Optimal Control of Hepatitis C Antiviral Treatment Programme Delivery for Prevention amongst a ...

17 Pages·2011·0.56 MB·English

Optimal Control of Hepatitis C Antiviral Treatment
Programme Delivery for Prevention amongst a
Population of Injecting Drug Users
Natasha K. Martin1,2*., Ashley B. Pitcher3*., Peter Vickerman1,2, Anna Vassall2, Matthew Hickman1
1DepartmentofSocialMedicine,UniversityofBristol,Bristol,UnitedKingdom,2DepartmentofGlobalHealthandDevelopment,LondonSchoolofHygieneandTropical
Medicine,London,UnitedKingdom,3Centred’AnalyseetdeMathe´matiqueSociales,L’E´coledesHautesE´tudesenSciencesSociales,Paris,France
Abstract
Inmostdevelopedcountries,HCVisprimarilytransmittedbyinjectingdrugusers(IDUs).HCVantiviraltreatmentiseffective,
and deemed cost-effective for those with no re-infection risk. However, few active IDUs are currently treated. Previous
modelling studies have shown antiviral treatment for active IDUs could reduce HCV prevalence, and there is emerging
interest in developing targeted IDU treatment programmes. However, the optimal timing and scale-up of treatment is
unknown, given the real-world constraints commonly existing for health programmes. We explore how the optimal
programmeisaffectedbyavarietyofpolicyobjectives,budgetconstraints,andprevalencesettings.Wedevelopamodelof
HCVtransmissionandtreatmentamongstactiveIDUs,determinetheoptimaltreatmentprogrammestrategyover10years
fortwobaselinechronicHCVprevalencescenarios(30%and45%),arangeofmaximumannualbudgets(»50,000–300,000
per1,000IDUs),andavarietyofobjectives:minimisinghealthservicecostsandhealthutilitylosses;minimisingprevalence
at10years;minimisinghealthservicecostsandhealthutilitylosseswithafinaltimeprevalencetarget;minimisinghealth
servicecostswithafinaltimeprevalencetargetbutneglectinghealthutilitylosses.Thelargestprogrammeallowedfora
givenbudgetistheprogrammewhichminimisesbothprevalenceat10years,andHCVhealthutilitylossandheathservice
costs, with higher budgets resulting in greater cost-effectiveness (measured by cost per QALY gained compared to no
treatment).However,iftheobjectiveistoachievea20%relativeprevalencereductionat10years,whileminimisingboth
healthservicecostsandlossesinhealthutility,theoptimaltreatmentstrategyisanimmediateexpansionofcoverageover
5–8years,andislesscost-effective.Bycontrast,iftheobjectiveisonlytominimisecoststothehealthservicewhileattaining
the20%prevalencereduction,theprogrammeisdeferreduntilthefinalyearsofthedecade,andistheleastcost-effective
ofthe scenarios.
Citation:MartinNK,PitcherAB,VickermanP,VassallA,HickmanM(2011)OptimalControlofHepatitisCAntiviralTreatmentProgrammeDeliveryforPrevention
amongstaPopulationofInjectingDrugUsers.PLoSONE6(8):e22309.doi:10.1371/journal.pone.0022309
Editor:PieterH.M.vanBaal,ErasmusUniversityRotterdam,TheNetherlands
ReceivedSeptember10,2010;AcceptedJune24,2011;PublishedAugust11,2011
Copyright: (cid:2)2011 Martin et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.
Funding:NKMissupportedbytheScottishGovernmentHepatitisCActionPlanandCentreforResearchonDrugsandHealthBehaviourNIHRGrant.ABPis
supportedbythe‘DyXi’projectoftheSYSCOMMprogrammeoftheFrenchNationalAgencyforResearch(grantnumberANR-08-SYSC-008).PVissupportedbya
MRCNewInvestigatorAward.Thefundershadnoroleinstudydesign,datacollectionandanalysis,decisiontopublish,orpreparationofthemanuscript.
CompetingInterests:Theauthorshavedeclaredthatnocompetinginterestsexist.
*E-mail:[email protected](NKM);[email protected](ABP)
.Theseauthorscontributedequallytothiswork.
Introduction syringe exchange, and opiate substitution therapy aim to prevent
transmission by reducing unsafe injecting [4]. However, despite
Hepatitis C virus (HCV) is a comparatively common blood- increases in intervention exposure, public health surveillance
borne disease with 130–170 million people (2–3%) globally indicates that substantial decreases in HCV prevalence have not
infected[1].Itisoneoftheleadingcausesofchronicliverdisease been achieved [8].
worldwide,andisthefastestgrowingcauseoflivertransplantation Antiviral treatment (peginterferon-alfa and ribavirin) for HCV
indevelopedcountries[2].Ifleftuntreated,about7–18%progress has been established as effective and results in viral clearance in
to liver disease within 20 years, which can result in liver failure, about45–80%ofcases,dependingongenotype[9–11].Economic
cirrhosis, hepatocellular carcinoma, anddeath[3]. evaluations have found treatment cost-effective for a population
The primary mode of transmission in developed countries is withnore-infectionrisk[12].Since2002,guidelinesintheUSand
amongst active injecting drug users (IDUs) where it is easily UK do not exclude active IDUs from treatment eligibility, given
transmittedthroughneedleandsyringesharing.IntheUK,most the growing evidence that IDUs exhibit a similar response to
developedcountries,andmanyotherdevelopingcountrieswithout treatment,andcouldbejustascompliantwithtreatmentasex-or
marked iatrogenic HCV risk (such as South Asia), over 80% of non-IDUs [11–13].Despitethese recommendations andthehigh
newcasesareattributedtoinjectingdrugs,with15–90%ofIDUs proportionofIDUsinfected,veryfew(v3–4%)activeIDUshave
testing positive for HCV antibodies [4–7]. Public health ever been treated [14,15]. Recent mathematical modelling has
interventions such as health education and advice, needle and predicted that antiviral treatment can be an effective prevention
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OptimalControlofHepatitisCTreatment
measure amongst IDUs, with modest and achievable levels of annualbudgetconstraintsandtwobaselineprevalences(30%and
treatmentresultinginsubstantialreductionsininfectedprevalence 45%).
[16–19]. Hence, treatment of injectors could have substantial
benefitsinrelationtoreducingongoingtransmission(despitesome
Methods
of the difficulties in delivering the treatment and potential loss of
sustained viralresponse (SVR)). Model background and assumptions
Ideally,fromaneconomicperspective,anyintervention,suchas
Infection with HCV leads to a relatively short (weeks to
HCVtreatment,thathasbeenfoundtobecost-effectivewouldbe
months)acutestage,whichmayleadtoaprolongedchronicstage
fullyfundedimmediately.Inreality,however,fundingandaccess
lasting for decades [28]. A fraction (about 26%) of acute
totreatmentmaynotbeprovidedasprogrammesarealwaysfaced
infections are spontaneously cleared by the individual [29]. Due
with a number of constraints. Globally, there are many
totheshortdurationoftheacutestage,thenumberofinfections
interventions that fall under the WHO willingness to pay
caused by people with acute HCV who spontaneously clear is
thresholds, but remain underfinanced by countries and face
small, and we neglect it for model simplicity. Those who
substantialbudgetconstraints.Forexample,inAustraliafreeHCV
spontaneouslyclearbecomesusceptibleagain,andtheremaining
treatment is available under the national health care system, but
fraction who do not spontaneously clear progress to the chronic
thegovernmentacknowledgesacapacitylimitwithinthespecialist
infection stage. There is controversy around the possibility of
hepatitisCtreatmentserviceswhichrestrictsthenumbersthatcan
sterilisingimmunityfollowingexposuretoHCV.However,given
undergo treatment each year [20]. In the UK, NICE (National
that immunity following exposure to HCV is uncertain, and
Institute for Health and Clinical Excellence) issues guidelines on
previous models have shown that, if present, this population is
which treatments should be offered, but different regions
relatively small, we neglect it for the purposes of this model
frequently offer different levels of treatment. Global (and
[16,17].
sometimes national) institutions frequently respond to this issue
Antiviraltreatmentleadstoasubstantialreductioninviralload
by setting targets, either for coverage or to achieve specific
inthefirstfewweeks(evenamongsomeeventualnonresponders)
reductions in prevalence. Indeed, the low coverage of HCV
[30]. Hence, we assume that active IDUs currently on treatment
treatment across the UK has resulted in the development of a
arenon-infectious.Duetothelackofdefinitiveevidencetosuggest
number of national action plans (Scotland, Wales, and England)
otherwise, we assume that the chances of spontaneous clearance
which aim to expand treatment coverage over the next 5 years
are equal for na¨ıve (those who have never been infected) and re-
[20–22]. These action plans do not specifically target current
infected IDUs. Furthermore, we assume that the probability of
IDUs,despitethegrowinginterestintargetingantiviraltreatment
treatment success is the same between na¨ıve and re-infecteds,
to IDUs as a means of prevention, and general movements for
which is supported by experimental evidence [31]. Finally, we
moreactiveIDUstobetreated[20,23].Thisstudyexamineshow
assume that treatment failures return to the chronically infected
constraints(annualmaximumbudgets)andobjectives(prevalence
population and are eligible for retreatment as a simplifying
targets, desire to minimise health utility losses) are likely to
assumption, as alternative dosing and treatment durations are
influence optimal timing and intensity of scale-up, and the
availableforthisgroup[32].Nonetheless,overourrelativelyshort
subsequent costs, impact, and cost per QALY (quality-adjusted
timescale(10years)andwiththelowleveloftreatmentexamined
lifeyear) gained in eachscenario.
in this manuscript, simulations tracking nonresponders show that
As such, our aim is not to perform an economic evaluation of
negligible levels ofinfecteds are retreated [16].
antiviral treatment, but to inform policy makers on how the
optimal treatment strategy and programme cost-effectiveness
(measured by cost per QALY gained) may be affected when Details and explanation of the model
different constraints are applied. This type of analysis (using WemodelthetransmissionofHCVamongstactiveIDUs,using
optimal control theory to determine the optimal resource asystemofordinarydifferentialequationssimplifiedfrom[16,17].
allocation as an epidemic progresses) has been used before in We utilise a three compartment model, tracking susceptible,
infectious disease prevention [24–27]. Hence, our aims and chronicallyinfected,andtreatedIDUs.SusceptibleIDUsbecome
mathematical techniquesarewellestablishedininfectiousdisease infected through sharing needles with infected IDUs. About one
literature, although its specific application to HCV is novel. quarterspontaneouslycleartheinfection,andbecomesusceptible
Importantly, few have managed to present this technique to a again.Theremainingthree-quartersprogresstochronicinfection.
broader audience (outside of mathematical modellers) and with Chronicinfectedscanbetreated,withacertainchanceofsuccess,
real-world budget constraintsandobjectives. andeitherfailtreatmentandreturntotheinfectioncompartment,
Weparameteriseourepidemicmodelwithrecentdatafromthe or clear thediseaseandbecome susceptible again.
UK, and our cost coefficients with current UK costs of antiviral In our model, X denotes the number of susceptible IDUs
treatment and HCV infection. We then examine the optimal (including those who have cleared the infection), C denotes the
treatment strategy for different economic and policy objectives, number of both chronically infected and acutely infected IDUs
which range from ‘ideal’ public health objectives (where health which will proceed to chronic infection, and T denotes the
service costs and HCV health utility loss or just prevalence is number of IDUs in treatment, t is time in years, and where
minimised)to‘lessideal’butperhapsmorerealisticscenarioswith N~total population~XzCzT. The equations describing the
a specific policy objective of reducing prevalence by a specific HCV transmissionare:
percentage by the end of the 10 year timeframe. The specific
scenarios we examine are: 1) minimising health service costs and
dX C
healthutility(QALY)loss;2)minimisingprevalence;3)minimising ~h{p(1{d) XzvaT{mX, ð1Þ
dt N
health service costs and health utility loss while achieving a final
time prevalence reduction of 20%; 4) minimising health service
costs while achieving a final time prevalence reduction of 20%
dC C
(and neglecting health utility loss). This is done for a variety of ~p(1{d) X{uCzv(1{a)T{mC, ð2Þ
dt N
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OptimalControlofHepatitisCTreatment
dT dx
~uC{vT{mT, ð3Þ ~m(1{x){p(1{d)cxzvat, ð8Þ
dt dt
with initial conditions X(0)~X , C(0)~C , T(0)~0. For the
0 0
initial conditions, we assume that the epidemic is at steady state dc
with no treatment,as isthecaseinmost placesintheUK[8]. ~p(1{d)cx{uczv(1{a)t{mc, ð9Þ
dt
Equation (1) represents the rate of change of the size of the
susceptible population, where new IDUs enter at a fixed rate h.
The second term in Equation (1) models the infection of
dt
susceptible IDUs, which is proportional to the number of ~uc{vt{mt: ð10Þ
dt
susceptibles, the fraction of the population chronically infected,
andtheinfectionratep.Theacuteinfectionspontaneouslyclears
Since xzczt~1, we can substitute t~1{x{c into Equations
in a proportion d, who return to the susceptible pool. The
(8) and (9) and get rid of (10) so that we are left with the smaller
remaining infected fraction which do not spontaneously clear,
1{d,progresstochronicinfection.ThethirdterminEquation(1) system
represents IDUs who exit treatment at a rate v, with successful
treatment proportion a. dx
Equation (2) models the rate of change of the number of ~mzva{(mzva)x{vac{p(1{d)cx, ð11Þ
dt
chronically infected IDUs. The first term represents those who
enter from the susceptible pool, which is proportional to the
number ofsusceptibles, thefractionofthepopulationchronically dc
infected, the infection rate (p), and the fraction who do not ~v(1{a)(1{x{c){uczp(1{d)cx{mc, ð12Þ
dt
spontaneously clear the acute infection (1{d). The fraction of
nonresponders to treatment, (1{a), return from treatment
with initialconditions
proportional torate v.
The second term in Equation (2), {uC, represents the
movement of infected IDUs into treatment. The proportion of x(0)~x w0, ð13Þ
0
infecteds put on treatment per year as an instantaneous rate
represented by u, can vary through time and is the function we
would like to optimise with respect to, given the constraints c(0)~c w0, ð14Þ
0
described later.
Equation (3) represents the rate of change of the number of where x ~mX =h, c ~mC =h and x zc ~1. We wish to find
0 0 0 0 0 0
IDUscurrentlyintreatment.InfectedIDUsentertreatmentatthe the function u(t) over the time horizon 0ƒtƒt such that we
f
rate u as discussed for Equation (2). IDUs exit treatment minimise theobjective functional
proportional totherate v.
Ineachofthepopulations,IDUsleave(duetodeathorceasing ðtf
injection)proportional totherate m. J(u):~ ½(b zb )ucNz(f zf )cNzcu2(cid:2)dt, ð15Þ
1 2 1 2
0
System and Objective Functional
subject totheconstraints
X C T
Ifweletx~ ,c~ ,t~ ,sothatthestatevariablesarenow
N N N
the fraction of the population in each compartment, then 0ƒuƒ1, ð16Þ
Equations (1)–(3) become
dx~h(1{x){p(1{d)cxzvat, ð4Þ b1ucNzcu2ƒM, ð17Þ
dt N
where M is a positive constant representing the maximum
programmespendrate,andb ,b ,f ,f andcarethefollowing
dc hc 1 2 1 2
~p(1{d)cx{uczv(1{a)t{ , ð5Þ decreasing functions of time:
dt N
b (t)~b e{rt, ð18Þ
dt ht 1 10
~uc{vt{ , ð6Þ
dt N
b (t)~b e{rt, ð19Þ
where N evolves accordingtotheequation 2 20
dN~h{mN: ð7Þ c(t)~c e{rt, ð20Þ
dt 0
Since the population is assumed to be in steady state, then
N~h=m.Making thissubstitution, Equations (4)–(6)become f (t)~f e{rt, ð21Þ
1 10
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OptimalControlofHepatitisCTreatment
Optimal control
f (t)~f e{rt: ð22Þ The optimal control problem is minuJ(u) subject to (11)–(14),
2 20
(16)and(17).Inotherwords,weseekthefunctionu(t)suchthat
Theparametersb ,b ,c ,f ,f andrarepositiveconstants. J(u) isminimisedsubject tothestate equations (11) and(12),the
10 20 0 10 20
The constant r is the instantaneous discount rate and can be initial conditions(13) and(14) andtheconstraints(16) and(17).
calculated using theformula WeemployPontryagin’sminimumprinciple[34]todetermine
necessaryconditionsthatmustbesatisfiedbyanoptimalcontrol,if
oneexists.Theexistenceofanoptimalcontrolandcorresponding
r~{log(1{r), ð23Þ
optimal states is guaranteed for this optimal control problem as
criteria (a),(b),(c) of Theorem 4.1 and criteria (d’) and (e’) of
where r istheannual effectivediscount rate.
Corollary 4.1 foundin[35, chap.3] aresatisfied.
WedividedthecostassociatedwithHCVintotwotypes.First,
The optimal control theory associated with various types of
we have the programme costs. These are the costs required to
inequalityconstraints,includingtheonesconsideredhere,canbe
search for and treat patients. Secondly, we have the costs
foundin[36].Forageneral introductorytextonoptimalcontrol
associated with infection. These are the costs incurred by the
methodsappliedtobiologicalsystems,wedirectinterestedreaders
health service when managing patients who are untreated and
towardsthetextbyLenhartandWorkman[37].Thefirststepisto
remain infected. Together, programme (antiviral treatment and
form the Hamiltonian associated with this optimal control
search) and infection costs add up to the health service costs
associated with HCV. More precisely, the function b represents problem, i.e.
1
theaveragecostofantiviraltreatmentperperson.Thesearchcosts
related to finding, diagnosing, and recruiting active IDUs onto H~(b zb )ucNz(f zf )cNzcu2H
1 2 1 2
treatmentarerepresentedbythetermcu2.Asiscommonlydone
zl (mzva{(mzva)x{vac{p(1{d)cx)
inoptimalcontrolmodelsofinfectiousdiseases,weuseaquadratic x
ð24Þ
function of the control to represent increasing marginal costs zl (v(1{a)(1{x{c){uczp(1{d)cx{mc)
c
associated with achieving high treatment coverage levels [24,33].
The function f1 represents the average annual infection cost per zL(b1ucNzcu2{M),
chronically infectedIDU.
where l , l and L are functions of t. The multiplier L is an
To be able to incorporate health utility losses into the optimal x c
adjoining Lagrange multiplier that must satisfy
control framework, we monetarise the QALY losses associated
with HCV infection and antiviral treatment. We detail this
approachfullyinthesection‘Economicparameters’.Inparticular, L(t)§0, if b ucNzcu2~M, ð25Þ
1
b represents the monetarisation of the QALY loss per antiviral
2
treatment,andf representsthemonetarisationoftheQALYloss
2
peryearassociatedwithHCV.Theparametersb ,b ,f ,and
10 20 10 L(t)~0, if b ucNzcu2vM: ð26Þ
f representtheinitial valuesof these costs at t~0. 1
20
The first constraint 0ƒuƒ1 means that we cannot treat at a
To find the characterisation of the optimal control, by Pontrya-
ratehigherthan100%oftheinfectedpopulationperyear,andthe
gin’sminimumprinciplewejustneedtofindtheuthatminimises
treatment rate cannot be negative. The second constraint
theHamiltonian.Sincethecontroluappearsquadraticallyinthe
b ucNzcu2ƒM limits the spending rate of the treatment
1 Hamiltonian,thentodothis wesimplyset
programme (antiviral treatment and search costs) so that the
programmespendingratecanneverbegreaterthan»M peryear.
LH
M will be referred to as the annual or yearly budget. As health ~0, ð27Þ
servicecostsrelatedtoinfection,f cN,areunrelatedtothedirect Lu
1
programme budget, we do not incorporate these into the budget
whichgives theequation
constraint.
Withourmodel,weprojecttheoptimaltreatmentprogramme
overtime,thecorrespondingprevalencereductionsandinfections (b1zb2)cNz2cu{clczb1cNLz2cuL~0: ð28Þ
averted with each programme, total programme costs, and total
infection costs. We divide our total programme costs by the Ifb1ucNzcu2vM,then(28),withL~0andtheconstraint(16),
number of infections averted to calculate cost per infection provides theoptimal control characterisation (call itu(cid:3)):
averted. We also calculate net monetary benefit (monetarised
benefits associated with QALYs gained minus net health services cl {(b zb )cN
u(cid:3)~minfmaxf0, c 1 2 g,1g: ð29Þ
costs) related to each programme compared to no treatment. 2c
Finally,wecalculatethehealthservicecostperQALYgainedfor
each programme. Due to the prevention impact of antiviral If b ucNzcu2~M, then the optimal control characterisation is
1
treatment, it is important to capture the future benefits of the derived bysimply solvingthisquadratic equationfor u toobtain
treatmentprogramme.Forthesecalculations,wesimulateeach10
year treatment programme, and then continue calculating qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
discountedhealthservicecostsandQALYsforafurther40years, {b cNz (b cN)2z4cM
u(cid:3)~minfmaxf0, 1 1 g,1g, ð30Þ
in order to determine the onward prevention impact of the 2c
different 10 year treatment programmes. Hence, the total time
horizoninthecostperQALYgainedcalculationis50years.We where we have taken the positive root because the negative root
calculatethecostperQALYgainedascomparedtoabaselineof always yields a negative value of the fraction in (30) since the
no treatment. parametersareallpositive.Duetothecontrolconstraint(16),the
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OptimalControlofHepatitisCTreatment
optimalcontrolisforcedtoliewithinitsadmissiblebounds0and If instead we wish to simply minimise final time prevalence
1. subjecttoourbudgetconstraint,wecandothisbyaddingtheterm
We know from (26) that L~0 when the constraint (17) is not wc(tf)totheobjectivefunctionalandmakingthepositiveconstant
tight. When the budget constraint is tight, the function L can be wverylarge,e.g.w~1010 sothattheothertermsintheobjective
determined from (28)with u~u(cid:3) asdefined in (30),sothat arenegligiblebycomparison.Mathematically,thisadditiontothe
objective functional leads to a different final time boundary
condition on l ,i.e. l (t )~w.
cl {2cu(cid:3){(b zb )cN c c f
L~ c 1 2 : ð31Þ
b1cNz2cu(cid:3) Uncertainty analysis
To examine how the uncertainty in the biological and cost
parametersalterstheoptimalcontrolsolution,weperformaLatin
Itisimportanttokeepinmindthatthesearchcostcoefficientc
Hypercube Sampling (LHS) of the cost parameters (b , b , f ,
cannot at anytime be zero or else the optimal control problem is 10 20 10
f , c ) and biological parameters (a , a , m and d) over a
singular. Singular optimal control problems require different 20 0 1 2=3
uniformrangeofvalues.Foreachofthe100parametersetsinour
solution methods (see[38–40], and[37]).
sampling, we calculate the corresponding infection rate (p) and
The corresponding differential equations for the adjoint
new injector rate (h) which gives the desired untreated endemic
variables (l and l ) are determined from partial derivatives of
x c prevalence(30%or45%),andretainsatotalof1,000IDUsinthe
theHamiltonian:
population. With each of these parameter sets, we calculate the
corresponding optimal control solution and infected prevalence
dl LH reduction.
x~{ , ð32Þ
dt Lx
Sensitivity analysis
Asensitivityanalysiswasperformedtodeterminehowsensitive
dlc~{LH, ð33Þ the infected prevalence is to variations in the epidemiological
dt Lc parameters and in the presence of treatment. This allows us to
identify which parameters play the most significant role in the
disease dynamics, as well as how the prevalence sensitivity varies
fromwhichweobtain theadjoint equations
between baseline prevalence scenarios. For the analysis, the
control function u is considered a constant function of time
dl
x~(mzva)l zp(1{d)cl zv(1{a)l {p(1{d)cl , ð34Þ ranging from zero to one. We then assess the variability in our
dt x x c c prevalenceat10years(thetimescaleunderconsideration).Wedo
this by again utilising LHS to select 1,000 combinations of the
inputparametersd,m,a ,a andu.Foreachsetofparameters,the
1 2
dlc~{(b zb )u(cid:3)N{b u(cid:3)NL{(f zf )Nzval infection rate, p, is then calculated given the untreated endemic
dt 1 2 1 1 2 x ð35Þ prevalence.Thevalueforhiscalculatedfrommtoretainatotalof
zp(1{d)xl zv(1{a)l {p(1{d)xl zu(cid:3)l zml : 1,000 IDUs. We then solve the system of ordinary differential
x c c c c
equations usingMATLAB,andtracktheprojectedprevalenceat
The adjointequations have thefinal conditions year10witheachparameterset.WethencalculateaPartialRank
Correlation Coefficient (PRCC) to assess the relative importance
ofeachparameterindetermininginfectedprevalence.PRCCsare
l (t )~0, ð36Þ
x f widely used in sensitivity analyses in systems biology and disease
transmission models to determine the importance of a parameter
on a given output while fixing the other parameters at their
lc(tf)~0: ð37Þ expected value [24,41–43]. The larger the absolute value of the
PRCC,themoreinfluenceaparameterhasonprevalence,witha
PRCC magnitude greater than 0.5 and a p-value of v0.05
The state and adjoint equations and their four associated indicating the output is sensitive to changes in the input
boundary conditions constitute a two-point boundary value parameter.
problem which we solve in MATLAB using the forward-backward
sweep numericalmethod [37]. Discussion of parameter estimates
In order to force a specific final time prevalence, we must Biological parameters. We obtain the model parameters
remove the final time condition lc(tf)~0. Instead, an iterative from the relevant literature on injecting drug use and HCV
bisectionmethodisusedtofindthevalueoflc(tf)thatyieldsthe treatment as well as epidemiological data collected in the UK
desired final timetarget prevalence c(tf). (Table 1). We present results for two common baseline chronic
Numerical solutions with a final time prevalence specified are prevalences:30%and45%(approximatelyequivalentto40%and
run with tf~10 years. Numerical solutions without a final time 60% antibody prevalences, respectively). The exit rate is
prevalence specified are run with tf~50 years, and results determined by the sum of the cessation of injecting rate (calcu-
presentedforthefirst10years.Theuseofanexpandedtimerange latedbyusingtheaverageinjectingduration)andtheIDUdeath
(tf~50) when there is no final time prevalence specified is rate. We use an average injecting duration of 11.5 years
necessarytoensurethatprematureterminationoftreatmentisnot (corresponding to a cessation rate of 8.7% per year) found in
recommended due to the lack of consideration for onward [44]. This estimate for injecting duration is similar to an average
transmissionaftertheinitialdecadeofinterest.Asnewtreatments estimate across England and Wales, and so is likely to be
will likely be introduced in the next decade (with different cost representativeofmanyareasintheUK[45].Arecentstudyfound
implications), itisreasonable tofocuson a 10yeartimeframe. anIDUmortalityrate(duetooverdose,suicide,andothercauses)
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OptimalControlofHepatitisCTreatment
Table1. Biologicalparameter valuesused inthe numericalsimulations.
Parameter Definition Value Units Source
a Proportioninfectionscuredbytreatment 0.625a - [11,13]
v 1/treatmentduration 1.992b peryear [11,13]
d Proportioninfectionsspontaneouslyclear 0.26 - [29]
p Infectionrate 0.1834–0.2334 peryear Fitto30%and45%infectionprevalences[44]
m Exitrate(throughdeathorcessation) 0.095c peryear [44–46]
h Newinjectorentrancerate 95 per1,000IDUsannually Givenvaluetoretainpopulationof1,000IDUs
aAverageofthegenotype1curerate(a1~0:45)andthegenotype2/3curerate(a2=3~0:8).
bExitratecalculatedfromtheaverageofthegenotype1treatmentlengthforrespondersandnonresponders:(a1|48z(1{a1)|12)weeksandthegenotype2
treatmentlength,24weeks.
cBasedonacessationrateof8.7%peryear,andanIDUdeathrateof0.75%peryear.
doi:10.1371/journal.pone.0022309.t001
of0.75%peryear[46].TherateofIDUsenteringthepopulation high coverage rates, due to the assumed increased difficulty in
is calculated from the exit rate to retain 1,000 IDUs in the recruitmentanduptake[33].AllIDUswouldneedtobeantibody
population. tested. The cost of a dried blood spot antibody test is
Treatmentdurationandsuccessdependsonthespecificgenotype approximately »19.84,so»19,840 inapopulation of1000IDUs.
of HCV being treated. In general, SVR is high for IDUs with The additional cost of RNA PCR for those who are chronically
genotype 2 or 3 (75–85%), and lower for genotype 1 (40–50%) infected and will enter treatment is incorporated in the antiviral
[9,11,13,47].Therecommendeddurationoftreatmentforgenotype treatment costs. About one quarter of those who are antibody
2/3 is 24 weeks for both responders or nonresponders, and for positive have spontaneously cleared the acute disease, and would
genotype1thedurationis48weeksforrespondersand12weeksfor needaPCRtesttoconfirmtheirnegativestatus.Atworst,inour
nonresponders[11,13].IntheUnitedKingdom,abouthalfofthe population of 30–45% prevalence, this amounts to about 150
infections are genotype 1, with the remaining half 2 or 3 [11]. IDUsperyear.Atacostof»70.77forthePCRtest,themaximum
Hence, we take an average between the genotype 1 and 2/3 costof PCR testingis»10,615.
parameters for the treatmentsuccess parameter(a), as well as the With current dried blood spot testing technology, HCV testing
treatmentduration(1=v).Finally, [29]performeda meta-analysis canbeimplementedbylocaloutreachservices(suchasneedleand
and found that 26% of infections lead to spontaneous clearance, syringeprogrammes),howevertheimplementationofanintensive
whichweusefortheparameterd.Theparameterpisdetermined testing programme would likely require staff and training, and
for the two different prevalence scenarios by assuming that HCV potentially overheads. There is no published cost estimation for
amongstIDUsisinsteadystate,areasonableassumptionforHCV needleandsyringeprogrammes(inparticulartrainingandstaffing)
amongstIDUstheUK[44]. in the UK [51] so estimates were taken from two other countries
Economic parameters. A summary of the economic (Canada and Ukraine) and costs translated to the UK using the
parameters can be found in Table 2, with the specific maximum 2009PurchasingPowerParity(PPP)IndexprovidedbytheWorld
annualbudgetsconsideredshowninTable3.Programmetreatment Bank[52].Jacobsetal.[53]reportedtheyearlycosttorunalocal
costsareconstructedfromcurrentdrugpriceestimatesandcosting needleandsyringeexchangeprogramme(includingstaff,training,
studies.Dependingonthetypeofpeginterferonalfaused(2aor3b) andoverheads,butexcludingsyringecosts)inEdmonton,Canada
andbody weight, thedrug costsofcombinationpeginteferonand as $253,553 CAD (2009). The programme distributed 565,754
ribvirin treatment is between »12,496–14,221 per person [11]. In needles in a year, but did not document the number of visits per
addition to drug costs, we include patient evaluation, tests, year.Noestimatesweregivenforthenumberofneedlesdistributed
screening, and consultations during and after treatment. Cost- at a time, however the mean number of syringes collected per
effectivenessanalysesforHCVtreatmenthaveestimatedthesetotal contact has been estimated at around 20 [54], leading to
coststobe»760(in2008/2009GBP)forinvestigationsofapatient approximately28,288contactsperyear,or$8.96CADpercontact.
who is considered for treatment, and »810.32–1,084.30 for Usingthe2009PPPconversionfactor(approx.1.76)[52]andthe
consultations depending on treatment length [12]. Therefore, the averageexchangeratein2009(1.78CADto1GBP,www.x-rates.
totalcostsofdeliveringantiviraltreatmentare»14,066–16,064per com),thisresultsinanaveragecostpercontactof»8.86(2009).In
year,andweuseameanvalueof»15,065peryear,varyingthisin thisstudy,alocalvanwasusedforoutreach,whichislikelysimilar
the uncertainty analysis. We calculate the cost per treatment by tothekindofprogrammewhichwouldbeusedintheUKiftheaim
multiplying the yearly treatment cost (which we denote b ) and wastoattainhighcoverage.Wethereforeuseitasthebaseforour
yr
averagetreatmentduration(1/v). programmecostestimation.
The parameter c represents the cost of ensuring 100% A similar analysis on data from the first year of a needle and
0
treatment coverage in the first year, which would involve an syringeprogrammeinUkraine[55]includesbuildingpurchaseand
extensivetestingandrecruitmentprogramme.Programmesearch constructioncosts,givingahighercostpercontactof»22.10(2009).
costsaredifficulttoestimate,asthereisnopublishedliteratureon Of this amount, nearly 25% is comprised of the first year capital
cost analysis of IDU recruitment to services by coverage level. non-reoccurringcosts.However,thisislikelytobeanappropriate
Systemic reviews examining costs of vaccination scale-up pro- upper bound as in subsequent years, especially in situations
grammesshowalackofgoodmethodologicalstudiesandrigorous targetinghighcoverageorwithlowprevalence,amediacampaign
cost analyses [48–50]. Nevertheless, it is reasonable to assume might be necessary. Hence, the repeated inclusion of this 25%
increasingmarginalcostwhenattemptingtoincreasecoverageto excesscost(forcapital,media,orother)seemsappropriate.
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OptimalControlofHepatitisCTreatment
Table2. Economicparameters.
Parameter Definition Scenario Value Source
b10 Antiviraltreatmentcostspertreatment all »15,065=v [12,56,73]
b20 MonetarisedQALYlosspertreatment A,B »5,800=v Seetext
C »0 -
f10 HCVinfectioncostsperyear all »657 [12,56,73]
f20 MonetarisedQALYlossforHCVperyear A,B »3,800 Seetext
C »0 -
c0 Recruitmentandtestingcost(peryearperunitu2) all »40,000 Littledata,seetext[53,55]
r Discountingrateforcostsandhealthutilitylosses(annual) all 3.5% [60]
M Maximumannualbudget(per1,000IDUs) all »50,000–300,000 -
ScenarioA:minimisinghealthservicecostsandHCVhealthutilitylosses(measuredinmonetarisedQALYloss).ScenarioB:minimisinghealthservicecostsandHCV
healthutilitylosseswithafinaltimeprevalencetarget.ScenarioC:minimisingonlyhealthservicecostswithafinaltimeprevalencetarget.
doi:10.1371/journal.pone.0022309.t002
In total, for u~1 (treating at a rate of 100% of the IDU These costs escalate markedly in the later stages of disease, but
population per year), the total cost of programme outreach and given the long timescale of disease progression to cirrhosis
testing of 1,000 IDUs would be »39,315, with a maximum (decades) and the average injecting duration (approximately 11
estimate of »52,555. We therefore use c ~»40,000 and varythis years)itisassumedallinfectedactiveIDUsareeitherinthemild
0
from»39,000 to »53,000 intheuncertainty analysis. or moderate stage.
The infection costs (f ) associated with a person with mild to InordertoquantifythequalityoflifereductionforactiveIDUs
10
moderatechronicHCVareapproximately»657peryear[12,56]. withHCVandforactiveIDUswithHCVundergoingtreatment,
Table3.Netcosts,netQALYsgained,andcostperQALYgainedascomparedtonotreatmentprogrammeforthe30%and45%
HCVprevalence scenariosand various optimisationprogrammes.
30%HCVprevalence 45%HCVprevalence
Maxannual Net NetQALY Cost(»)per Net NetQALY Cost(»)per
Scenario budget(») costs1(») gain2 QALYgained costs1(») gain2 QALYgained
A 50,000 220,028 135 2148 - - -
100,000 299,210 287 2346 - - -
150,000 2256,459 462 2555 327,996 293 1,120
200,000 2520,789 668 2780 359,397 413 870
250,000 - - - 332,703 550 605
300,000 - - - 227,253 710 320
B 50,000 - - - - - -
100,000 249,338 203 2243 - - -
150,000 246,459 216 2215 288,304 233 1,237
200,000 244,793 224 2200 316,510 251 1,263
250,000 - - - 335,742 263 1,277
300,000 - - - 349,527 272 1,285
C 50,000 - - - - - -
100,000 240,348 159 2254 - - -
150,000 233,130 145 2228 213,173 179 1,192
200,000 223,268 137 2169 190,679 163 1,167
250,000 - - - 183,950 154 1,192
300,000 - - - 183,212 148 1,239
Theoptimal10yearprogrammeisdeterminedforeachscenario,andthencostsandQALYsarecalculatedforafurther40years(fora50yeartimehorizon)inorderto
accountfortheonwardpreventionbenefitsofthetreatmentprogramme.ScenarioA:minimisinghealthservicecostsandHCVhealthutilitylosses(measuredin
monetarisedQALYloss).ScenarioB:minimisinghealthservicecostsandHCVhealthutilitylosseswithafinaltimeprevalencetarget.ScenarioC:minimisingonlyhealth
servicecostswithafinaltimeprevalencetarget.
1Netcosts=healthcarecostsover50yearswiththe10yeartreatmentprogramme-healthcarecostsover50yearswithnotreatment.Healthcarecostsaredefinedas
programme(antiviraltreatmentandsearch)costsaswellasHCVinfectionrelatedcosts.
2NetQALYgain=QALYsgainedover50yearswiththe10yeartreatmentprogramme-QALYsgainedover50yearswithnotreatment.
doi:10.1371/journal.pone.0022309.t003
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OptimalControlofHepatitisCTreatment
we monetarise the QALY loss in each case. This approach examineseveralbudgetscenarios,fromamaximumyearlybudget
excludesnon-healthbenefitsassociatedwiththeintervention(such of »50,000to»300,000per 1,000 IDUs.
asimpactonproductivity),butnonethelessallowsustoarriveatan Parameter ranges for the uncertainty analysis. During
estimate of the net monetary benefit of the programme. In the the uncertainty analysis, each parameter istaken tobe uniformly
UK,thisispossibleusingthe‘willingness-to-pay’thresholddefined distributed, with ranges as shown in Table 4 for the biological
by NICE, which essentially determines the amount the UK’s parametersvaried.Forthecostparameters,therangesforb ,b ,
10 20
National Health Service is willing to pay for a treatment. The f , and f were taken to be plus and minus 10% of the values
10 20
current willingness-to-pay threshold (i.e. the UK’s monetarised giveninTable1.Duetothehighuncertaintyinc ,therangeused
0
value of a QALY) is approximately »20,000 per QALY gained was from»39,000to»53,000.
[57].Estimatesforthehealthstateforanon-orex-IDUwithmild
chronic HCV infection are around 0.78 QALYs per year,
Results
depending on the evaluation method [12]. Healthy active IDUs
tendtohavealowerbaselinequalityoflifethannon-orex-IDUs, Ifonlyhealthservicecostsareminimised(withnorequirement
with estimates at around 0.85 QALYs per year [56], i.e. a 15% to reduce health utility losses or reach a final time prevalence
reductioninqualityoflifefromthehealthynon-IDUstate.Weuse reduction), the optimal solution is to treat no one. In each of the
this15%reductiontocalculatethehealthstatereductionforactive scenarios considered below, we add further considerations to this
IDUswithHCV,inlinewithothereconomicevaluationsofHCV baseline case.
in active IDUs [56]. Hence, we estimate that the health state for
anactiveIDUwithHCVis0.66QALYsperyear,i.e.anabsolute
Scenario A: Minimising health service costs and HCV
reduction of 0.19 QALYs per year from the healthy active IDU
health utility losses (f ,b w0)
state, meaning that the (monetary) benefit from successfully 2 2
treating a chronically infected IDU is approximately »3,800 per Thisscenariorepresentsan‘ideal’situation,wherepolicymakers
year (0.19 QALYs/year|»20,000/QALY). In other words, are motivated to minimise health service costs and health utility
»3,800 is the estimated monetary benefit foregone per year by losses associated with HCV, limited only by budget restrictions.
nottreatinganinfectedIDU,henceitisessentiallyanopportunity The optimal programme is to spend the maximum possible
cost, represented by themonetarised value of QALYslost due to amounteachyearonthetreatmentprogramme,whichsucceedsin
non-treatment. On the other hand, being on HCV antiviral reducing infectionsandQALY losses.
treatment also results in a QALY reduction (from infection level) If the objective is to minimise health service costs and health
ofabout0.10QALYsperyearforthedurationoftreatment[12], utilitylossrelatedtoHCVinfection,Figure1showsthepotential
hence we assume the net reduction from the healthy active IDU impact of the optimal treatment programmes for various
state to an active IDU with HCV on treatment is approximately maximum yearly programme budgets with a 30% baseline
0.29 QALYs per year, and therefore the cost in terms of prevalence. In all budget scenarios, the optimal number treated
monetarised value of QALYs lost due to treatment is approxi- increases over time, due to the discounting treatment costs and
mately»5,800peryear.ThesemonetarisedvaluesofQALYslost subsequentabilitytoincreasetreatmentallocation.Figure1shows
due to either infection or treatment make up the HCV health that depending on the maximum annual budget (»50,000 to
utilitylosses.Duetothelackofevidencesurroundingutilityvalues »200,000 per 1,000 IDUs annually), the number of treatments
following treatment of those with mild (in particular, asymptom- allocated yearly varies from 7 to 37. With a »100,000 maximum
atic) HCV, we use published estimations that SVR from mild annualprogrammebudget,theprevalencedecreasesfrom30%to
HCV resultsina return tothenormalhealth state [58,59]. about 21% at year 10 with a total of 43 cases (per 1,000 IDUs)
Health costs and utility losses are discounted at an annual averted. This equates to a total programme spend per infection
effective rate of 3.5% per year, meaning that our cost weights avertedof»23,597.Theinfectionrelatedcostsreach»1.42million
decrease through time [60]. This allows for the correct cost by 10 years. Net monetary benefit at 10 years is »291,088. The
adjustment when using a maximum implementation yearly costperQALYgainedwitha50yeartimehorizonis2»346,with
budget. All costs are presented in UK pounds (GBP, ») in fiscal negativenetcostsandpositivenetQALYsgainedascomparedto
year 2008/2009 values and updated with the Hospital and no treatment, indicating that the programme is cost saving
Community Health Services Pay and Prices Index [61]. We (Table 3).
Table4. LHSsensitivityanalysisonthe modelwith constantcontrol (n~1,000).
Parameter Range(Min,Max) Source PRCCa10yrprevalence(30%baseline) PRCCa10yrprevalence(45%baseline)
a1 (0.40,0.50)b [11,13,47] 20.6839* 20.6904*
a2=3 (0.75,0.85)b [9,11,13] 20.7153* 20.6762*
m (0.05,0.1429)c [44–46,74] 0.4271* 0.7234*
u (0,1) - 20.9992* 20.9990*
d (0.22,0.29) [29] 20.0387 20.0338
aPartialRankCorrelationCoefficient,
baiscalculatedasanaverageofthegenotypetreatmentsuccessrates,a1anda2=3.
cTheexitrateiscalculatedfromtherangeofthegenotype1treatmentlengthforrespondersandnonresponders:(a1|48z(1{a1)|12)weeksandthegenotype2
length:24weeks.
*denotesap-valueofbelow0.05.
doi:10.1371/journal.pone.0022309.t004
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OptimalControlofHepatitisCTreatment
When the budget constraint isincreased to »200,000 per year, Increasingthebudgetto»200,000decreasesthedurationofthe
the prevalence decreases to below 12% within a decade. optimalprogrammetofiveyears,withalowerleveloftreatments
Additionally, 96 infections are averted by 10 years, resulting in a (26–30)forthefirstfouryears,tailingoffwith5treatmentsinthe
reductionincostperinfectionavertedto»20,945.Ascomparedto fifthyear.Thisstrategyincreasestheprogrammecostto»830,051,
the lower budget scenario, the increased programme costs are but averts more infections (50 by year 10), resulting in a lower
partly offsetbyreduced infection-related costs of»1.17million at programme cost per infection averted of »16,737. Furthermore,
10 years. Net monetary benefit increases to »630,111. Addition- theinfectionrelatedcostsarereducedto»1.36million.However,
ally, increasing the budget saves more in terms of cost over a 50 the cost per QALY gained is slightly higher than in the lower
yeartimehorizonandgainsmoreQALYs,resultinginacostper budget scenario, at 2»200 due to the higher programme cost
QALY gainedof 2»780,indicating thatthisprogrammeismore (Table 3).
cost-effective thanonewithalowerbudget(Table3).Resultsare With a 45% baseline prevalence scenario, the optimal pro-
qualitatively similar for the 45% baseline prevalence scenarios, grammeisstillaninitialprogramme,buttheprogrammeduration
however with the same yearly budget the relative reduction in to reduce prevalence by 20% is longer (Supporting Information
prevalence is smaller, there are fewer infections averted, and the Figure S2). With an annual budget of »200,000, the programme
costperavertedinfectionishigher(SupportingInformationFigure
spans7yearsinsteadofthe5yearsinthelowerprevalencescenario.
S1). Furthermore, the programme results in fewer QALY gains,
Total programme and infection costs are substantially higher, at
and thecostper QALYgained ishigher, at »870(Table 3).
»1.35 million and »2.1 million, respectively. Additionally, fewer
Overall, increasing the yearly budget results in greater short-
infections are averted (40 in 10 years), and the cost per infection
termreductionsinprevalence,increasedinfectionsaverted,lower
averted is nearly double that of the 30% prevalence scenario.
programme cost per infection averted, and substantial reductions
Similarly,thecostperQALYincreasesto»1,263(Table3).
in infection related costs at year 10 due to the subsequent
preventioneffect. Foragivenyearlybudget,theimpactishigher
Scenario C: Final time prevalence reduction only
in lowerprevalence areas.
(f ~b ~0, c(t )~0:8c )
Theoptimalprogrammestrategyandresultspresentedaboveare 2 2 f 0
identicaliftheobjectiveisinsteadtominimiseprevalenceattheend In the least ‘ideal’, but perhaps most relevant and ‘real-world’
ofyear10whilebeingconstrainedbythesamebudgetrestrictions. scenario, policymakers may be motivated solely by a political
Inotherwords,theprogrammewhichminimisesprevalenceatthe commitmenttoreduceprevalencebyaspecificamount,neglecting
endofyear10istheonewheretheentirebudgetisspenteveryyear, thelossofhealthutilitiesassociatedwithinfection.Ascomparedto
andthisstrategyalsohasthelowestcostperQALYgained. Scenario B, where there is a bias towards early and intensive
treatment to reduce the cumulative number of infections,
Scenario B: Minimising health service costs and HCV neglecting health utility losses results in very different optimal
programmes (Figure 3). In the 30% prevalence scenario, with a
health utility losses with a final time prevalence target
(f ,b w0, c(t )~0:8c ) maximumbudgetat»200,000peryear,theoptimalprogrammeis
2 2 f 0 implemented in the final three years only, increasing treatment
Inaless‘ideal’scenario,policymakerscouldbemotivatedbya
coveragefrom11%to16%.Thisresultsintreating13–36people
politicalconstraintaswell,andspecifyaneedtoreduceprevalence
peryear,costingtheprogrammeonly»478,369,butinfectioncosts
by a specific amount within 10 years. Therefore, in this case we
reach»1.62million.Sincethecostsassociatedwithinfectionsare
examine the optimal timing and intensityof a programme where
much lower when neglecting loss of health utility, the optimal
thedesireistoachieveaspecificprevalencereductionbythefinal
strategy shifts toward achieving the target prevalence with fewer
time, while also minimising total health service costs and HCV
treatments, whereas with health utility losses included in the
health utility losses. These objectives are again constrained by
objective,theoptimalstrategyseekstodecreaseagreaternumber
yearly budget restrictions. In these simulations, the final time
of the infections by treating earlier instead of later. Hence, the
prevalence is specified as a necessary condition, such that the
prevalenceatt ~10yearsisreducedbyarelative20%(so,from numberofinfectionsavertedisonly8per1,000IDUsatyear10,
f
andthecostperinfectionavertedissubstantiallyincreasedtoover
30% to24% or45%to36%).
»62,324. Importantly, in this scenario, the programme does not
Numerical solutions indicate that the best strategy with a final
result ina netmonetary benefit at 10years, with costs exceeding
timetarget including costsrelated tohealth statereductions isan
benefits throughout the decade. The cost per QALY gained
initial, intense programme (Figure 2). In the 30% baseline
(2»169) is slightly higher than Scenarios A and B, with fewer
prevalence scenario, the »50,000 maximum budget scenario is
QALYsgained, andhighercosts (Table 3).
not sufficient to result in a 20% relative reduction in prevalence.
With an annual budget of »100,000, the optimum is an 8 year Decreasing the yearly budget to »100,000 results in a longer
programmeofincreasingtreatmentcoverage(expandingfrom4% programme, lasting years 5 through 10, increasing treatment
to8%ofinfectedIDUs),treating13–17peopleperyearinthefirst coverage from 5% to 8%. Reducing the budget increases pro-
seven years where the full budget is spent, and 7 people in the grammecosts(to»560,241),butdecreasesinfectioncoststo»1.57
eighth year (Figure 2). This programme results in an initial swift million. Furthermore, decreasing the maximum annual budget
decrease in prevalence, slightly overshooting the 10 year target increases the treatment programme duration, resulting in more
prevalence in year 8, and eventually rebounding to the target by infections averted (nearly 16 per 1,000 IDUs by year 10).
theendofyear10.Theprogrammeaverts37infectionsresulting Additionally, the cost per infection averted is substantially less,
in a programme cost per infection averted of »19,888. The total atjustover»35,087.Thisindicatesthatalthoughahigherannual
costs of the programme (treatment and search) reaches just over budget canachieve thesameprevalencereductionwithashorter
»740,803withtheinfectionrelatedcostsreaching»1.44million.In programmeduration,achievingearlier prevalencereductions (for
this scenario, net costs as compared to no treatment are still example, by treating fewer but initiating the programme earlier)
negative, but fewer QALYs are averted as compared toScenario resultsinmoreinfectionsavertedandareducedcostperinfection
A, and cost per QALY gained is higher at 2»243, however the averted. The qualitative shape of the programme (delayed until
programme isstill cost-saving(Table 3). final years) isunchangedifdiscounting isneglected.
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OptimalControlofHepatitisCTreatment
Figure 1. Scenario A: Minimising health service costs and HCV health utility losses. Simulations are with a 30% baseline prevalence,
showing(A)programmecoverage,(B)prevalencereductions,(C)numberoftreatments,(D)totalhealthservicecosts(comprisedofprogrammecosts
and infection costs), (E) infections averted, and (F) net monetary benefit. Parameters used are as shown in Tables 1–2, with f ~3,800,
20
b ~5,800|(1=v),t ~50,andwithnofinaltimeprevalenceconstraint.
20 f
doi:10.1371/journal.pone.0022309.g001
For the 45% prevalence scenario, the optimum is also a late higherthaninscenariosAandB,orforanyofthelowprevalence
initiated programme, but with a longer programme duration to scenarios,atabout»1,200.
achieve the target prevalence (Supporting Information Figure S3).
Theprogrammesbegininyears4–8andescalateuntilthefinalyear.
Cumulative programme costs are roughly equal (at the »200,000 Uncertainty analysis of optimal control solution
annualbudget)and15%higher(atthe»150,000annualbudget)than Theimpactofuncertaintyofourparametersontheoptimalcontrol
in the 30% prevalence scenario, and the cost per QALY gained is andprevalencereductionsforvariousmaximumbudgetscenariosare
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