV(a,0)).Bitingmaylead

toachangeindiseasestatus(i.e.asusceptibleindividualbecomes exposed)howeverbitingdoesnotaffectmovementsfromexposed toinfectiousclasses.ThesehappeninstantaneouslyattherateV

whichisindependentofTSLB,asisbiologicallyrepresentative.This meansthecontinuousdisease-dynamicsofthesystemare main-tained.

Table4

Thefourcasesofdeathandbiteratesunderconsideration.

(i) (ii)

(a) Ageandbitestructure hasnoeffect.Thisis thestandard Ross–Macdonald model(1.1)

Age-structured populationonly

(b) Bite-structured populationonly

Fullage-and bite-structuredvector population

Asthestep-size,h,tendstozeroandAandTbecomelarge,the discretisedsystemofODEsconvergestoPDEsystem(2.1).Ideally valuesofh,AandTcanbefoundsuchthattheMOLapproximates projectedepidemicoutcomeswellbutthatMandQarenotsolarge thatsimulationdurationbecomesinfeasible.

Inordertocompensateforthelossofthetailofthedistribution causedbyusingmaximumage,A,andmaximumTSLB,T,scaling isusedduringsimulationsothatthevolumeundertheprojected surfacerestrictedbytheseboundsisthetotalvectorpopulation sizerequired.

5. Results

TheODEsgeneratedbytheMOLweresolvedthroughtimewith MATLAB’sode45tosimulatethedynamicsofanepidemic. Simu-lationshavebeenperformedforthefourdifferentcasesgivenin Table4tocomparetheeffectsofageand/orbitestructureupon diseasedynamicsofvector-bornedisease.

5.1. Choosingamortalityfunction

Inordertobeabletoincluderealisticlifeexpectancies,the pre-ciseformofthevectordeathrateasafunctionofage,dV(a)mustbe

chosencarefully.TheRoss–Macdonaldmodel(1.1)givesthevector deathrateas:

dV(a)=d1 (Case(i))

where d1 is constant; this gives rise to exponentially

dis-tributed life expectancies. To improve upon this original rate, age-dependent mortality is chosen such that the death rate increaseswithagetoincorporatetheconceptofsenescence.Crude suppositionmayleadtothesimplestcasethatisage-dependent, wheredV(a)=d2a,howeverdataforsenescenceinfemale,

blood-fedmosquitoes(Styeretal.,2007)suggeststhatlifeexpectancies mightbeassumedtobelogisticallydistributedandsothedeath ratemaygivenintheform:

dV(a)=d3

1

1+ed4(−a+d5) (Case(ii)) Inthiscased3correspondswiththemaximumvalueofthefunction,

d4controlsthesteepnessandd5,wherethe“switching”behaviour

occurs.Case(ii)hasalreadybeenparameterisedusing experimen-tal data available for the mosquito Aedes aegypti in laboratory conditions(Styeretal.,2007).Unfortunatelytheparameterisation foundin thestudy is deemedtobe unrepresentativefor these purposesaslaboratorybredmosquitoeshaveahighermeanlife expectancythanthosefoundinthewild,howeverStyeretal.(2007) diddevelopamodelandgeneratedavarietyofparameterisations forotherlifeexpectancies.

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0 10 20 30 40 50 60 0

0.05 0.1 0.15 0.2 0.25

age (days)

death rate d

V

(a)

case (i) case (ii) other case (ii) possibilities

Fig.3.Examplesofthetypesofpossibledeathratewhichallgiveamean14daylife expectancy.Forcase(i),d1=1/14andforcase(ii),d3=0.15,d4=0.22andd5=7.51. Someotherpossibleparametercombinationsforcase(ii)areshownby:dotted(with

d3=0.14,d4=0.14andd5=6.69)anddashed(withd3=0.22,d4=0.22andd5=10.9).

directly.Someofthedifferentshapedfunctionswhichhaveamean lifeexpectancyof14daysareshowninFig.3.

Thesemortalityfunctionsaretheinstantaneousdeathratesat agivenage,a.Amongsttheliterature(ParhamandMichael,2010; Styeretal.,2007;Bellan,2010)andinsurvivalanalysisthese func-tionsmaybealsoreferredtoasmortalityhazardsorhazardrates. Mortalitywithagemaydifferacrossspeciesand soitis impor-tanttoemphasisethatthefunctiongivenhereforthevectordeath ratemaynotbeappropriateforothervectors,subspeciesor envi-ronments.ThePDEmodeldevelopedhereisabletocopewitha genericdeathrate,assumedingeneraltobeage-dependent.

Inordertocompareandcontrastbetweendifferentmortality rateformulationsinsimulation(i.e.cases(i)and (ii)),themean survivaltimeiskeptconstantat14days(Chitnisetal.,2008).

5.2. Choosingabiteratefunction

Littleinformationisavailableforthederivationofthebiterate function,˛().Unlikevectormortality,whichcanbeestimatedin avarietyofways,studieshaveconcluded(atbest)an approxima-tionofthemeantimebetweenblood-mealsformosquitoesand conductedsomewhatinconclusivecost-beneﬁtanalysesofpossible feedingpatternsoftsetse(HargroveandWilliams,1995).

Itisconjecturedthatthebiterate,˛()maybedecomposedinto 2elements:

1.ˇ,arateparameterthatdeterminesthatmaximumrateatwhich avectormayobtainblood-mealsorencounterhosts.Thisterm assumesthatthevectorcanalwaysﬁndasuitablehostfor bit-ingoralternativelyitcouldbepresumedthattheprobabilityof ﬁndingasuitablehostisabsorbedintothisrate.

2.r(), theprobabilitythat thevectorwilltakeablood-mealif it encountersa host giventhat it last tooka blood-meal(or matured)daysago

Intheabsenceof information,thesimplestcase istotakea constantbiterate:

r()=ra (Case(a))

leadingtoexponentiallydistributedtimesbetweenfeeding. How-ever by considering thebiological imperative, a vector willbe unlikelytobiteagainimmediatelyaftertakingoneblood-mealand thedesiretobiteshouldincreaseuntilsaturation(deﬁnedhere astheprobabilityofavectorbitingbeingone,shouldthevector

0 2 4 6 8 10

0

0.2 0.4 0.6 0.8 1

TSLB (days)

biting rate,

α

(

τ

)

constant (case (a))

linear

logistic (case (b))

step function

Fig.4. Examplesofdifferentbitingfunctionsallwiththesamemeantimetobite(4 days)including(HargroveandWilliams,1995)feeding/non-feedingpattern repre-sentedbyastepfunction;thisisalimitingcaseofthelogisticcase.

ﬁndasuitablehost).Thereforeitwillbeassumedthattheremay bevarioustypesofsuitablecandidatesforthebiterateincluding logisticallydistributedtimebetweenbites:

r()= 1

1+erc(−+rd) (Case(b))

Otherfunctionshavebeenposed,suchasaﬁxedperiodof non-feedingfollowing a blood-meal(Hargroveand Williams, 1995), however,fornowtheseotherformulationswillbeputasideboth foreaseofimplementation(itishardertoformulatethePDEmodel withnon-continuousbitingfunctionssuchasthisHeavisidestep function)andasthereisnocompellingevidencetosuggestsuch functionsgivea more realistic representation ofvector feeding behaviour.SomeofthepotentialfeedingratesareshowninFig.4. Thelogisticcase(case(b))isparticularlyaptasitmayapproximate eitheralinearfunctionoraHeavisidestepfunctionbyasimple changeinparameterchoice.

Thewaitingtimesuntilabiteoccurscanbeconsideredtobe independentrandomvariablesgovernedbytheprobability func-tion,r(),andtherateparameter,ˇ,whichwillremainconsistent throughoutanychangesmadetor().r()isconstructedinsucha waythatthemeantimetobiteisthesameinbothcases(a)and(b). Theparametersusedinthesimulationhavebeenbasedon “typ-ical”valuesfromtheliteratureforahuman-mosquitopopulation withendemicmalaria(seeTable5),howeveritisnotedthat esti-matesforalmostallparametersvarygreatlyaccordingtovector species, locationand disease strain(Chitniset al.,2008 outline manyofthesevariations).Theinitialconditionsareimportant; sim-ulationswithdiseasearestartedfromanequilibriumdistribution ofvectorsacrossageandTSLBclassesinthesusceptiblepopulation. Diseaseisintroducedviainfectedindividualsinthehost popula-tiononly,sothatdiseaseentersthevectorpopulationina“natural” way(i.e.uponvectorsfeeding).Thisavoidstheproblemofneeding toknowwhereinfectionliesinthevectorpopulation.

5.3. Effectofageandbitestructure

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Table5

ParametersusedinthePDEmodelsimulation.Allparametervaluesweretakenfrommid-rangeestimatesformalariafromtheliterature(seeChitnisetal.,2008)unless speciﬁedotherwise.Cases(i),(ii),(a)and(b)aredescribedinTable4.

Parameters Description Value

NH Populationsizeofhosts,non-reservoirhosts,andvectorsrespectively 1000a

m 500a

NV 5000a

˛0 Averagevectorfeedingrate 0.25days−1

ˇ Maximumvectorfeedingrate 0.5days−1b

ra Parameterincase(a) 0.25b

rc Parameterincase(b) 6b

rd Parameterincase(b) 3b

bH,dH Percapitabirth/deathrateofhosts 0.02yr−1

bV Percapitabirthrateofvectors 0.0714days−1

dV Averagevector“natural”deathrate 0.0714days−1

d1 Parameterincase(i) 0.0714days−1c

d3 Parameterincase(ii) 0.15c

H Incubationrateofinfectioninhostsandvectorsrespectively 0.1days−1

V 0.1days−1

H Recoveryrateofhosts1and2andvectorsrespectively 0.002days−1

V 0

DH Disease-inducedmortalityrateinhosts 0yr−1

pH Probabilityoftransmissionfromvectorstohosts 0.022

pV Probabilityoftransmissionfromhosttovector 0.31

A Maximumage 60days

T MaximumTSLB 60days

h AgeandTSLBstep-size 0.167days

a_These_values_were_selected_for_this_simulation_based_on_a_{“village”}_of₁₀₀₀_people_and_a_{human/mosquito}_population_ratio_of_1:5_(as_given_for_areas_with_lower_malarial

prevalencebyChitnisetal.(2008)).

b _This_estimated_feeding_pattern_retains_a_mean_time_to_bite_of₄_days.

c _Computed_to_retain_an_average_life_expectancy_1/_d V.

infectedvectorsacrossages;andTSLBsandtheprevalenceinboth hostandvectorpopulations.

ThedistributionofthevectorpopulationacrossagesandTSLBs isdependentonthevectordeathrate,dV,andbiterate,˛;thiscan

berepresentedgraphicallyusingacontourplot.Ineachsimulation

thedistributionofthevectorpopulationchangedaccordingtothe deathandbitefunctionsused,howeverthetotalvector popula-tionsizewaskeptconstant(NV=5000).Thevectorpopulationsize

iscolour-codedonalog-scaleforthefourcasesunder examina-tioninFig.5.Undertheassumptionoflogisticallydistributedtime

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Fig.6. Endemicdiseasedistributioninthevectorpopulation.ParametervaluesaretakenfromTable5withsimulationsrunfor4000days.(a)The2Drepresentationofthe distributionofinfectedvectorsinallfourcases,zoomedintoshowonlyvectorswithTSLBlessthan20days.(b)and(c)Thesameinformationas3Dplotsoftheageand TSLB-independentcase(ia)andtheage-andTSLB-dependentcase(iib)respectively.

betweenbites(case(b)),thepopulationisshiftedgreatlytowards lowerTSLBs. Logisticallydistributed life expectancies(case (ii)) reducethetailendofthedistribution.

Fig.6demonstratestheeffectofageandbitestructureupon thedistributionof infectioninthevector population. Itis seen thatforexponentiallydistributedlifeexpectancy(case(i)),there isadistinct“tail”intotheolderages,whichisnotpresentunder thelogisticdistribution(case(ii)).Likewise,incase(b)the major-ityoftheinfectedpopulationhasalowerTSLBthaninthecase (a),theTSLB-independentcase.Agestructurenotonlysigniﬁcantly reducesthenumbersofoldervectorsbutthereislesstotalinfection incase(ii)thanincase(i).Theimpactofbitestructureonthe dis-tributionofvectorsisparticularlystriking;inthelogisticcase(case

(b)),distinctbandsofinfectionareseencorrespondingtovectors whichfedandagesaround4and8.Foroldervectorstheseeffects smearout.

The dynamicsof hostinfection mayalsodiffer substantially betweendifferentcases(seeFig.7).Introducinganage-dependent deathrateleadstolargereductionsinprevalenceinboththehost andvectorpopulations(seeTable6),whereastheeffectofa TSLB-dependantbiteratearemorecomplex;theTSLB-dependentbiting herecausesmoreinfectionwhenlifeexpectancyisexponentially distributedbutless inthelogisticcase(again inbothhostsand vectors). The results are highly stratiﬁed by age-dependence, whereas TSLB-dependent effects are noticeable but less dramatic.

Table6

ThetotalpercentprevalenceatequilibriuminbothhostandvectorsforthefourcasesofthePDEmodelusingparametersfromTable5.

(i) (ii)

Host Vector Host Vector

(a) 53.4 13.0 29.2 4.8

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0 1000 2000 3000 4000 400

500 600 700 800 900 1000

# susceptible

(ia) (ib) (iia) (iib)

0 1000 2000 3000 4000

0 100 200 300 400 500 600

# infectious

time (days) host population

Fig.7.Dynamicsofdiseaseprevalenceinthehostpopulationunderthefourcases. ParametervaluesaretakenfromTable5.

6. Conclusions

Theeffectsofageandbitestructureinvectorpopulationscan leadtonon-negligiblechangesindiseaseprevalenceinbothhost andvectorpopulations.Resultsshowitispossibletogenerate sig-niﬁcantdifferencesinhostinfectionlevelsineachofthefourcases. Thisdemonstratesthatitisquitepossibleforthedistributionsof thevectorbiteanddeathratestoplayakeyroleindisease transmis-siontothehostpopulation,ashasbeendiscussedintheliterature forvectorsenescence(Styeretal.,2007;Bellan,2010).

Itisconcludedthatagestructureplaysalargeroleindisease prevalenceinbothvectorsandhumanswithage-dependant mor-talitybeinglinkedwithlowerlevelsofinfection.Simulationsusing otherchoicesofdeathfunction(notshownhere)indicatethatthis resultholdsforarangeofplausibleage-dependentfunctions.This ﬁndingechospreviousconclusionsthateffectofvectorsenescence indiseasetransmissionisofgreatimportance.

Anotherkeyresultofthesesimulationsis thedistributionof infectionwithinthevectorpopulationduetofeedingpatterns.The prevalenceinthevectorpopulationmaybesimilarbetweencases (a)and(b),however,thedistributionofinfectioninthevectorsis stillsigniﬁcantlydifferent.Incase(b),wherethebiteratewastaken tobeTSLB-dependent,noticeablebandsinvectorinfection num-berswereproducedatlowageandTSLB.Thisresultisaremnant ofthebitingprocess;whilstthereareseveralprocesses(including latencyperioddistributionanddeaths)governingtheappearance andlocationofthebands,theyarisesthroughtheoverlappingof thebitingpoissonprocessofmanyindividuals.Forotherchoicesof bitingfunctionwiththesamemeantimebetweenbites,thiseffect isalsoobserved(resultsnotshownhere)althoughasthevariance increasesitbecomeslessapparent.

Thisshift in distribution poses questions about theefﬁcacy of mosquito controlssuchas shortening vectorlife expectancy viacontrolsusingWolbachia.Wolbachiaisamaternallyinherited bacteriumwhich can reducethe lifeexpectancy of mosquitoes includingthosewhichcarrymalariaanddengue(Iturbe-Ormaetxe etal.,2011)aswellasother diseasevectorssuchas thetsetse (Medlocketal.,2013).Itisthoughttohavegoodpotentialasaform

ofvector-bornediseasebiocontrol.UndertheODEmodelthismay eliminatesomeofthe“infectiontail”seenforolderages,however underage-dependentmortality,themajorityofvectorinfection occursinyoungerindividuals.

Thereisaninterestingrelationshipbetweenbitestructureand diseaseprevalencewhichchangesdependentuponagestructure. Forexponentialvectorlifeexpectancies,imposinglogisticfeeding patternsyieldsahigher prevalence(in bothhosts andvectors), whereasintheage-dependentcase,thesamefeedingpatternsgive lowerprevalence.Bitestructureisstronglylinkedtoagestructure. Byconsideringtheextremecasewherevectorsbiteexactlyevery 4daysandsurviveexactly7days,itisseenthattherewouldbe nopossibilityfortransmissionasvectorsmustbiteoncetoacquire infectionandasecondtimetotransmit.Attheotherendofthe spectrum,withaconstantbitingfunction(˛=0.25)andthesame 7daysurvival,itismorethanpossibletohavesustaineddisease spreadprovidedthattransmissionprobabilitiesweresufﬁciently high.

Finallythesimulationsshowthatunderthismalaria-like param-eterisation theeffectsof agestructure inthevector population overshadowthoseofbitestructureintermsofprevalence, how-evervectorfeedinghasdistinctramiﬁcationsfordistributionsof infectedvectorswhichshouldnotbedisregarded,especiallywhen modellingcontrolstrategies.

7. Discussion

ThePDEmodelintroducedprovidesanextensionofcurrentthe ODEmodelssuchas(1.1)and it emphasises theimportanceof incorporatingvectorageingwithinmodels.Whilstnotexplicitly demonstrated,reducingvectorlifeexpectancymaynoteliminate muchoftheinfectioninthevectorpopulation.Obviously,the man-ner in which thedistributionis changed willaffect how much infectionisremovedfromthevectorpopulationand,consequently, thehostpopulation.

Furtherworkusing this model couldexamine theeffects of alteringvariousotherparameterswhichcanbephysicallychanged in order to explore the efﬁcacy of different types of control. In particular controls which impactupon age-structure (mass-spraying),bite-structure(bednets)orboth(insecticide-treatedbed nets)couldbeexaminedwithanewperspectiveusingthisnovel methodology.

Thisnewmodelstructuremayalsoenablemoredetailedstudy ofvectorspeciessuchasthetsetse(Glossina)forwhich feeding is intrinsicallylinked tosurvival;tsetse must blood-fedor else theywillstarve.Herethereisaclearrelationshipwithstarvation andTSLBwhichithasnotbeenpossibletomodelmechanistically before.

Inthesimulationsperformedhere,theage-dependent death rate(caseii)wasextrapolatedfromdatapertainingto laboratory-bredmosquitoes.Itwouldbeexpectedthatwildvectorpopulations mayhaveslightlydifferentshaped-distributionsfrom laboratory-bredonesandthesameistruebetweendifferentspeciesaswell; thismayhaveanimpactontheresults.Thebitingfunctionwas constructedfromthe limitedinformation available inthe liter-ature about the feedingfrequency of vectors. Gaining a better understandingoftheshapeofthedistributionresultingfromthis biologicalprocesswouldleadtogreaterconﬁdenceintheresults generatedbythemodel.

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Whilstintroducingage-structureinthevectorpopulationvia a PDEmodel is unusualit hasbeen donebefore(Adler, 1976). Itis,however,morecommontoseeage-structuredhost popula-tionsinhost–vectormodels.Ifamodelincludesadeath-dependent mortalityfunctionwithoutasimilarTSLBbite-rateitispossible thatdiseaseprevalencemaybeunder-estimated.Theintroduction of TSLB-structure hereis believed tobe novelfor an epidemi-ological model and reveals that lower prevalences are seen in boththehostandvectorpopulationswithnobitestructurethan with.

It is computationally expensive to perform the simulations requiredtosolvethisPDEsystemandsoitisnecessarytoassess whethertheadvantagesoftheextrainformationgainedoutweigh the disadvantage of increased computation time. The obvious advantagesofthemoresimplemodelsaretransparency, mathe-maticaltractabilityandcomputationalcheapness.Integratingmore oftheknownbiologythroughthisPDEmodeldoesindicatethat simplemodelsmaynotcaptureimportantcomplexitiescausedby vectorsenescenceand bitingpatterns; inparticulartherelative efﬁcacyofcontrolmeasures.

Thereiscertainlymuchtobesaidforretainingenoughsimplicity toreallyelucidatetheeffectofeachparameteronamodelandkeep mathematicaltractability.However,ifkeyfeaturesoftheinherent biologicalsystemaremissing,itishardtoperceivewhetherthese modelsreallyperformsatisfactorilyinpredictingdiseasedynamics. Inallmathematicalmodelling,thereisabalancing actbetween exceedinglycomplex,“realistic”models,whichmaybeesotericand difﬁculttoanalyse,andsimplemodels,whichmaymisskeyfactors contributingtodiseasetransmission.

Acknowledgements

ThisworkwasconductedbyKSRfundedthroughanEPSRCPhD grantandlaterbyWAMP.MJKwassupportedbyERA-netanihwa grant(LIVEepi)withfundingfromDefra.

AppendixA. SupplementaryData

Supplementary data associated with this article can be found,intheonlineversion,athttp://dx.doi.org/10.1016/j.epidem. 2015.02.006.

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