VMB interventions, similar to vaccination programs, induce both direct and indirect effects on HIV transmission dynamics. Protective effects from reduced susceptibility of the women using VMB are classified as direct while the other protective or detrimental effects which result from the intervention-induced changes in transmission are classified as indirect effects. These include effects from reduced infectiousness of VMB users, reduced transmissibility of drug resistant HIV strains, and decreased HIV prevalence reducing the risk of contacts with infected partners.

We studied the mechanisms which contribute to the VMB benefits for each gender during the short-, intermediate-, and long-term phases of intervention implementation. Since the intervention is introduced in the female subpopulation, the number of female infections due to reduced susceptibility should be impacted from the moment that VMB usage begins. On the other hand, men benefit from the combination of reduced infectiousness of the HIV-positive users and reduced transmissibility of drug-resistant HIV. A decline in the number of male infections should be expected later in time, once enough HIV-positive women are using VMB and a fraction of them have developed resistance. Intuitively we would expect an initial female advantage in benefits regardless of the evaluation method. In contrast, we found that the initial advantage with respect to the cumulative indicator (*C*_{I}) heavily depended on pre-existing population settings at the start of the intervention such as HIV acquisition risks per sex act. The same factors do not affect the fractional indicator (*F*_{I}). The influence of the pre-enrollment screening which prevents the usage of VMB by HIV-positive women is initially strong because it controls the rate of drug-resistance development and affects the risk of transmision from women to men in case of bi-directional protection. This influence diminishes over an extended intervention period for HIV epidemics with basic reproductive number *R*_{0} > 1 in the absence of the VMB (). The pre-enrollment screening effectiveness (θ) remains a factor if the reproductive number *R*_{0} < 1 in which case the number of new HIV infections decreases substantially over time () and the impact of the intervention is limited in time.

The analytical assessment of the initial and asymptotic QBR provides an insight into the dynamical trends of the gender-specific benefit distribution. However, investigators and decision makers are more often interested in the public-health impact of the interventions over fixed periods of 5, 10, or 20 years after their initiation. We studied numerically how different epidemic and intervention parameters may influence the conclusions based on such analysis and identified parameter combinations which make an intervention designed and initiated in women to provide more benefits to men. We demonstrated that the choice of the evaluation period could taint the results, especially if it is not presented in connection with the dynamical trends in the HIV epidemic. Our analysis of the regions of male advantage (RMA) indicated complicated temporal correlations (–) between the QBRs and a series of pre-intervention (e.g., HIV acquisition risks per act) and intervention parameters (e.g., VMB efficacy mechanisms, rates of resistance development and reversion). When taken out of the complete dynamical picture, some of the observed patterns seem unexpected or counterintuitive but they all have reasonable explanations when the interaction between the mechanisms embedded in the mathematical model are discussed. In general, the RMA grow over the 25 year period after the introduction of VMB. However, the expansion rate significantly slows down toward the end of the period when the HIV epidemic, modified by VMB usage, approaches its new equilibrium state. Although independent from the intervention, the relative acquisition risk per act, male versus female, plays an important role in determining the size of the RMA. Its decline leads to a sizable reduction in the likelihood of male advantage associated with the cumulative indicator (*C*_{I}) while the RMA associated with the fractional indicator (*F*_{I}) grows (see , ).

These results highlight the importance of the combined effects of some epidemic and intervention parameters for the conclusions based on modeling studies. Investigators often spend some effort justifying the choice of individual parameter ranges but do not elaborate on how suitable their integrated parameter space is for the particular problem or environment. This is especially true in modeling biomedical interventions, where attention is focused on the changes in the epidemic inflicted by a newly developed product, while the “original” epidemiological settings are often neglected. This paper presents evidence demonstrating that the appropriateness of the pre-intervention settings has to be carefully evaluated. Long-term benefit distributions of VMB interventions introduced in communities where the HIV epidemic has low basic reproductive number (

*R*_{0} ≈ 1 or smaller) are quite different from those where the reproductive number is high (

*R*_{0} 1) because the interventional effects are limited in duration and magnitude by the declining epidemics. Moreover, the likelihood of male advantage changes when HIV acquisition risks for the genders are varied relative to each other. Therefore, the pre-intervention epidemic environment must be studied carefully before being used as a baseline for intervention evaluations.

Next, we would like to address some simplifying assumptions incorporated in our modeling study and their impact on the reported results. In the epidemic model (), we average the transmission rates over different phases of HIV infections and aggregate possible effects of ARV treatments in HIV acquisition risks per act. These factors will introduce symmetric effects on the gender benefits and therefore have little influence on the QBR and RMA. Risk compensation due to VMB usage is not considered but could impact the QBR especially if condoms, which provide protection to both gender, are replaced by VMB which protects only women. An inclusion of this factor will further decrease the likelihood of male advantage. The model allows former VMB users who have developed drug-resistance to reverse back to wild type dominance and assumes that the reversion rate is the same for acquired and transmitted resistance. Data collected in patients with failing ARV therapy suggest that such reversion can be expected from 12 to 16 weeks after therapy interruption, while data from ARV-naive people shows that transmitted drug-resistance can persist for more than 4 years [

12]. However, the uniform treatment of all female drug-resistant cases has little impact on the epidemic because the prevalence of transmitted compared to acquired drug resistance among HIV-positive women is extremely small. As a result, our analysis slightly overestimates the impact of resistance reversion on the HIV epidemic.

A major conclusion of this analysis is that the cumulative and fractional indicators may disagree on important issues related to the distribution of the benefits from a VMB intervention. Therefore, these indicators must undergo a careful consideration before being used as a basis for public health projections. Our analysis shows that for many feasible parameter settings, simulated VMB interventions prevent more infections in one gender but avert a larger fraction of infections in the other. The results are especially difficult to interpret when HIV acquisition risks per act are varied (see ) because the RMA associated with *C*_{I} and *F*_{I} over 25 years are mutually exclusive but in combination they cover almost completely the entire parameter space. This discrepancy is embedded in the theoretical formulations of the QBRs, which is based on the difference in the number of new HIV infections in the presence and absence of VMB usage. However, the HIV epidemics without intervention, which are used as reference points, do not have identical dynamical behavior, but express significant variation with respect to pre-intervention parameters. For instance, when the female HIV acquisition risk is higher (β_{w} > β_{m}) the HIV epidemic produces substantially more female than male infections. Therefore, significantly more infections must be prevented in women to avert the same percentage infections as in men. It is difficult to say which indicator is more relevant and informative when benefit distribution is discussed. The reduced sensitivity of the fractional indicators to the pre-intervention parameters makes it more suitable for studies where those parameters are not precisely estimated. We suggest that both indicators must be evaluated and compared in any formal analysis.

Moreover, we believe that other indicators, such as the relative reduction in HIV prevalence and incidence must be considered to avoid the ambiguity associated with indicators based on the absolute number of prevented infections. We illustrate our concern with an example of a hypothetical VMB intervention which reduces the susceptibility of the users by 56%, infectiousness by 59.5%, and is used by 42% of the women in the community. The comparison of the number of new HIV infections accumulated each year with and without intervention shows that during the initial 10–15 years the use of VMB implies fewer new infections per year (). However, during the next 15–20 years the pattern reverses and the number of new infections in the scenario with VMB surpasses those in the scenario without intervention. Is that enough to conclude that the HIV epidemic will worsen after the initial 15 years and should we recommend this product only for a limited period of time? Obviously, that is not the case because the simulated intervention provides good bi-directional protection which remains unchanged over time. The alarming readings of this indicator are the result of the intervention being effective in slowing down the HIV epidemic, which after 15 years has significantly affected the population size (). The big size difference between the HIV-negative subpopulations leads to more new infections in the intervention scenario which does not imply that the community will be better off if the product is withdrawn at that point. In fact, other indicators, such as HIV prevalence and incidence, continue to support the positive VMB impact up to 50 years after its introduction (). Therefore, QBR based on HIV prevalence and incidence can be a useful addition to any intervention evaluation, as shown for the gender-specific effects in . This example confirms our conclusion that the assessment of ongoing and future VMB (or other biomedical) interventions must be based on more comprehensive analyses than calculations of infections prevented over a fixed period of time. Although focused on the gender-specific effects, this study raises the awareness that a careful consideration of the pre-intervention epidemic conditions as well as the dynamical behavior of the qualitative indicators must be integrated in all studies reporting results obtained through mathematical modeling.

A. Intervention scenarios

The sets of parameter values and ranges used in the intervention scenarios simulated throughout the paper are summarized in . Parameters not included in each scenario description are fixed at their baseline values (see ).

| **Table 2**Simulated intervention scenarios by figure |

B. Model description and technical details

Model equations and initial conditions The model is formulated by the following system of differential equations:

Entry rates Λ

_{w} = μ

_{w}N_{w} and Λ

_{m} = μ

_{w}N_{m} are selected to balance the departure rates in population which have not been exposed to HIV. Here

represents the sexually active women (men) in the population. The rate of resistance development

*r*_{1} is a product of the maximum rate of resistance development and the probability of systemic absorption. Since no VMB resistance data is available we study a wide range for the likelihood of systemic absorption (from 0 to 100%) and assume that if absorbed VMB causes drug-resistance after an average period which varies from 1 year to never (i.e. infinite). The resulting range for

*r*_{1} (from 0 to 1) is used in the analysis (see and ). All other variables and parameters are described in section 2 and .

Here ρ_{w} (ρ_{m}) is the average number of sex partners that women (men) have per year, *n*_{w} (*nm*) is the average number of sexual acts that women (men) have per year, β_{w} (β_{m}) is the female (male) HIV-acquisition risk per act, α_{s} (α_{i}) measures the efficacy of VMB in reducing susceptibility (infectiousness) of VMB users, and α_{r} is the relative fitness of drug-resistant HIV strains compared to the wild-type HIV.

VMB interventions are initiated at time

*t* = 0 in populations with

*N*_{w}(0) =

*N*_{m}(0) = 1, 000, 000 and equal HIV-prevalence (

*P*) in men and women. VMB is initially used by proportion

*k*_{1} of the HIV-negative women and reduced proportion of (1−θ)

*k*_{1} of the HIV-positive women. The state of HIV epidemic at the start of a VMB intervention is set as follows:

Simulation of a VMB intervention presents variations in population size, HIV prevalence and incidence as well as HIV infections prevented per year for a VMB intervention which reduces the susceptibility of the users by 56%, infectiousness by 59.5%, and is used by 42% of the women in the community.

Equilibrium population and basic reproductive number *R*_{0} in absence of VMB intervention Our analysis shows that the benefit distribution of VMB intervention strongly depends on the pre-intervention settings. Basic reproductive number

*R*_{0} is a key characteristic which is used to determine the long-term outcome of the HIV epidemic. If

*R*_{0} > 1 the infection persists and the infected population stabilizes at an endemic fixed point while if

*R*_{0} < 1 the infection naturally dies out and the population stabilizes at its disease-free equilibrium. In absence of VMB the system

(3) reduces to 4 equations with variables

*S*_{w},

*S*_{m},

*I*_{w}, and

*I*_{m}. The disease-free equilibrium of the system is given by:

From the condition for local stability of the disease-free equilibrium we calculate

where

*b*_{m} = ρ

_{m}(1−(1−β

_{m})

^{nm/ρm}) and

*b*_{w} = ρ

_{w}(1−(1−β

_{w})

^{nw/ρw}). When

*R*_{0} > 1 the system

(3) in absence of VMB posses an endemic equilibrium given by:

Estimates of the initial values of QBR To estimate the initial values of the the ratios

*C*_{I} and

*F*_{I} we calculate the limits

. Let

_{w}(Δ

*t*) (Σ

_{w}(Δ

*t*)) be the cumulative number of new HIV infections in women over the initial period Δ

*t* with (without) VMB usage. Similarly,

_{m}(Δ

*t*) (Σ

_{m}(Δ

*t*)) is the cumulative number of new HIV infections in men over the initial period Δ

*t* with (without) VMB usage. Therefore,

*C*_{Iw}(Δ

*t*) = Σ

_{w}(Δ

*t*) −

_{w}(Δ

*t*) and

*C*_{Im}(Δ

*t*) = Σ

_{m}(Δ

*t*) −

_{m}(Δ

*t*) are the number of infections prevented in women and men over the period Δ

*t*. Using these notations we obtain:

Here the barred variables are calculated in the presence of VMB while the non-barred variables are calculated assuming no VMB usage. To approximate the forces of infections (λ) we use that (1−

*x*)

^{n} ≈ 1−

*nx* when

*nx* 1. Replacing the initial conditions

(5) the following hold:

Thus,

. Assuming equal sexual activity for both genders (

*n*_{m} =

*n*_{w}) we end up with the expression for

*C*_{I} in

(1). The expression

is obtained similarly.

Asymptotic estimates of the QBR Let

be the equilibrium population (susceptible women, HIV-positive women, susceptible men, HIV-positive men) reached in the absence of the VMB, while

represents the equilibrium population assuming an VMB usage. Here

aggregates both susceptible female classes

and

*S*_{w},

aggregates sexually active HIV-positive female classes, and

aggregates HIV-positive male classes

*I*_{m} and

*I*_{rm}. When the system is already at equilibrium the amount of the new infections (inflow) for each gender is equal to the number of the individuals leaving the infected classes (outflow). Therefore, the number of infections prevented in women

*C*_{Iw}(Δ

*t*) and men

*C*_{Im}(Δ

*t*) over the period Δ

*t* can be estimated by the difference in the outflow with and without VMB usage:

Provided that

(

*R*_{0} > 1) the fractions of the infections prevented in women

*F*_{Iw}(Δ

*t*) and men

*F*_{Im}(Δ

*t*) over the same period are:

Since these estimates remain unchanged over time the asymptotic limits of the quantitative indicators will be:

We study the public-health benefits of interventions of vaginal microbicides.

We evaluate gender-specific impact using two indicators based on prevented HIV.

Our analysis exposes complicated correlations between parameters and indicators.

Comparison of infections prevented over a fixed period of time may be misleading.

We recommend more comprehensive analysis based on additional indicators.