Modeling Human Papillomavirus transmission. Impact of a quadrivalent vaccine.

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Modeling Human Papillomavirus transmission. Impact

of a quadrivalent vaccine.

Laureen Ribassin-Majed, Rachid Lounès, Stéphan Clémençon

To cite this version:

Laureen Ribassin-Majed, Rachid Lounès, Stéphan Clémençon. Modeling Human Papillomavirus

trans-mission. Impact of a quadrivalent vaccine.. 2010. �hal-00555716v3�

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quadrivalent vaccine. Ribassin-MajedL.

1

,LounesR.

1

,Clémençon S.

2

.

1 Mathématiques AppliquéesParis 5(MAP5)CNRS UMR 8145, Université Paris Descartes, Paris, France

2 LTCI Telecom ParisTech, CNRS 5141, Paris, France.

∗

E-mail: laureen.majed@parisdescartes.fr

Abstract

HumanPapillomavirusisthemostfrequentsexuallytransmittedinfection. HumanPapillomavirus(HPV) istheprimarycauseofcervicalcanceranditsprecursorlesions. TwoprophylacticvaccinesagainstHPV infections areavailable. Mathematicalmodelscanbeused to compareseveralvaccine strategies. Con-sequently, mosteectivevaccinestrategycanbeenlightenedandselected. Nevertheless,proposedHPV transmissionmodelsinthelitteraturehavebecomeverycomplexwhilesomeinputvaluesremainunknown orbadlyestimated. Ouraimwasto assessthevariabilityin theoutcomevariablethat isduetothe un-certaintyinestimatingtheinputvalues. WecarriedoutandcalibratedaSusceptible-Infected-Susceptible modelofheterosexualtransmissionofHumanPapillomavirusinfectionsforserotypes6/11/16/18which arecoveredbythequadrivalentvaccine. Immunityobtainedfromvaccinationwasconsidered. Thebasic and vaccinatedbasicreproduction numberswereexpressed. Model predictionsensitivityto parameters uncertainty hasbeenassessed using thePartial RankCorrelation Coecients. Three scenarios of vac-cination havebeencompared consideringestimatedHPV infectionprevalences. Sixposteriorparameter sets among onemillion combinationtested best tted epidemiologic data. Sensitivity analysis showed thatthesigniciancelevelofuncertaintywaslinkedtothelengthofdierentserotypeHPV infectionsin modelpredictions. Deterministic modelingofHPV infectiontransmissionallowedusto compare poten-tialeciencyof3vaccinationscenarios. Additionalvaccinationofthehalfofmenwhoenterannuallyin the sexuallyactivepopulationled to thesameresultswhencompared to anexclusivelargevaccination rate of women (who enter annuallyin thesexuallyactive population). Sensitivityanalysis showed the importance of clearance ratein the precisionof model predictions,therefore eortshaveto been made to focusdata collectionconcerningdurationofHPV infections. Furthermore,usefulnessofmen's vacci-nationdependsonwomen'svaccinationrate.

Keywords: HumanPapillomavirus,dynamicmodel,sensitivityanalysis,vaccine.

Introduction

HumanPapillomavirus (HPV)is themost frequentsexuallytransmitted infection. At least70percent of sexuallyactivemen and women acquireHPV infection at somepointsin theirlives[29]. Eightyper cent ofHPV infectioncasesare clearedin afewmonthsfrom thebodybythe immune systemwithout treatment, the rest 20%infection become persistent. One hundreddierent HPV serotypeshavebeen identied, therearelowriskserotypeswhichareresponsibleforbenign anogenitallesions,and highrisk serotypes which can induce precancerous and cancerouslesionsin the cervix. Serotype16 is the most common indevelopedcountries[4, 25]. Epidemiological studiesonHPV infections establishtherole of thesevirusesastheprimarycauseofcervicalcancer[22]. Theseinfectionsarealsothecauseofanogenital cancers,headandneckcancers,anogenitalwartsandrecurrentrespiratorypapillomatosisamongwomen andmen. Invasivecervicalcanceristhemostcommoncanceramongwomenworlwide[26]. Itisestimated thatHPV infectionsareresponsibleforapproximately500,000cervicalcancercasesworldwideeachyear [24]. VaccinationagainstHPVinfectionsrepresentsaneectivewaytodecreasecervicalcancerincidence,

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behighlyecientin"naive"women[8].

HPVtransmissionmodelshavebecomeverycomplex. Severaldeterministicmodelshavebeendeveloped to assessthe potential impactof vaccination against HPV; Hugheset al [15] developped aSIR model of heterosexualtransmissionwhich included3sexual activitygroups,theirobjectivewasto explorethe eect of a mono-valent high-risk HPV vaccine on the steady-state endemic prevalence of HPV 16 in the population; Barnabaset al[2]explored the eect ofamultivalentHPV vaccineusing aSIRmodel whichincludedsexualbehaviour,smokingandage;Elbashaetal[11]simulatedtheprogressionofHPV disease in the population using 9 compartments, the used SIR model included 2 groups of serotype, sexual behaviourand 17 age-groups. Taira et al [30] assessed HPV vaccination programs using a SIS modelregardingoneserotypestratiedbyageandsexualactivity.

Models cited above were based on numerical simulations with few analytical results. The variability of model predictions due to theuncertaintyin estimating theinput valueswasrarely explored. While some input parameters are usually unknown and are estimated in the calibration of the model, other parametersareassessedusingepidemiologicaldata. Uncertaintyanalysismaybeusedto investigatethe predictionimprecisionintheoutcomevariablethatisdue totheuncertaintyinestimatingthevaluesof theinputparameters[16].

In another paper, Elbasha computed the basic and vaccinated reproduction number of a simple SIR modelregardingoneHPV-serotypetransmission[9]. Thebasicreproductionnumber

R

0

is athreshold quantity which determines if anepidemic can spread in a population ordie out. It is dened by the expected numberofsecondarycasesofHPV producedbyaninfectedindividualduringitsentireperiod ofinfectiousness,in acompletelysusceptiblepopulation[7].

SIRmodelsareusedassumingthatindividualswhoclearHPVinfectionsbecomeimmunetoanewHPV infection. WhileecientprotectiveimmunityagainstHPV followingarstinfectionremainsuncertain [17], SIS models may be employed. In this paper, wepresent aSusceptible-Infected-Susceptible (SIS) deterministicmodel ofheterosexualtransmissionofHPV.

We developped and parametrized a two-sex model of HPV infection transmission in a sexuallyactive population. Weincludedthefour serotypesofHPV whicharecoveredbythequadrivalentvaccine. The basicandvaccinatedreproductionnumbersaregivenforthemodelconsideringthefourHPVserotypes. We assessedthesensitivityof model predictions toparameter uncertainty. Weestimated the potential impact of a quadrivalent HPV-vaccine on the occurrence of HPV infections comparing 3 vaccination scenarios.

Method

HPV model structure The model with vaccination

The model describesHPV infection transmission in aheterosexually active population. We developa deterministicmodelusingaSusceptible-Infected-Susceptible(SIS)structureandconsideringvaccination. Themodelincludes 2classesofHPV genotypes: HPV-16/18(high-oncogenicrisktypes)andHPV-6/11 (low-risktypes). Apossibleco-infection6/11/16/18wasalsotakenintoaccount(gure1).

Non-vaccinated (resp. vaccinated) women enter thesexuallyactive populationin thesusceptible com-partment

X

00

(resp.

V

00

) at aconstant rate [(1-

ϕ

f

)Λ

] (resp. [

ϕ

f

Λ

]) and leave allcompartmentsat rate

µ

. Non-vaccinated(resp. vaccinated) men enter thesexuallyactive population in the susceptible compartment

Y

00

(resp.

W

00

)ataconstantrate[(1-

ϕ

m

)Λ

](resp. [

ϕ

m

Λ

])andleaveallcompartmentsat rate

µ

. Then,womencanmoveintoinfectedcompartments(iftheyhaveaninfectedcontactwithaman) in non-vaccinated population (resp. vaccinated):

X

01

for women infected with HPV 6/11(resp.

V

01

),

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(resp.

V

11

)(detailinTable1). Inthesameway,non-vaccinatedandvaccinatedmencanmovetoinfected compartments. We assumethat vaccinatedpeople canbeinfected. The degreeofvaccine protectionis

τ

, therelativeriskofavaccinatedindividualexperiencingabreakthroughinfectionis(1-

τ

). Weassume that vaccinatedandinfected individualscantransmitHPV asmuch asnon-vaccinatedindividuals. We assumethatvaccineimmunitydoesnotdecreaseduringtheirsexuallyactivelife. Womenand menwho clearHPVinfectionleaveinfectedcompartmentsandgobacktothesusceptiblecompartmentsorinfected compartmentswithotherserotype. VariablesandparametersaredescribedinTable1.

Demographicandbiologicalparametersarestrictlypositive.

TheordinarydierentialequationsthatrepresentthiscompartmentalmodelarepresentedinAppendix. Basic and Vaccinated Reproduction Number

Inthe abscenceof vaccination,

ϕ

m

= 0

and

ϕ

f

= 0

aswell as

V

00 = V

01 = V

10 = V

11 = W

00 = W

01 =

W

10 = W

11 = 0

. Thesystemofdierentialordinaryequationsisasfollows:

dX

00 dt

=

Λ −

σ

f

N

m

(Y

01 + Y

10 + Y

11 )X

00 + δ

01 X

01 + δ

10 X

10 + δ

11 X

11 −

µX

00 dX

01 dt

=

σ

f

N

m

Y

01 X

00 −

σ

f

N

m

(Y

11 + Y

10 )X

01 −

δ

01 X

01 + δ

10 X

11 −

µX

01 dX

10 dt

=

σ

f

N

m

Y

10 X

00 −

σ

f

N

m

(Y

11 + Y

01 )X

10 −

δ

10 X

10 + δ

01 X

11 −

µX

10 dX

11 dt

=

σ

f

N

m

Y

11 X

00 +

σ

f

N

m

(Y

11 + Y

01 )X

10 +

σ

f

N

m

(Y

11 + Y

10 )X

01 −

(δ

10 + δ

01 + δ

11 + µ)X

11

(1)

dY

00 dt

=

Λ −

σ

m

N

f

(X

01 + X

10 + X

11 )Y

00 + δ

01 Y

01 + δ

10 Y

10 + δ

11 Y

11 −

µY

00 dY

01 dt

=

σ

m

N

f

X

01 Y

00 −

σ

m

N

f

(X

11 + X

10 )Y

01 −

δ

01 Y

01 + δ

10 Y

11 −

µY

01 dY

10 dt

=

σ

m

N

f

X

10 Y

00 −

σ

m

N

f

(X

11 + X

01 )Y

10 −

δ

10 Y

10 + δ

01 Y

11 −

µY

10 dY

11 dt

=

σ

m

N

f

X

11 Y

00 +

σ

m

N

f

(X

11 + X

01 )Y

10 +

σ

m

N

f

(X

11 + X

10 )Y

01 −

(δ

10 + δ

01 + δ

11 + µ)Y

11

Y

∗

11

)= (

Λ

µ

,0,0,0,

Λ

µ

,0,0,0)istheDFE.

Thebasicreproductionnumber

R

0

isathresholdquantitywhichdeterminesifanepidemiccanspreadin apopulationordieout. ItisdenedbytheexpectednumberofsecondarycasesofHPVproducedbyan infectedindividualduring itsentireperiodofinfectiousness,inacompletelysusceptiblepopulation[7]. WeusetheNext GenerationMatrix(NGM) [32]tocompute

R

0

.

R

0

isequaltothespectralradiusof

F

1 V

−1

, X

11 ∗∗

, Y

00 ∗∗

, Y

01 ∗∗

, Y

10 ∗∗

, Y

11 ∗∗

, V

00 ∗∗

, V

01 ∗∗

, V

10 ∗∗

, V

11 ∗∗

, W

00 ∗∗

, W

01 ∗∗

, W

10 ∗∗

, W

11 ∗∗

)

=

(1 − ϕ

f

)

Λ

µ

,

0, 0, 0, (1 − ϕ

m

)

Λ

µ

,

0, 0, 0, ϕ

f

Λ

µ

,

0, 0, 0, ϕ

m

Λ

µ

,

0, 0, 0

Thevaccinatedreproductionnumbertakesintoaccountvaccineprotection. Followingthesamemethod usedforthebasicreproductionnumbercomputation (NextGenerationMatrix),

R

v

=

1 (min(δ

01 , δ

10 ) + µ)

pR

m

R

f

with

R

k

= σ

k

[(1 − ϕ

k

) + (1 − τ )ϕ

k

]

,fork=f,m. Also:

R

2 _v

= R

0 q

[(1 − ϕ

m

) + (1 − τ )ϕ

m

][(1 − ϕ

f

) + (1 − τ )ϕ

f

].

Note that terms inside brackets are less than one,

R

v

< R

0

. The term under the square root shows howmuchvaccinationreduces

R

0

. This parameteris veryimportantbecauseit representsathreshold quantity and bringing it below one could allow the eradication of endemicity of HPV. The level of impact that is necessaryto achieveepidemic elimination depends onthe combined eects of male and female vaccinationprograms. Consideringthe basicreproduction numberpreviously obtained,we plot thecriticallevelofmalevaccinecoveragethat isnecessaryto achieveepidemic eliminationaccordingto female vaccinationrate (gure 2). The impactof female-only vaccinationhasto bemore than74%to achieveHPV elimination.

Model simulations

First,weprogramthesystemwithout vaccinationin Scilab software. Wesolveitusing aRunge-Kutta method. Inputparameterswere evaluatedusing publisheddata. Therate ofexit ofthe sexuallyactive populationcanbeestimatedastheoppositeofthedurationofsexuallyactivelife[14]. Hughesetal[15] have estimated the average durationof sexually active life to 15 years. Assuming that thesize of the populationin themodelisconstant,thenumberofnewrecruitsintothesexuallyactivepopulation(per year)wasestimatedtobe30,000. Weperformedareviewoflitteraturetondpublishedepidemiological dataonHPVprevalencesandaveragedurationofHPVinfectionsforthe4serotypes6/11/16/18ineach gender. We used available epidemiological data regarding general population. US data were used to estimate prevalencesof HPV infection [23, 27]. The annualclearance rateis estimated astheopposite oftheaveragedurationoftheinfection(inyears)[14]. Weassumedthatclearanceratesweresimilarin maleandfemaleandaccordingtovaccinestatus. However,clearanceratesvariedaccordingtoserotypes. Clearance rates in presence of multiple infections were dened asthe clearance rate corresponding to the longest infections. The mean durations of HPV infection estimated in the litterature were dier-entaccordingto theexploredpopulation. Therefore,type-specic clearancerateswereassigned usinga prioruniformdistributionbetweentheminimumandmaximumestimatesfoundinthelitteraturereview [11, 13, 15,19,20,21,28,31].

Twoannualinfectionratesweredenedinmaleandfemaleandweresimilarforallserotypes. The infec-tionratewasthesameforasusceptibleindividualorforsomeonealreadyinfectedwithotherserotypes. Publishedestimationsofinfectionratescouldnotbeemployedastheydependedonthecaracteristicsof modelsused. Consequently,these parametersweregenerated from auniform distribution on[0,5]. See Table2.

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judged to produce acceptable t when the associated model prediction fell simultaneouslywithin pre-speciedtargetsdenedusingtheepidemiologicaldataofprevalence. Theoutputsofthemodelreached the target if they were inside intervalsof

±

10% of inputs. Inputs were thesize of the 8model com-partments. Amongthemillion randomlysampledcombinationsofparameters,6setsof naturalhistory parameters met ourpredined goodness-of-t criteria. Model simulations were based onone posterior parametersetthat wasidentiedduring modeltting.

Sensitivity analysis

An uncertainty analysiswasperformed. First, westudied the impactof a20%parametervariation on model predictions. We considered variations of new recruit and retirment rate of the sexually active populationtogether,then variationsofclearanceratesand infectionratestogether, nallyvariations of initial prevalences. Each time, thepredictions ofthe model were compared to the pre-speciedtarget. Then,insensitivityanalysis,weidentiedthemostinuentialparametersonmodelpredictionscomputing PartialRankCorrelationCoecient(PRCC)[5 ]. Calculationof PRCCenablesthedetermination ofthe statistical relationships between each input parameter and each outcome variable while keepingall of the other input parameters constant. The magnitude of the PRCC indicates the importance of the uncertainty in estimating the value of the outcome variable. However, in this analysis we only kept outcome variableswhichweremonotonically relatedto theinputparameters. Inthis analysis,weused Rsofware(www.r-project.org).

Vaccine characteristics

Base-case vaccine characteristics were assumed to be as follows: reduction in susceptibility to HPV 6/11/16/18(vaccineecacy)was90%,vaccinedurationislifelong,vaccinatedpeoplewhichareinfected are asinfectiousas thenon-vaccinatedinfected people. Wecompared 3scenariosof vaccination(Table 2) consideringasignicantreductionofHPV-16/18infectedmenandwomen. Wecalculatedhowmany yearswere necessaryafterintroductionof vaccinationto havethe sizeof HPV-16/18infected compart-mentsbelow10,000.

Results

Model t and validation

Ofonemilliondierentcombinationsofparameterssampledfromtheuniformdistributions,6parameter setsproducedmodelresultswithintheprespeciedtargets(Table3).

These6combinationsweredierent. Ineachofthe6combinations,a10%variationofoneparameterwhile keepingtheothers constant didnotproducedoutput in thepre-dened target. Thethird combination wasusedin theanalysesthat follow. We couldassessa

R

0

value at 1.73. As expected, this valuewas above1becauseHPVinfectionshavereachedanendemicstate. Thisvaluedidnotgiveanestimationfor thetimewhichwasnecessarytoeradicateHPVinfections. Inthesectionforvaccinescenario,weestimate howmanyyearsareneededin ordertoobserveasignicantdiminution ofHPVinfectedindividuals.

Sensitivity analysis

Inarststep,weassessedtheeect ofparametervariationsin ascaleof20%(increaseordecrease)on thepredictionsofthemodel. Whenconsideringprevalenceparameters,predictionsofthemodelachieved thepre-speciedtarget. Nonetheless, modicationregardingtheratesofentranceand withdrawalfrom

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andinfectionratevariationsledtopredictionsoutsidethetarget.

In a second step, we conducted a sensitivity analysis using PRCC. Monotonicity between each input variablesandoutputvariableswasassessedconsideringscatterplots. Onlyoutcomevariableswhichwere monotically related to the input parameters were used to compare the PRCC. We computed PRCC betweeneach4inputparameters(femaleinfectionrate,meninfectionrate,HPV-6/11clearancerateand HPV-16/18infectionrate)andthe8outputvariables(sizeofthe8non-vaccinatedcompartments). The relativeimportanceoftheinputvariablescouldbedirectly evaluatedbycomparingthesePRCC(Table 4).

ConsideringsignicantresultsofPRCC,itcanbefoundthattheuncertaintiesinestimatingthevaluesof clearancerateforHPV6/11andHPV16/18arethemostimportantinaectingthepredictionprecisionof susceptiblepopulation. Femaleinfectionrateestimation uncertaintiescontributetopredictionprecision of HPV-6/11 infected men and women. In this case, PRCC relating to men are smaller that PRCC relating to womenbut it canbeenexplainby thenon-monotonous relationformen infectionratewith alloutputvariables. Inthis case,itcouldimplicatethatthePRCCislow.

Vaccine scenarios

Inthecaseof alowvaccinecoverage forwomen(50% ofwomenwhoenter annuallythesexuallyactive population)andwithoutmen'svaccination(scenario 1),50yearswerenecessary,aftervaccine introduc-tion, to observe lessthan 10,000HPV-16/18 infected women (gure 3). Introductionof men's vaccine in scenario2reducedbyhalf this time. Thethird scenariowascaracterisedby ahigh vaccinecoverage among women (90% of womenwho enter,annually, thesexuallyactivepopulation) and theabsence of men's vaccination. In this case, wefound the sametime again that with thesecond scenarioin which half ofmenandwomen,whoenterannuallythesexuallyactivepopulation,were vaccinated.

Discussion

Two prophylactic vaccines against HPV infections are proposed to young women in several countries. In the United States, the Centers for Disease Control and Prevention (CDC) recommend vaccination for girls and women 11 to 26 years old with quadrivalent vaccine, in order to preventcervical cancer, pre-cancerous lesionsand genitalwarts caused byserotypes6,11, 16 and18. InEurope,several coun-tries recommend vaccination against HPV infection, vaccinationagainst HPV starts at dierent ages, between9and14years[1]. Actually,thequestionofvaccinationforboysisbeingstudied[3,12,18,10]. Mathematical models areuseful toappreciatetheimpact ofprophylacticvaccinationagainstHPV and theeectiveness of vaccinationstrategies, forinstanceintroductionof boy'svaccination. Previously, no SISmodelincludingthefourHPV serotypescoveredbythequadrivalentvaccinehavebeendevelopped. OnlyTairaetal[30]havepublishedaSISmodel includingonlyone serotypeofHPV.Inthispaper,we developped adeterministicSIS model ofheterosexually HPV transmission includingthefour serotypes coveredbythequadrivalentvaccine. Wederivedexplicitformulaforthebasicandvaccinated reproduc-tionnumbersthatcharacterizeswhethertheepidemicwillbecontainedfollowingvaccinationornot. We foundthat thebasicreproductionnumberis

R

0 =

r

_σ

f

(min(δ

01 , δ

10 ) + µ)

σ

m

(min(δ

01 , δ

10 ) + µ)

andthevaccinatedreproductionnumberwasassessed:

R

v

= R

0 q

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rateswereabovemaleinfectionratesbecausethetransmissionriskfromaninfectedmantoasusceptible womanishigherthanfrom aninfectedwomantoasusceptibleman[2,11,15]. Theestimatedinfection rateswerehardlycomparablewiththosefoundinthelitteraturebecausemostofthepublishedmodelsare stratiedonsexualbehaviorandage[2,11,15]. Parametersassessedinthesemodelsaretheprobability oftransmission. Sexualbehaviorisintroducedusingaveragerateofsexualpartnerchangeandamixing matrice which describes how partnerships between men and women are formed. The clearance rates (Table2) werenear tothelowervaluesfoundin thelitterature. Theycorrespondedtolongerdurations ofinfection. Furthermore,sensitivityanalysisshowedthatclearancerateshaveanimportantimpacton modelpredictions. In publishedstudies, infected women areseenevery6months toassess theaverage durationofHPV infection,thisperiod implicatesanuncertaintywithrespecttotheexacttimeofHPV clearance[28,31,33]. Thus,moreaccurateepidemiologicaldataonthedurationofHPVinfectionscould improvetheprecisionofmodelpredictions.

Introduction of vaccination in the model allowed us to compare 3 scenarios for vaccination. In the rst scenario, weconsidered that 50% of women who enter annually in the sexuallyactivepopulation werevaccinated. Since vaccinerecommandationsin US arevaccinationat12 yearsold (andacatch-up programm for 13-26 years old girls) this scenario corresponded to half of the 14 years old girls, who enterannuallyinthemodelprotectedbythevaccine. Introductionofmen'svaccinationbesideswomen's vaccination(scenario2vsscenario1)allowsto obtainatwiceasfastdiminution ofHPV-16/18infected individual number. Nevertheless, we found the same fastness with an exclusive high female vaccine coverage (90%) (scenario 3). Therefore, men's vaccination eectiveness has to be discussed according to vaccine coverage acquired for women. These resultscome from a simplied model and have to be conrmedbydevelopingamodelincludingageandsexualbehaviour.

Acknowledgments

Theauthors would liketothankHector deArazoza,Kamel Senouci, Adrien Dozol andBilalMajed for theirinvaluablehelp.

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5,Oct2000.

[23] C.M. Nielson. Type-specic prevalenceandpersistenceof humanpapillomavirusin womenin the unitedstateswhoarereferredfortyping asacomponentofcervicalcancerscreening. AmJObstet Gynecol, 200(3):245,2009.

[24] SPagliosi.Humanpapillomavirusandcervicalcancer.Technicalreport,WorldHealthOrganization. [25] C.Pannier-Stockman,C.Segard,S.Bennamar,J.Gondry,J-C.Boulanger,H.Sevestre,S.Castelain, and G. Duverlie. Prevalence of hpv genotypes determined by pcr and dna sequencing in cervical specimensfromfrenchwomenwithorwithoutabnormalities.JClinVirol,42(4):353360,Aug2008. [26] D.M.Parkin,P.Pisani, andJ.Ferlay. Estimatesoftheworldwideincidenceof25majorcancersin

1990. IntJCancer,80(6):827841,Mar1999.

[27] H.E.Ralston, Z. Li,RCMcGlennen, WLHellerstedt, and LS Downs Jr. Type-specic prevalence andpersistenceofhumanpapillomavirusinwomenintheunitedstateswhoarereferredfortypingas acomponentofcervicalcancerscreening.Americanjournalofobstetricsandgynecology,200(3):245, 2009.

[28] HarrietRichardson,MichalAbrahamowicz,Pierre-PaulTellier,GailKelsall,RoxaneduBerger,Alex Ferenczy,FrançoisCoutlée,andEduardoLFranco.Modiableriskfactorsassociatedwithclearance oftype-specic cervicalhumanpapillomavirusinfectionsinacohortof universitystudents. Cancer EpidemiolBiomarkers Prev,14(5):11491156,May2005.

[29] KSyrjnen,M.Hakama,S.Saarikoski,M.Vyrynen,M.Yliskoski,S.Syrjnen,V.Kataja,andO. Cas-tren. Prevalence,incidence,andestimatedlife-timeriskofcervicalhumanpapillomavirusinfections inanonselectednnishfemalepopulation. SexTransm Dis,17(1):159,1990.

[30] Al VTaira, ChristopherP Neukermans,andGillian D Sanders. Evaluatinghumanpapillomavirus vaccinationprograms. EmergInfectDis,10(11):19151923,Nov2004.

[31] Helen Trottier, Salaheddin Mahmud, José Carlos M Prado, Joao S Sobrinho, Maria C Costa, Thomas E Rohan, Luisa L Villa, and Eduardo L Franco. Type-specic duration of human pa-pillomavirus infection: implications for human papillomavirusscreening and vaccination. J Infect Dis,197(10):14361447,May2008.

[32] P. Van den Driessche and James Watmough. Reproduction numbers and sub-threshold endemic equilibriaforcompartmentalmodelsof diseasetransmission. Math Biosci,180:2948,2002.

[33] C. B. Woodman, S. Collins, H. Winter, A. Bailey, J. Ellis, P. Prior, M. Yates, T. P. Rollason, and L. S. Young. Natural history of cervical humanpapillomavirus infection in young women: a longitudinalcohortstudy. Lancet,357(9271):18311836,Jun 2001.

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Theordinarydierentialequationsthatrepresentthecompartmentalmodelincludingvaccinationare:

dX

00 dt

=

(1 − ϕ

f

)Λ −

σ

f

N

m

(Y

01 + Y

10 + Y

11 + W

01 + W

10 + W

11 )X

00 + δ

01 X

01 + δ

10 X

10 + δ

11 X

11 −

µX

00 dX

01 dt

=

σ

f

N

m

(Y

01 + W

01 )X

00 −

σ

f

N

m

(Y

11 + W

11 + Y

10 + W

10 )X

01 −

δ

01 X

01 + δ

10 X

11 −

µX

01 dX

10 dt

=

σ

f

N

m

(Y

10 + W

10 )X

00 −

σ

f

N

m

(Y

11 + W

11 + Y

01 + W

01 )X

10 −

δ

10 X

10 + δ

01 X

11 −

µX

10 dX

11 dt

=

σ

f

N

m

(Y

11 + W

11 )X

00 +

σ

f

N

m

(Y

11 + W

11 + Y

01 + W

01 )X

10 +

σ

f

N

m

(Y

11 + W

11 + Y

10 + W

10 )X

01 −

(δ

10 + δ

01 + δ

11 + µ)X

11 dY

00 dt

=

(1 − ϕ

m

)Λ −

σ

m

N

f

(X

01 + X

10 + X

11 + V

01 + V

10 + V

11 )Y

00 + δ

01 Y

01 + δ

10 Y

10 + δ

11 Y

11 −

µY

00 dY

01 dt

=

σ

m

N

f

(X

01 + V

01 )Y

00 −

σ

m

N

f

(X

11 + V

11 + X

10 + V

10 )Y

01 −

δ

01 Y

01 + δ

10 Y

11 −

µY

01 dY

10 dt

=

σ

m

N

f

(X

10 + V

10 )Y

00 −

σ

m

N

f

(X

11 + V

11 + X

01 + V

01 )Y

10 −

δ

10 Y

10 + δ

01 Y

11 −

µY

10 dY

11 dt

=

σ

m

N

f

(X

11 + V

11 )Y

00 +

σ

m

N

f

(X

11 + V

11 + X

01 + V

01 )Y

10 +

σ

m

N

f

(X

11 + V

11 + X

10 + V

10 )Y

01 −

(δ

10 + δ

01 + δ

11 + µ)Y

11 dV

00 dt

=

ϕ

f

Λ − (1 − τ )

σ

f

N

m

(Y

01 + Y

10 + Y

11 + W

01 + W

10 + W

11 )V

00 + δ

01 V

01 + δ

10 V

10 + δ

11 V

11 −

µV

00

(2)

dV

01 dt

=

(1 − τ )

σ

f

N

m

(Y

01 + W

01 )V

00 −

(1 − τ )

σ

f

N

m

(Y

11 + W

11 + Y

10 + W

10 )V

01 −

δ

01 V

01 + δ

10 V

11 −

µV

01 dV

10 dt

=

(1 − τ )

σ

f

N

m

(Y

10 + W

10 )V

00 −

(1 − τ )

σ

f

N

m

(Y

11 + W

11 + Y

01 + W

01 )V

10 −

δ

10 V

10 + δ

01 V

11 −

µV

10 dV

11 dt

=

(1 − τ )

σ

f

N

m

(Y

11 + W

11 )V

00 + (1 − τ )

σ

f

N

m

(Y

11 + W

11 + Y

01 + W

01 )V

10 + (1 − τ )

σ

f

N

m

(Y

11 + W

11 + Y

10 + W

10 )V

01 −

(δ

10 + δ

01 + δ

11 + µ)V

11 dW

00 dt

=

ϕ

m

Λ − (1 − τ )

σ

m

N

f

(X

01 + X

10 + X

11 + V

01 + V

10 + V

11 )W

00 + δ

01 W

01 + δ

10 W

10 + δ

11 W

11 −

µW

00 dW

01 dt

=

(1 − τ )

σ

m

N

f

(X

01 + V

01 )W

00 −

(1 − τ )

σ

m

N

f

(X

11 + V

11 + X

10 + V

10 )W

01 −

δ

01 W

01 + δ

10 W

11 −

µW

01 dW

10 dt

=

(1 − τ )

σ

m

N

f

(X

10 + V

10 )W

00 −

(1 − τ )

σ

m

N

f

(X

11 + V

11 + X

01 + V

01 )W

10 −

δ

10 W

10 + δ

01 W

11 −

µW

10 dW

11 dt

=

(1 − τ )

σ

m

N

f

(X

11 + V

11 )W

00 + (1 − τ )

σ

m

N

f

(X

11 + V

11 + X

01 + V

01 )W

10 + (1 − τ )

σ

m

N

f

(X

11 + V

11 + X

10 + V

10 )W

01 −

(δ

10 + δ

01 + δ

11 + µ)W

11

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N

f

= X

00 + X

01 + X

10 + X

11 + +V

00 + V

01 + V

10 + V

11 N

m

= Y

00 + Y

01 + Y

10 + Y

11 + +W

00 + W

01 + W

10 + W

11 N

= N

f

+ N

m

.

Nisthesize ofthesexuallyactivepopulation. Wehave

N

0 _{= 2Λ − µN.}

Sinceatequilibrium

N

∗

_{= 2}

Λ

µ

.

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I 6/11 V S I 6/11 I 6/11/16/18 I 16/18 I 16/18 I 6/11/16/18 Entranceinto thesexuallyactive

population

Figure1. Transferdiagram oftheHPV model. Thedierentcompartmentsrepresentindividuals in eachstateofHPVinfection(roundedupcornerforvaccinatedpopulation,S: non-vaccinatedand susceptible,V: vaccinatedandsusceptible,I6/11: infectedwithHPV-6or/andHPV-11,I 16/18: infectedwithHPV-16or/andHPV-18,I 6/11/16/18: infectedwithHPV-6or/andHPV-11andHPV-16 or/andHPV-18). Thearrowsrepresenttheowbetweenthese states(boldlinesrepresententranceinto thesexually-activepopulation,solidlinesrepresentinfection, dashedlinesrepresentclearanceand regression,dotted linesrepresenttheexit ofthemodel).

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 female vaccine coverage

male v

accine co

v

er

age

Figure 2. Gender-specic vaccineimpact necessary to achieve epidemiceliminationwhen

R

2

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vaccination. Att=0introductionofvaccine,starsrepresentthescenario1,solidlinerepresentsthe scenario2,dashedlinerepresentsthescenario3.

(16)

11

(t)(

W

11

(t)) Infectedmen withHPV6/11/16/18 Demographicparameters

Λ

Newrecruitsinto thesexuallyactivepopulation

µ

Deathorremoveratefrom thesexuallyactivepopulation Biologicalparameters

σ

f

Infection rateforwomen

σ

30,000

1

2 µN †

Deathorremoveratefrom thesexuallyactivepopulation 6% [15] (peryear)

µ

Naturalhistory

‡

Parameterrange

Infectionrateforwomen

σ

f

0-5 Assumption Infectionrateformen

σ

m

0-5 Assumption ClearancerateforHPV 6/11(

δ

01

),16/18(

δ

10

) 0.6-2 [11,13,15,20,21,28,31]

ClearancerateforHPV6/11/16/18(

δ

11

)

= δ

10

longestduration Vaccines

Degreeofvaccineprotection

τ

90% [8] Vaccinationrate Female Male

Scenario1 50% 0%

Scenario2 50% 50%

Scenario3 90% 0%

∗

compartmentsizelargeenoughtoapplyadeterministicmodel

†

assumptionto haveaconstantpopulationsizein themodel

‡

Thenaturalhistoryparametersareannualtransitionrates

Table3. Combinationsof parameterswhichproduct resultswithintheprespeciedtarget

1 2 3

†

4 5 6

Infectionrateforwomen* 1.02 1.14 1.49 1.57 2.36 2.37 Infectionrateformen* 0.75 0.68 0.90 0.95 1.75 1.52 ClearancerateHPV-6/11* 0.64(18.8) 0.65(18.5) 0.87(13.8) 0.91(13.2) 1.55(7.7) 1.46(8.2) ClearancerateHPV-16/18* 0.62(19.4) 0.63(19.0) 0.84(14.3) 0.88(13.6) 1.5(8.0) 1.42(8.4) *Annualrates

Durationofinfectionareinparentheses(inmonths)

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*theresultsaresignicantatthe0.001level