Mathematical Insights in Evaluating State Dependent Effectiveness of HIV Prevention Interventions

Bull Math BiolDOI 10.1007/s11538-013-9824-7

O R I G I NA L A RT I C L E

Mathematical Insights in Evaluating State DependentEffectiveness of HIV Prevention Interventions

Yuqin Zhao · Dobromir T. Dimitrov · Hao Liu ·Yang Kuang

Received: 1 May 2012 / Accepted: 28 January 2013© Society for Mathematical Biology 2013

Abstract Mathematical models have been used to simulate HIV transmission andto study the use of preexposure prophylaxis (PrEP) for HIV prevention. Often a sin-gle intervention outcome over 10 years has been used to evaluate the effectivenessof PrEP interventions. However, different metrics express a wide variation over timeand often disagree in their forecast on the success of the intervention. We developa deterministic mathematical model of HIV transmission and use it to evaluate thepublic-health impact of oral PrEP interventions. We study PrEP effectiveness withrespect to different evaluation methods and analyze its dynamics over time. We com-pare four traditional indicators, based on cumulative number or fractions of infectionsprevented, on reduction in HIV prevalence or incidence and propose two additionalmethods, which estimate the burden of the epidemic to the public-health system. Weinvestigate the short and long term behavior of these indicators and the effects of keyparameters on the expected benefits from PrEP use. Our findings suggest that public-health officials considering adopting PrEP in HIV prevention programs can makebetter informed decisions by employing a set of complementing quantitative metrics.

Keywords HIV transmission · HIV prevalence or incidence · PrEP interventions ·ODE model

Y. Zhao (�) · H. Liu · Y. KuangDepartment of Mathematics, Arizona State University, Tempe, AZ 85287, USAe-mail: [email protected]

H. Liue-mail: [email protected]

Y. Kuange-mail: [email protected]

D.T. DimitrovStatistical Center for HIV/AIDS Research and Prevention, Fred Hutchinson Cancer Research Center,Seattle, WA 98109-1024, USAe-mail: [email protected]

Y. Zhao et al.

1 Introduction

In 2010, evidence from two different randomized clinical trial suggested that pre-exposure prophylaxis (PrEP) products based on antiretroviral drug Tenofovir takenorally as a pill (oral PrEP) or applied topically in the form of gel (vaginal micro-bicides) can help prevent HIV. First, the CAPRISA 004 trial demonstrated a 39 %(95 %CI, 6 % to 60 %) overall decrease in HIV incidence among women in theVMB arm of the trial who were advised to use the product before and after each sexact (Karim et al. 2010). Later, the Global iPrEx trial of a daily use of a combina-tion of two oral antiretroviral drugs, emtricitabine and tenofovir disoproxil fumarate(FTC/TDF) demonstrated a 43.8 % efficacy (95 %CI, 15 % to 63 %) to reduce HIVacquisition among men-who-have sex with men (MSM) (Grant et al. 2010). Althoughthese products await confirmation from another trial to move toward licensure or labelchange, there is already a broad discussion on what will be population-level benefitsfrom wide-scale PrEP use in high prevalence settings.

Mathematical models have been used to simulate HIV transmission and to studythe use of chemoprophylaxis among MSM (Desai et al. 2008; Supervie et al. 2010).Deterministic mathematical models of HIV heterosexual transmission stratified bygender have been analyzed in Dimitrov et al. (2010, 2011), Abbas et al. (2007), Vis-sers et al. (2008), Pretorius et al. (2010), Wilson et al. (2008). Often a single inter-vention outcome based on cumulative number or fractions of infections prevented, onreduction in HIV prevalence or incidence has been used to evaluate the effectivenessof PrEP interventions. These indicators express a wide variation over time and oftendisagree in their forecast on the success of the intervention (Dimitrov et al. 2010;Wilson et al. 2008). Therefore, the conclusions of many modeling studies are signifi-cantly influenced by the choice of the evaluation method and the period of evaluation.In particular, it has been pointed out that indicators based on prevented infectionstend to show mixed results over time due to their sensitivity to changes in populationdynamics (Dimitrov et al. 2011).

In this paper, we develop a deterministic mathematical model of HIV transmis-sion to evaluate the public-health impact of oral PrEP interventions, compare PrEPeffectiveness with respect to different evaluation methods, and analyze its dynamicsover time. We compare four traditional evaluation methods including relative reduc-tion in HIV prevalence and incidence, which avoid the ambiguity associated withcommonly used indicators based on the absolute number of prevented infections. Weconsider two additional methods, which estimate the burden of the epidemic to thepublic-health system. We then investigate the short term and long term behavior ofthese indicators and the effects of key parameters on the expected benefits from PrEPuse. The effects of demographic, behavioral, and epidemic parameters on the PrEPimpact are studied in a multivariate sensitivity analysis.

2 Model Description and Analysis

In our models (see Fig. 1), the population is divided into two major classes, PrEPusers (superscript p, Sp + Ip) and those that do not use PrEP (S + I ), and further

Evaluating Effectiveness of HIV Prevention Interventions

Fig. 1 Flow diagram of a PrEPintervention for the model (1)formulation

stratified according to their HIV status into susceptible (S, Sp) and infected (I , Ip).Individuals who develop AIDS are accumulated in nonsexually active class A. Indi-viduals join the community (reaching sexual maturity) and departure from the sexu-ally active population at constant rates (Λ, μ). A proportion k of the new recruits startusing PrEP. PrEP users are assumed to strictly follow the prescribed regimens. Themodel which assumes that PrEP reduces both susceptibility and infectiousness of theusers (“dual-protection” model) is formulated by the following system of differentialequations:

dSp

dt= kΛ − (1 − αs)β

SpI

N− (1 − αs)(1 − αi)β

IpSp

N− μSp,

dS

dt= (1 − k)Λ − β

IS

N− (1 − αi)β

IpS

N− μS,

dIp

dt= (1 − αs)β

ISp

N+ (1 − αs)(1 − αi)β

IpSp

N− (μ + d)Ip,

dI

dt= β

IS

N+ (1 − αi)β

IpS

N− (μ + d)I.

(1)

Since the differential equations for these four compartments are independent fromthe AIDS class (A), we do not include it in our ODE system. Here, N = Sp + S +Ip + I represents the sexually active population and αs (αi ) measures the efficacy ofPrEP in reducing susceptibility (infectiousness) of PrEP users. The cumulative HIVacquisition risk per year β is calculated based on the HIV risk per act (ba) with aHIV-positive partner and the average number of sex acts per year (n):

β = 1 − (1 − ba)n.

Cumulative acquisition risk (β) is an increasing function with respect to HIV-acquisition risk per act ba and average number of sexual acts per year n.

PrEP is introduced at time (t = 0) in a population with N(0) = 1,000,000 andHIV-prevalence (P ). We assume that PrEP is initially adopted by a fraction k1 of theindividuals and that the initial fraction of HIV-positive individuals is reduced by θ asa result of preenrollment HIV screening:

Sp(0) = k1(1 − P)N(0),

S(0) = (1 − k1)(1 − P)N(0),

Ip(0) = (1 − θ)k1PN(0),

I (0) = (1 − (1 − θ)k1

)PN(0).

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Table 1 Parameter description and baseline values

Par. Description Value Ref.

d HIV carrier’s annual rate ofprogression to AIDS

0.1302 fitted (Mid-yearpopulation estimates2011)

Λ Annual rate at which individualsbecome sexually active

38094 calc. (Mid-year populationestimates 2011)

1μ Time (in years) to remain sexually

active

10.0250 fitted (Mid-year

population estimates2011)

ba HIV acquisition risk per act 0.0030 fitted (Mid-yearpopulation estimates2011)

n Number of sexual acts per year perindividual

65.8494 fitted (Mid-yearpopulation estimates2011)

β Cumulative HIV-acquisition risk β(n, ba) calc.

N(0) Initial sexually active population 106 assumed

P Initial HIV prevalence 0.166 Mid-year populationestimates (2011)

k1 Initial PrEP coverage 0.2 assumed

θ Reduction in the initial fraction ofHIV positive individuals as a resultof preenrollment screening

0.5 assumed

k Proportion of the new recruits thatstart using PrEP

k = k1 assumed

αs Efficacy of PrEP in reducingsusceptibility of PrEP users

0.5 assumed

αi Efficacy of PrEP in reducinginfectiousness of PrEP users

0.5 assumed

The initial distribution of the epidemic classes may not be critical for the asymp-totic behavior of the system, but it is essential for the impact indicators calculatedover fixed periods of time after the start of the intervention. To isolate the impactof the choice of the evaluation method, we simplify the intervention schedule andassume instantaneous uptake of PrEP at a predetermined level (k = k1).

Clinical trials in HIV prevention are usually designed to evaluate the efficacy ofthe tested products in reducing susceptibility (αs ) only. Although a reduction in infec-tiousness (αi ) is plausible, it is not verifiable since all HIV positive participants areimmediately withdrawn from the product. Moreover, the biomedical products cur-rently in testing are based on antiretroviral drugs, and are not recommended for useby infected individuals due to the risk of drug resistance development. Therefore,the majority of the modelers assume unidirectional PrEP protection (αi = 0), whichmeans that using PrEP has no effect on the infectiousness or that infected individualsdo not take PrEP anymore. This scenario may also represent the idea of control of thePrEP usage by the HIV-positive individuals since fast removal of the infected usersfrom PrEP is the equivalent of setting αi = 0. To address that possibility, we consider

Evaluating Effectiveness of HIV Prevention Interventions

a “single-protection” model in which the variable Ip is removed from the baselinemodel as follows:

dSp

dt= kΛ − (1 − αs)β

SpI

N− μSp,

dS

dt= (1 − k)Λ − β

SI

N− μS,

dI

dt= β

SI

N+ (1 − αs)β

SpI

N− (μ + d)I

(2)

with initial conditions:

Sp(0) = k1(1 − P)N(0),

S(0) = (1 − k1)(1 − P)N(0),

I (0) = PN(0).

In our analysis, HIV epidemics are simulated in presence and in absence of PrEP.If PrEP is not available, the “no intervention” model is reduced to the followingsystem:

dS

dt= Λ − β

SI

N− μS,

dI

dt= β

SI

N− (μ + d)I

(3)

with S(0) = (1 − P)N(0) and I (0) = PN(0).

2.1 Modeling Assumptions

Several important assumptions are incorporated into the model:

• The HIV prevalence in the whole population is representative for the HIV preva-lence among each gender, i.e., the chance to have a HIV-positive partner is propor-tional to the total HIV prevalence.

• Individuals are assumed to have a fixed number of sex acts per year.• Sexual behavior of an individual does not change if he/she starts using PrEP but

sexual activity stops once AIDS is developed.• The use of PrEP reduces both HIV susceptibility and infectiousness and by this

reduces the HIV acquisition risk per sex act.• We assume perfect adherence to PrEP: individuals who start using PrEP continue

to follow the prescribed regimen indefinitely. However, the scenario with no reduc-tion of infectiousness due to PrEP (αi = 0) is equivalent to immediate withdrawalfrom PrEP after HIV acquisition.

• The use of other HIV prevention measures including condom use, male circumci-sion, and ARV treatments are not considered separately in our model. Their effectson HIV transmission are aggregated in the HIV acquisition risk per act.

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2.2 Equilibrium Analysis

The “no intervention” model (3), has two steady states: infection free equilibrium(Λ

μ,0) and endemic equilibrium ( 1

β−dΛ,

β−(μ+d)(μ+d)(β−d)

Λ) when β > (μ + d). They ex-

change stability when the basic reproduction number R0 = βμ+d

crosses the thresholdof one. In epidemiology, the basic reproduction number (sometimes called basic re-productive rate or basic reproductive ratio) of an infection is the mean number ofsecondary infections a typical single infected individual will cause in a populationwith no immunity to the disease in the absence of interventions to control the in-fection. It is often denoted by R0. This metric is useful because it helps determinewhether or not an infectious disease will spread through a population. In general, ifR0 > 1, then the infection free steady state is unstable and the infection persists inthe population while if R0 < 1 then the infection free steady state is stable and the in-fection may be eradicated. That is exactly the case for the HIV epidemic modeled bythe “no intervention” model (3). A simple application of the Dulac criterion ensuresthat model (3) does not generate nontrivial periodic solution.

The “single-protection” model (2) has an infection free steady state ( kμΛ, 1−k

μΛ,0).

The “dual-protection” model (1) has an infection free steady sate ( kμΛ, 1−k

μΛ,0,0).

However, the positive steady states for these two models, when they exist, are toocomplicated to be expressed explicitly.

2.3 Model Parametrization

We used demographic and HIV prevalence data representative for the sexually activepopulation (15–49 years old) in South Africa for the period between 2001 and 2011provided by the Statistical Institute of South Africa (Mid-year population estimates2011) to parameterize our models in the scenario without PrEP.

First, we estimate the recruitment rate in the sexually active population (Λ).We base our calculations on the approximated number of 15-year olds( population aged 15 to 19

5 = 51754005 = 1035100) and the total population size (27,172,400)

aged 15 to 49, in year 2011. In our model without PrEP, we assume initial total sex-ually active population to be N = 106. Therefore, we scale the estimated entrancerate to obtain the recruitment of the sexually active population (Λ) in our model:Λ = 106 · 1035100

27172400 ≈ 38094, which we use in the epidemic simulations.Next, we fit the projected HIV prevalence I

S+Iby the model without PrEP to the

2001–2011 prevalence data from South Africa (Mid-year population estimates 2011).We use the Matlab built-in function “fminsearch” to do the data fitting, with error

measurement∑n

i=1 |psi−pi |n

, where psi represents the HIV prevalence from modelsimulation, pi represents the HIV prevalence from data, and n represents the numberof data points. Starting with initial parameter values borrowed from published studies:ba = 0.0038 (Boily et al. 2009), n = 80 (Wawer et al. 2005; Kalichman et al. 2009),μ = 1/35 (AIDS epidemic update 2009), and d = 1/10 (Morgan et al. 2002; Porterand Zaba 2004), we obtain the following parameter set, which fits best the prevalencedata from year 2001 to year 2011: ba = 0.0030, n = 65.8494, μ = 0.0250, and d =0.1302 (with error of data fitting = 0.0737). Figure 2 shows the HIV prevalence data

Evaluating Effectiveness of HIV Prevention Interventions

Fig. 2 (a) HIV prevalence among sexually active population in South Africa for the period 2001–2011from data and fitted with the “no intervention” model; (b) Long-term projections of the HIV prevalencebased on fitted “no intervention” model

Fig. 3 (a) Long-term compartment dynamics of the “dual-protection” model and (b) comparison of theepidemic dynamics projected by the “dual-protection,” “single-protection,” and “no intervention” modelsusing baseline parameter values from Table 1

and the best-fitting estimates obtained by the “no intervention” model for the period2001–2011 (Fig. 2(a)) as well as its long-term projections (Fig. 2(b)).

2.4 Epidemic Projections

We present the epidemic dynamics obtained by the “dual-protection” model (1) us-ing the baseline parameter values from Table 1 in Fig. 3(a) and compare them withthe projections of the “single-protection” (2) and “no intervention” (3) models inFig. 3(b).

We observe that all simulations approach steady states after a period of 200 years.A 50 % efficacious PrEP, which reduces both susceptibility and infectiousness of itsusers will stabilize on disease-free equilibrium if PrEP is used consistently by 20 %

Y. Zhao et al.

Table 2 Indicator description

Indicator Name Description

CI (T ) Cumulative indicator Cumulative number of infections prevented over theperiod [0, T ] due to the usage of PrEP

FI (T ) Fractional indicator Fraction of infections prevented over the period[0, T ] due to the usage of PrEP

PI (T ) Prevalence indicator Reduction in HIV-prevalence at time t = T due to theusage of PrEP

aII (T ) Incidence indicator Reduction in the annual HIV incidence at time t = T

due to the usage of PrEP

CI (T ) Reduction indicator Reduction in the projected number of infections attime t = T due to the usage of PrEP

FI (T ) Fractional reduction indicator Fraction of the projected number of infections attime t = T reduced due to the usage of PrEP

of the all sexually active individuals. A unidirectional PrEP protection, simulated bythe “single-protection” model, will not be enough to eliminate HIV from the SouthAfrican population but will reduce the infected population significantly. However, itis not straightforward to evaluate the population level impact of PrEP from the long-term epidemic projections. Therefore, it is very important to find biologically reason-able metrics to quantify the effectiveness of PrEP interventions. We introduce severalindicators and study their behavior under different scenarios over various time peri-ods. We also compare the indicators readings for different interventions to understandhow the choice of metric and/or duration affects the relative public-health impact ofPrEP projected with mathematical models.

3 Effectiveness Indicators

The impact of PrEP in our analysis is evaluated by the quantitative indicators de-scribed in Table 2. The first four indicators are widely used in modeling studies toevaluate the impact of interventions over fixed periods [0, T ]. The cumulative and thefractional indicators measure the intervention effectiveness based on the infectionsprevented in scenarios with PrEP compared to scenarios without PrEP. The preva-lence and incidence indicators measure the reduction of the projected HIV prevalenceand incidence due to PrEP. We propose the last two evaluation methods based on thereduction of the number of infected individuals as they are closely related to the eco-nomic burden of the HIV epidemic on the public health system at community andstate level since the money allocated for HIV treatment is proportional to the size ofthe infected population.

Predictions of mathematical models based on quantitative indicators are often usedto estimate the effectiveness of novel interventions and to compare the expected ben-efits from different prevention options. The analytical conclusions in favor of specificoption are usually based on evaluations of the indicators over a few fixed periods ofintervention time, most likely 10 years but almost certainly between 5 and 30 years.

Evaluating Effectiveness of HIV Prevention Interventions

Fig. 4 Comparison of the indicators projections for two PrEP interventions over a 50-year period. In-tervention 1 assumes that θ = 0 and αs = αi = 0.5. Intervention 2 assumes that θ = 1, αs = 0.5, andαi = 0.9. All other parameters are fixed on their baseline parameter values from Table 1

However, all indicators vary over time and may express different preferences whenused to decide between comparable prevention programs. We illustrate the idea with acomparison of the indicator dynamics for two hypothetical PrEP interventions. Inter-vention 1 assumes no control of the PrEP use by HIV-positive individuals (θ = 0) and50 % PrEP efficacy in reducing both susceptibility and infectiousness (αs = αi = 0.5)while Intervention 2 requires a negative HIV test as a condition for prescribing PrEP(θ = 1) and better PrEP efficacy in reducing infectiousness (αs = 0.5, αi = 0.9). Eachof the incidence, prevalence, and fractional indicators shows increasing effectivenessof both interventions over 50 years after initiation of PrEP (Fig. 4) with more benefitsattributed to Intervention 1 initially, but higher impact of Intervention 2 in a long-term. However, they disagree on the timing when the advantage of the Intervention 1ends. For instance, a preference to Intervention 2 is given after 17 years of PrEP useif based on reduction in HIV incidence and after 22 years if based on reduction inHIV prevalence. The public-health impact of Intervention 1 measured in terms of cu-mulative fraction of prevented infections remains higher compared to Intervention 2for up to 32 years which is substantially longer than the evaluation periods used inthe majority of the quantitative analyses. Therefore, if PrEP is evaluated over periodsbetween 17 and 32 years the choice of quantitative indicator is critical. We take acloser look at the key drivers of those discrepancies in the indicators’ dynamics.

3.1 Indicator Expressions

To utilize the calculation of the cumulative indicators, we need to keep track of thecumulative number of new infections. For this reason, we add two equations to the“dual-protection” model (1):

d(IpNew)

dt= (1 − αs)β

SpI

N+ (1 − αs)(1 − αi)β

SpIp

N,

d(INew)

dt= β

SI

N+ (1 − αi)β

SIp

N,

(4)

and add an equation to the “single-protection” model (2):

d(INew)

dt= β

SI

N+ (1 − αs)β

SpI

N(5)

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with initial conditions INew(0) = IpNew(0) = 0, and INew(0) = 0, respectively. These

new variables (IpNew and INew) represent cumulative HIV infections in PrEP users and

non-users, respectively.If PrEP is not available, the “no intervention” model becomes:

dS

dt= Λ − β

SI

N− μS,

dI

dt= β

SI

N− (μ + d)I,

d(INew)

dt= β

SI

N

(6)

with initial conditions S(0) = (1 − P)N(0), I (0) = PN(0), and INew(0) = 0.We proceed with analysis of the behavior of the indicators assuming “dual pro-

tection.” For the remainder of the paper, we use [ ] to denote variables from themodel without PrEP (6) and [ ]DP for variables from the “dual-protection” modelwith PrEP (4). Using these notations, the qualitative indicators have the followingexpressions:

CI (T ) =∫ T

0

([d

dtINew(t)

]−

[d

dtI

pNew(t) + d

dtINew(t)

]

DP

)dt,

FI (T ) =∫ T

0 ([ ddt

INew(t)] − [ ddt

IpNew(t) + d

dtINew(t)]DP) dt

∫ T

0 [ ddt

INew(t)]dt,

PI (T ) = 1 − [ Ip(T )+I (T )Sp(T )+S(T )+Ip(T )+I (T )

]DP

[ I (T )S(T )+I (T )

]

aII (T ) = 1 − [∫ T +1T [ d

dtI

pNew(t)+ d

dtINew(t)]dt

Sp(T )+S(T )]DP

[∫ T +1T

ddt

INew(t) dt

S(T )]

,

CI (T ) = [I (T )

] − [Ip(T ) + I (T )

]DP,

FI (T ) = [I (T )] − [Ip(T ) + I (T )]DP

[I (T )] = 1 − [Ip(T ) + I (T )]DP

[I (T )] .

Since integral evaluated on derivative function can be simplified, previous expres-sions of the indicators are equivalent to the following:

CI (T ) = [INew(T )

] − [I

pNew(T ) + INew(T )

]DP,

FI (T ) = [INew(T )] − [IpNew(T ) + INew(T )]DP

[INew(T )] = 1 − [IpNew(T ) + INew(T )]DP

[INew(T )] ,

PI (T ) = 1 − [ Ip(T )+I (T )Sp(T )+S(T )+Ip(T )+I (T )

]DP

[ I (T )S(T )+I (T )

]

Evaluating Effectiveness of HIV Prevention Interventions

aII (T ) = 1 − [ IpNew(T +1)+INew(T +1)−(I

pNew(T )+INew(T ))

Sp(T )+S(T )]DP

[ INew(T +1)−INew(T )S(T )

] ,

CI (T ) = [I (T )

] − [Ip(T ) + I (T )

]DP,

FI (T ) = [I (T )] − [Ip(T ) + I (T )]DP

[I (T )] = 1 − [Ip(T ) + I (T )]DP

[I (T )] .

From these expressions, we can see that the indicators FI , PI , aII , and FI aredimensionless and attain value in [0,1], and they do not depend on the populationsize. The other two indicators CI and CI measure changes in population group sizes,and are not dimensionless.

3.2 Initial Dynamics of the Indicators

To understand the practical value of the qualitative indicators, we examine theirshort, intermediate, and long term dynamics. We begin with indicator approxima-tions shortly after the start of the intervention. Using the initial conditions definedabove, we obtained the following expressions associated with the initial indicators’behavior (details can be found in the Appendix):

CI ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0) dt,

FI ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1,

PI ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P)dt,

aII ≈ (1 − αsk1)[1 − (1 − θ)αik1

]{1 − (1 − αsk1)k1

[1 − (1 − θ)αik1

]}k1βP dt,

CI ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0) dt,

FI ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P)dt.

Here, we assume dt = 1 for the approximation for aII because the definition of theincidence indicator is on annual basis.

Note that the expression for the fractional indicator (FI ) depends only on the PrEPefficacy (αs , αi ) and factors related to the implementation of the intervention at itsstart such as initial coverage (k1) and the introductory control of the PrEP usage byHIV-positive individuals (θ ) but not on the demographic, behavioral, and epidemicparameters. Therefore, fractional indicator represents a metric of the “immediate im-pact of PrEP” on the HIV epidemic, which is independent of the specific populationand the status of the HIV epidemic in it. This metric accounts for the effects of the re-duced susceptibility (αs ) of the fraction k1 of the population which initially uses PrEPcombined with the reduced infectiousness (αi ) of a limited fraction (1 − θ)k1 of theinfected population when in contact with partners unprotected by PrEP (1 − αsk1).Clearly, if PrEP provides unidirectional protection (αi = 0) or none of the infectedindividuals is using PrEP (θ = 1), then the “immediate impact of PrEP” is given bythe product of PrEP efficacy and coverage (αsk1). The initial behavior of all otherindicators depend on the HIV prevalence (P ) at the time of PrEP introduction as wellas on the cumulative HIV-acquisition risk (β). Moreover, the cumulative (CI ) andreduction (CI ) indicators also depend on the initial population size (N(0)), which is

Y. Zhao et al.

Fig. 5 Initial growth rate of reduction (CI ) and fractional reduction (FI ) indicators with respect to (a) θ ,(b) k1, and (c) P . CI is denoted by green solid line, while FI is denoted by blue dashed line. All otherparameters are fixed on their baseline parameter values from Table 1 (Color figure online)

consistent with the fact that only indicators CI and CI measure changes on popula-tion group sizes, and are not dimensionless.

The initial rate of change of the indicators can be approximated as:

C′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0),

P ′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P),

aI ′I ≈ (1 − αsk1)

[1 − (1 − θ)αik1

]{1 − (1 − αsk1)k1

[1 − (1 − θ)αik1

]}k1βP,

C′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0),

F ′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P).

Notice that initially C′I ≈ C′

I and P ′I ≈ F ′

I . We study the sensitivity of the initialrate of change of the reduction indicators (CI and FI ) to some of the intervention(θ , k1) and epidemic (P ) parameters by bifurcation simulations (Fig. 5). These bi-furcation parameters were chosen because they are easier to evaluate at communitylevels compared to HIV-acquisition risk and PrEP efficacy. The graphs in Fig. 5(a)and (b) demonstrate that the growth of both indicators accelerates if more people starton PrEP (larger k1), but decelerates if the control of the PrEP usage by infected in-dividuals is more effective (larger θ ). The initial rate of change is more sensitive tok1 than to θ but it is clear that the growth rate of both indicators at the time of PrEPintroduction expresses qualitatively similar behavior with respect to the interventionparameters (θ , k1). In contrast, the graphs presenting the dependence on the initialHIV prevalence show serious discrepancies (Fig. 5(c)). The initial growth rate of thereduction indicator (CI ) increases when the HIV prevalence ranges from 0 to 50 %which includes all realistic values observed so far, particularly in Sub-Saharan Africa(Fig. 5(c)). In comparison, the increase in HIV prevalence within the same rangeimplies smaller growth rate of the fractional reduction indicator FI .

3.3 Asymptotic Behavior of the Indicators

In resource-constrained settings, it is unrealistic to expect the HIV epidemic will dieout without additional intervention. Therefore, in the following, we assume that the

Evaluating Effectiveness of HIV Prevention Interventions

basic reproduction number of the “no intervention” model is R0 > 1. The asymptoticHIV prevalence in this case is given by: [ I

S+I] = 1 − μ+d

β= 1 − 1

R0.

We want to point our that if the PrEP intervention is strong enough to cause theeradication of HIV in the population, i.e., the HIV epidemic approaches the disease-free equilibrium with the “dual-protection” model, then the asymptotic behavior ofall indicators is well determined: (i) the cumulative indicator will grow to infinity;(ii) the reduction indicator will stabilize at [I ] = R0−1

β−dΛ; and (iii) all other indicators

will approach one. Unfortunately, PrEP intervention alone is unlikely to be sufficientto eradicate HIV. In that case, we show that the asymptotic behavior of the indicatorscan be expressed in terms of the asymptotic proportion (p) of the HIV-positive sub-population, which have been infected while using PrEP (details are presented in theAppendix), where p is defined as follows:

p =[

Ip

Ip + I

]

DP.

Expressions for the asymptotic values of four of the indicators:

PI = 1 − R0 − 1−αs(1−p)(1−αs)(1−αip)

R0 − 1,

aII = 1 − R0(1−αs)(1−αip)

1−αs(1−p)− 1

R0 − 1,

CI =[R0 − 1

β − d− R0(1 − αip)(1 − αsk) − 1

β(1 − αip)(1 − αsp) − d

]Λ,

FI = 1 − R0(1 − αip)(1 − αsk) − 1

R0 − 1

β − d

β(1 − αip)(1 − αsp) − d

and the asymptotic rate of growth of the cumulative indicator:

C′I = (μ + d)

[R0 − 1

β − d− R0(1 − αip)(1 − αsk) − 1

β(1 − αip)(1 − αsp) − d

]Λ = (μ + d)CI

show that they are independent of the initial HIV prevalence (P ) and the control onthe PrEP use by HIV-positive individuals (θ ), which have been of critical importancefor the initial dynamics of the indicators. Cumulative indicators (CI and CI ) dependindirectly on the population size (N ), which determines the entry rate in the pop-ulation (Λ). Notice that C′

I = (μ + d)CI so the value of the reduction indicator isproportional to the annual number of new infections prevented due to PrEP use in along term. The rest of the indicators are not influenced by the population size (N ).Although recruitment parameters such as k and Λ are not explicitly present in someof the expressions above they may affect the asymptotic proportion of PrEP usersamong infected subpopulation (p).

Therefore, a good approximation of p is important for the evaluation of the asymp-totic levels of the indicators. Boundedness of the prevalence indicator PI < 1 and

Y. Zhao et al.

Fig. 6 (a) Asymptotic values of the proportion of the HIV-positive individuals who have been infectedwhile using PrEP (p) as a function of the PrEP efficacies (αs and αi ) assuming 10 % PrEP coverage(k = k1 = 0.1). (b) Long term dynamics of the quantitative indicators based on simulations with “du-al-protection” and “no intervention” models using parameters from Table 1

positivity of the annual incidence indicator both imply that

p <(1 − αs)[R0 − 1]

(1 − αs)αiR0 + αs

.

This provides a rough approximation for p. We study the variation of p when thereduction in susceptibility and infectiousness range from 0 to 100 % (Fig. 6(a)) forintervention coverage (k = 0.1) which is not sufficient to eradicate HIV even if thePrEP protection against HIV is perfect (αs = αi = 1). It shows that p depends greatlyon the reduction in susceptibility (αs ) and very little on the reduction in infectiousness(αi ). It is clear that the fraction of infections, which occur when using PrEP (p) rangesfrom zero, in case that PrEP provides complete protection against HIV (αs = 1) andno PrEP users ever get infected, to the level of the PrEP coverage (k) in case thatPrEP is completely ineffective (αs = 0) and infections are proportionally distributedamong PrEP users and nonusers.

Our next goal is to examine and compare the long-term behavior of the indicators(specifically those expressed as ratios) for a fixed PrEP intervention (Fig. 6(b)). Al-though qualitatively similar, the trajectories of the indicators show some importantdifferences. First, some indicators such as the reduction in HIV prevalence and thereduction in the infected fraction start at zero while others such as the fraction ofprevented infections and the reduction in HIV incidence initiate at positive values.Therefore, it is not surprising that the indicators reach a specific threshold of 20 %at times varying from 3 to 11 years after the introduction of PrEP. The times neededto report 50 % effectiveness are even farther apart. It takes the intervention 24 yearsand 33 years to reduce in half the expected HIV incidence and HIV prevalence, re-spectively. However, almost 90 years are necessary to reduce the cumulative numberof new infections by 50 %, i.e., such reduction is infeasible over traditionally usedevaluation periods of up to 30 years.

Evaluating Effectiveness of HIV Prevention Interventions

Fig. 7 Contour plots of the indicators (FI , PI , and aII ) over 10 and 30 years with respect to selectedintervention parameters αi and αs . All other parameters are fixed at the baseline parameter values fromTable 1

4 Evaluation of the Public-Health Impact PrEP

The initial and asymptotic behavior of the indicators are useful in understanding whatdrives the observed differences in their projections. However, from a public healthperspective it is more important to analyze the indicators values over more practicaltime intervals. Although no fixed standards exist, the majority of the quantitativestudies assume that preventive interventions are implemented for 10 years when their

Y. Zhao et al.

Fig. 8 Contour plots of the indicators (FI , PI , and aII ) over 10 and 30 years with respect to selectedepidemic parameters β and k1. All other parameters are fixed at the baseline parameter values from Table 1

effectiveness is evaluated. The same period is recommended by the World HealthOrganizations as an evaluation period when cost-effectiveness analyses are conducted(World Health Organization (WHO) 2003). Longer periods are investigated in fewstudies but always up to 30 years.

In this section, we explore the dependence of the indicator readings over 10 and30 years on key epidemic and intervention parameters. Clearly, the impact of PrEPis positively correlated with both reductions in susceptibility (αs ) and infectiousness

Evaluating Effectiveness of HIV Prevention Interventions

(αi ) regardless of what indicator is used to quantify it (Fig. 7). The slopes of thecontour plots in the PrEP efficacy parameter space show that if the susceptibilityefficacy(αs ) is relatively low (up to 30 %), all indicators are equally dependent ofboth αs and αi . However, with the increase of the PrEP protection against HIV, theinfluence of the reduction of infectiousness decreases significantly. The prevalenceindicator (PI ) projects the least effectiveness over 10 years of PrEP use. It predictsthat more than 70 % and 55 % PrEP efficacy is needed to achieve 20 % reductionin HIV prevalence with uni-directional (αi = 0) and bidirectional (αi = αs ) interven-tions, respectively. In comparison, 20 % reduction in the expected HIV infections ispossible with 65 % effective unidirectional and 55 % bidirectional PrEP while 20 %reduction in HIV incidence is feasible even if less than 45 % effective unidirectionaland 30 % bidirectional PrEP is used over 10 years. The order of predicted effective-ness by the prevalence and the fractional indicators is reversed over an evaluationperiod of 30 years. More than half of the parameter space results in more than 50 %reduction in HIV prevalence (PI ), an unreachable threshold as a reduction in expectedinfections (FI ).

All indicators increase with coverage k = k1 (Fig. 8). The prevalence and frac-tional indicators, in contrast to the incidence indicator, are sensitive to changes in thetransmission rate (β) with increasing influence of β on the fraction of prevented in-fection for larger evaluation periods. The maximum PrEP effectiveness over 10 yearsof PrEP use is predicted for complete coverage (k) and modest level of the transmis-sion rate (β) while over 30 years it is achieved for the lowest possible β .

5 Sensitivity Analysis

Finally, we explore the sensitivity of the indicators to changes in each parameter.Using the algorithm presented in Blower and Dowlatabadi (1994), we calculate thePartial Rank Correlation Coefficients (PRCC), which evaluate the monotonicity ofthe model outcomes (indicators) in terms of the model parameters. Values of PRCCcloser to ±1, imply stronger correlation between the output indicator and the inputparameter while the sign of the coefficients determines if the outcomes grow or de-crease with an increase of the input parameters.

We study separately the sensitivity of the indicators to the parameters (ba , n, μ,and d) which we fitted using data from South Africa. In our analysis, we choose1,000 random parameters combinations of those input parameters sampled uniformlyfrom their corresponding ranges: [0.0015,0.0045] for ba , [32.9247,98.7741] forn, [0.0125,0.0375] for μ, and [0.0651,0.1953] for d . Each range is chosen as[0.5,1.5]∗(baseline parameter value in Table 1). The rest of the parameters are fixedon their baseline values in Table 1. For each parameter set, we simulate the mod-els with “dual-protection” and no intervention and calculate PRCC matrix of all sixindicators CI , FI , PI , aII , CI , and FI for the first 10 years (standard analysis) aswell as for the 100 years (long-term analysis). Similarly, we investigate the indica-tors’ sensitivity to the remaining epidemic and intervention parameters (αs , αi , P , k1(k = k1), and θ ), uniformly sampled from their corresponding ranges: [0.25,0.75] forαs , [0.25,0.75] for αi , [0.083,0.249] for P , [0.1,0.3] for k1, and [0.25,0.75] for θ .Results are presented in Fig. 9.

Y. Zhao et al.

Fig. 9 Partial rank correlation coefficients (PRCC) between model parameters and the quantitative indi-cators over 10 and 100 years (Color figure online)

Correlations for 10-year intervention suggest that from the fitted parameters theindicators are most sensitive to the factors (ba and n), which determine the trans-mission rate β . However, their influence over time decreases. CI and CI are stillpositively correlated to the two factors while the rest of the indicators are negativelycorrelated to the two factors over 100 years, both dependencies are weak. The in-tervention outcomes are split into two groups with respect to their correlation withthe HIV induced mortality (d): the cumulative indicators being negatively correlatedwhile the rest being positively correlated with d . Similar discrepancy is observedwith respect to the influence of the initial HIV prevalence P over 10 years, but thecorrelations are reversed (positive—for the cumulative indicators and negative for therest). Interestingly, in that case the difference between the indicators disappears in along term. Note that although P appears in the initial conditions only, it continues tohave strong influence on all the cumulative and reduction indicators for more than 10years while its impact on the fraction of prevented infection gets even stronger overtime.

Among the intervention parameters, PrEP coverage (k) and PrEP efficacies peract (αs and αi ) express strong positive correlation with all the indicators in a shortterm. It remains significant in a long term for all outcomes. This confirms that PrEPcoverage and protection level are critical to the intervention success regardless whichqualitative metric is used. In contrast, the influence of the initial control on the PrEPuse by HIV-positive individuals (θ ) reduces substantially in time.

Evaluating Effectiveness of HIV Prevention Interventions

The prevalence (PI ) and the annual incidence (aII ) indicators express almost thesame sensitivity to all parameters. Therefore, they should have consistent projectionswhen evaluating the impact of the intervention.

6 Conclusions

Precise evaluation of the expected public-health impact of biomedical interventionsfor HIV preventions becomes increasingly important with more prevention optionsentering the pipeline toward licensure. The practice shows that even if the productsare effective in reducing the individual risk of acquisition (individual efficacy) thebenefits from general usage (population effectiveness) may be limited by variety ofepidemic, behavioral, and intervention factors. In this paper, we analyzed differentevaluation metrics of population effectiveness of pre-exposure prophylaxis (PrEP)interventions. We compare four traditional indicators, based on cumulative numberor fractions of infections prevented, on reduction in HIV prevalence or incidence withother two metrics based on the reduction in infected subpopulation.

We demonstrated that the quantitative indicators have a distinct dynamical profileshortly after the start of PrEP intervention which modifies substantially over time.As a result, when calculated over a fixed period of time, these indicators may projectsignificantly different PrEP effectiveness and, therefore, influence the decision if par-ticular products are potentially good enough for implementation. In general, new pre-vention methods need to prove their effectiveness in randomized clinical trial (RCT),i.e., to demonstrate that the observed efficacy is significantly larger than zero (pos-itive 95 % confidence interval), before applying for licensure. In reality, developersand public-health officials try to avoid PrEP products with low efficacy because thecontrolled environments of the clinical trials are difficult to be replicated at commu-nity level. Another concern is that the availability of PrEP may affect sexual behaviorand encourage risk-sex practices. Therefore, minimal efficacy thresholds of 20 % orhigher are often included in the design of RCTs and similar levels of effectivenessis expected when interventions are modeled at population level (Grant et al. 2010;Partners PrEP Study Team 2011; Dimitrov et al. 2013). Other studies imply that 50 %biological efficacy is needed to guarantee significant public-health impact. The ques-tion remains what does 20 % or 50 % PrEP effectiveness mean? We have shown thewidely used evaluation metrics may disagree over practical intervals of time (Fig. 6).A reduction in HIV incidence at pre-specified levels seems most realistic as an in-tervention goal, but it is not easy to be estimated in the population. In contrast, areduction in HIV prevalence is easier to be recorded but more difficult to be achievedin a short term. The reduction in the number of new HIV infections, which is themost popular public-health metric, projects strong PrEP effectiveness initially butgrows slower than the other indicators over time.

Moreover, if used to compare the impact of PrEP interventions different indicatorsmay give preference to different options (Fig. 4). We recommend that public-healthofficials who consider PrEP to be integrated in HIV prevention programs should basetheir decision on a complex of quantitative metrics. Although is specifically focusedon HIV prevention, the same theoretical approach may be extended to model other in-fectious diseases, such as malaria, cholera, and tuberculosis (Mtisi et al. 2009; Mwasa

Y. Zhao et al.

and Tchuenche 2011) or evaluate the impact of interventions, such as male circumci-sion, vaccination, or quarantine strategies (Mubayi et al. 2010; Alsallaq et al. 2010;Hallett et al. 2011; Foss et al. 2003; Blower et al. 2001). The proposed metrics com-parison may be useful when the influence of different behavior factors, such as sex-ual disinhibition, smoking, or drinking on the epidemics dynamics is studied (Bhunuet al. 2011; Mubayi et al. 2011).

Presented results, assuming perfect adherence and instantaneous uptake, are likelyto give optimistic views of the potential impact of a PrEP intervention. Although,overall self-reported adherence in the concluded clinical trials is high it is unclear howconsistently PrEP will be used in real settings. That will depend among other thingson the delivery system (e.g., coital or daily use, oral, topical, or slow release ring),individual preferences, availability, acceptability, and cost of the product (Eisingerichet al. 2012; Greene et al. 2010). Perfect adherence and other simplifying assumptionsallowed us to support our observations on the indicators and their simulated dynamicswith analytical expressions, which were easier to be interpreted. We believe that morecomplex and realistic modeling setup will be more useful in projecting benefits dueto PrEP use but it is unlikely to resolve the differences between the interventionaloutcomes reported in this paper.

Acknowledgements D.D. is supported by a grant from the National Institutes of Health (Grant number5 U01 AI068615-03). Y.Z., H.L., and Y.K. are supported in part by DMS-0920744.

The authors thank the anonymous referees for many useful comments on an earlier draft.

Appendix

A.1 Approximations of the Indicators at the Start of the Intervention

Based on the initial conditions for the two models, we can obtain the approximatebehavior of the indicator at the start of the intervention. To facilitate this, we letIT (0) = I (0) + Ip(0) and ST (0) = S(0) + Sp(0). Recall that N(0) = S(0) + I (0)

when there is no PrEP user and N(0) = S(0) + Sp(0) + I (0) + Ip(0) when there arePrEP users.

As examples, we will provide the approximation computation for the preva-lence indicator, cumulative indicator, and fractional indicator. Recall that PI =1 − [ Ip+I

Sp+S+Ip+I]DP

[ IS+I

] . We will first estimate [ IS+I

] and [ Ip+ISp+S+Ip+I

]DP separately:

[I

S + I

]≈

I (0) + dIdt

(0) dt

I (0) + S(0) + ( dIdt

+ dSdt

)(0) dt

= I (0) + [β S(0)I (0)N(0)

− (μ + d)I (0)]dt

I (0) + S(0) + [Λ − μS(0) − (μ + d)I (0)]dt

= PN(0) + [βP (1 − P)N(0) − (μ + d)PN(0)]dt

N(0) + [Λ − μ(1 − P)N(0) − (μ + d)PN(0)]dt

= P + [βP (1 − P) − (μ + d)P ]dt

1 + ( ΛN(0)

− μ − dP )dt;

Evaluating Effectiveness of HIV Prevention Interventions

[Ip + I

Sp + S + Ip + I

]

DP

≈

IT (0) + ( dIp

dt+ dI

dt)(0) dt

N(0) + ( dSp

dt+ dS

dt+ dIp

dt+ dI

dt)(0) dt

= IT (0) + {[IT (0) − αiIp(0)][ST (0) − αsS

p(0)] βN(0)

− (μ + d)IT (0)}dt

N(0) + [Λ − μN(0) − dIT (0)]dt

= PN(0)

(1 +

{(1 − P)N(0)

[1 − (1 − θ)k1 + (1 − αi)(1 − θ)k1

][1 − k1

+ (1 − αs)k1] · β

N(0)− (μ + d)

}dt

)· 1

N(0) + (Λ − μN(0) − dPN(0)) dt

= P + {[1 − (1 − θ)k1αi](1 − αsk1)βP (1 − P) − (μ + d)P }dt

1 + ( ΛN(0)

− μ − dP )dt.

Then by the expression of prevalence indicator, we have

PI ≈ 1 −P+{[1−(1−θ)k1αi ](1−αsk1)βP (1−P)−(μ+d)P }dt

1+( ΛN(0)

−μ−dP )dt

P+[βP (1−P)−(μ+d)P ]dt

1+( ΛN(0)

−μ−dP )dt

= 1 − P + {[1 − (1 − θ)k1αi](1 − αsk1)βP (1 − P) − (μ + d)P }dt

P + [βP (1 − P) − (μ + d)P ]dt

= 1 − 1 + {[1 − (1 − θ)k1αi](1 − αsk1)β(1 − P) − (μ + d)}dt

1 + [β(1 − P) − (μ + d)]dt

≈

(−{[1 − (1 − θ)k1αi

](1 − αsk1)β(1 − P) − (μ + d)

}dt

)

{1 − [

β(1 − P) − (μ + d)]dt

}

≈ −{[1 − (1 − θ)k1αi

](1 − αsk1)β(1 − P) − (μ + d)

}dt

+ [β(1 − P) − (μ + d)

]dt

= [αs + (1 − θ)αi − (1 − θ)αsαik1

]k1β(1 − P)dt

= [αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P)dt.

Thus, the slope at which the prevalence indicator increases at the beginning of theintervention can be approximated by

P ′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1β(1 − P).

By the expression of cumulative indicator, we have

CI ≈

[INew(0) + d

dtINew(0) dt

]

−[INew(0) + d

dtINew(0) dt + I

pNew(0) + d

dtI

pNew(0) dt

]

DP

Y. Zhao et al.

=[

0 + βS(0)I (0)

N(0)dt

]−

[0 +

(β

S(0)I (0)

N(0)+ (1 − αi)β

S(0)Ip(0)

N(0)

)dt

+ 0 +(

(1 − αs)βSp(0)I (0)

N(0)+ (1 − αs)(1 − αi)β

Sp(0)Ip(0)

N(0)

)dt

]

DP

= dtβ(1 − P)PN(0) − dt[β(1 − k1)(1 − P)

(1 − (1 − θ)k1

)PN(0)

+ (1 − αi)β(1 − k1)(1 − P)(1 − θ)k1PN(0)

+ (1 − αs)βk1(1 − P)(1 − (1 − θ)k1

)PN(0)

+ (1 − αs)(1 − αi)βk1(1 − P)(1 − θ)k1PN(0)]

= dtβ(1 − P)PN(0)

· {1 − [(1 − k1)

(1 − (1 − θ)k1

) + (1 − αi)(1 − k1)(1 − θ)k1

+ (1 − αs)k1(1 − (1 − θk1)

) + (1 − αs)(1 − αi)k1(1 − θ)k1]}

= dtβ(1 − P)PN(0)

· {1 − [(1 − αsk1)

(1 − (1 − θ)k1

) + (1 − αsk1)(1 − αi)(1 − θ)k1]}

= dtβ(1 − P)PN(0){1 − (1 − αsk1)

(1 − αi(1 − θ)k1

)}

= dtβ(1 − P)PN(0){1 − [

1 − αsk1 − αi(1 − θ)k1 + αsk1αi(1 − θ)k1]}

= [αs + (1 − θ)αi − (1 − θ)αsαik1

]k1βP (1 − P)N(0) dt

= [αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0) dt.

Thus, the slope at which the cumulative indicator increases at the beginning of theintervention can be approximated by

C′I ≈

[αs + (1 − θ)αi(1 − αsk1)

]k1βP (1 − P)N(0).

By the expression of fractional indicator, we have

FI ≈ 1 − [INew(0) + ddt

INew(0) + IpNew(0) + d

dtI

pNew(0)]DP

[INew(0) + ddt

INew(0)]

= 1 −[

0 +(

βS(0)I (0)

N(0)+ (1 − αi)β

S(0)Ip(0)

N(0)

)dt

+ 0 +(

(1 − αs)βSp(0)I (0)

N(0)+ (1 − αs)(1 − αi)β

Sp(0)Ip(0)

N(0)

)dt

]

DP

· 1

[0 + βS(0)I (0)

N(0)dt]

= 1 − [β(1 − k1)(1 − P)

(1 − (1 − θ)k1

)PN(0)

+ (1 − αi)β(1 − k1)(1 − P)(1 − θ)k1PN(0)

+ (1 − αs)βk1(1 − P)(1 − (1 − θ)k1

)PN(0)

Evaluating Effectiveness of HIV Prevention Interventions

+ (1 − αs)(1 − αi)βk1(1 − P)(1 − θ)k1PN(0)] · 1

β(1 − P)PN(0)

= 1 − [(1 − k1)

(1 − (1 − θ)k1

) + (1 − αi)(1 − k1)(1 − θ)k1

+ (1 − αs)k1(1 − (1 − θ)k1

) + (1 − αs)(1 − αi)k1(1 − θ)k1]

= [αs + (1 − θ)αi − (1 − θ)αsαik1

]k1

= [αs + (1 − θ)αi(1 − αsk1)

]k1.

A.2 Asymptotic Approximations of the Indicators

To approximate the asymptotic behavior of indicators, we need to study the steadystate of the two models. What is challenging is to study the steady state of the baselinemodel using PrEP (4). From

d(Sp + S + Ip + I )

dt= Λ − μ

(Sp + S + Ip + I

) − d(Ip + I

) = 0

we obtain that at steady state

N = Λ − d(Ip + I )

μ.

From

d(Sp + Ip)

dt= kΛ − μ

(Sp + Ip

) − dIp = 0

we obtain that at steady state,

Sp = kΛ − (d + μ)Ip

μ.

Moreover, from

d(S + I )

dt= (1 − k)Λ − μ(S + I ) − dI = 0

we obtain that at steady state,

S = (1 − k)Λ − (d + μ)I

μ.

In the following, let m = μ + d and I = Ip + I . Then at steady state,

d(I )

dt= (1 − αs)β

SpI

N+ β

SI

N+ (1 − αs)(1 − αi)β

SpIp

N+ (1 − αi)β

SIp

N− mI

= [I + (1 − αi)I

p](1 − αs)β

Sp

N+ [

I + (1 − αi)Ip]β

S

N− mI

= [I + (1 − αi)I

p][

S + (1 − αs)Sp] β

N− mI

Y. Zhao et al.

= [I + (1 − αi)I

p][ (1 − k)Λ − mI

μ+ (1 − αs)

kΛ − mIp

μ

]β

N− mI

= 0.

Hence,

β[I + (1 − αi)I

p][(1 − k)Λ − mI

μ+ (1 − αs)

kΛ − mIp

μ

]= mI

Λ − dI

μ.

Let p = [ Ip

Ip+I]DP = [ Ip

I]DP, and assume that I = Ip + I �= 0 at steady state. Then

β(I − αipI )

[(1 − k)Λ − m(1 − p)I

μ+ (1 − αs)

kΛ − mpI

μ

]− mI

Λ − dI

μ= 0,

βI (1 − αip)

[(1 − k)Λ − m(1 − p)I

μ+ (1 − αs)

kΛ − mpI

μ

]= mI

Λ − dI

μ,

β(1 − αip)

[(1 − k)Λ + (1 − αs)kΛ

m− (1 − p)I − (1 − αs)pI

]= Λ − dI ,

β(1 − αip)(1 − αsk)

μ + dΛ − β(1 − αip)(1 − αsp)I = Λ − dI ,

I =β(1−αip)(1−αsk)

μ+d− 1

β(1 − αip)(1 − αsp) − dΛ = R0(1 − αip)(1 − αsk) − 1

β(1 − αip)(1 − αsp) − dΛ.

Notice that in the model with no intervention (6), the basic reproduction number isR0 = β

μ+d, and the endemic steady state is ( 1

β−dΛ,

R0−1β−d

Λ). By the definitions of theindicators, now we can find the expressions to approximate the asymptotic behaviorof CI and FI :

CI =[R0 − 1

β − d− R0(1 − αip)(1 − αsk) − 1

β(1 − αip)(1 − αsp) − d

]Λ;

FI = 1 − R0(1 − αip)(1 − αsk) − 1

R0 − 1

β − d

β(1 − αip)(1 − αsp) − d.

Now consider the prevalence indicator, we want to find [ Ip+ISp+S+Ip+I

]DP at thesteady state.

dIp

dt+ (1 − αs)

dI

dt

= (1 − αs)β(Sp + S)I

N+ (1 − αs)(1 − αi)β

(Sp + S)Ip

N

− (μ + d)[Ip + (1 − αs)I

]

= (1 − αs)β[I + (1 − αi)I

p]Sp + S

N− (μ + d)

[Ip + (1 − αs)I

] = 0.

Evaluating Effectiveness of HIV Prevention Interventions

Hence,

Sp + S

N= μ + d

(1 − αs)β

1 − αsI

Ip+I

1 − αiIp

Ip+I

= R−10

1

1 − αs

1 − αs(1 − p)

1 − αip;

Ip + I

N= 1 − μ + d

(1 − αs)β

1 − αsI

Ip+I

1 − αiIp

Ip+I

= 1 − R−10

1

1 − αs

1 − αs(1 − p)

1 − αip.

By the definition of prevalence indicator, the asymptotical behavior of the prevalenceindicator can be determined by

PI = 1 − 1 − R−10

11−αs

1−αs(1−p)1−αip

1 − R−10

= 1 − R0 − 1−αs(1−p)(1−αs)(1−αip)

R0 − 1.

For the cumulative indicator at the steady state, we have

d

dtCI =

[d

dtINew

]−

[d

dt

(I

pNew + INew

)]

DP,

d

dtCI = [

(μ + d)I] − [

(μ + d)(Ip + I

)]DP

=(

β − (μ + d)

β − d− β(1 − αip)(1 − αsk) − (μ + d)

β(1 − αip)(1 − αsp) − d

)Λ

= (μ + d)

(R0 − 1

β − d− R0(1 − αip)(1 − αsk) − 1

β(1 − αip)(1 − αsp) − d

)Λ.

We see that the cumulative indicator keeps increasing, and eventually the rate ap-proaches a constant.

Finally, we look at the annual incidence indicator

aII (n) = 1 − [ IpNew(n+1)+INew(n+1)−(I

pNew(n)+INew(n))

Sp(n)+S(n)]DP

[ INew(n+1)−INew(n)S(n)

]

= 1 − [ ddt

(IpNew(n+q)+INew(n+s))

Sp(n)+S(n)]DP

[ ddt

INew(n+r)

S(n)]

with some q, s, r ∈ [0,1],

then at steady state, we have

aII = 1 − [ ddt

(IpNew+INew)

Sp+S]DP

[ ddt

INew

S]

= 1 − [ (μ+d)(Ip+I )Sp+S

]DP

[ (μ+d)IS

] = 1 − [ Ip+ISp+S

]DP

[ IS]

= 1 − R0(1−αs)(1−αip)

1−αs(1−p)− 1

R0 − 1.

Y. Zhao et al.

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