# Analytic Insights Into the Population Level Impact of Imperfect Prophylactic HIV Vaccines

The population level implications of imperfect HIV vaccines were studied using a mathematical model. A criterion for determining the utility of a vaccine at the population level is introduced, and 2 useful summary measures, namely, vaccine utility (ϕ) and vaccine infection fitness (ψ), are derived and shown to characterize the population-level utility once vaccine efficacies are determined. The utility of the vaccine alone does not guarantee a substantial impact, however, because the effectiveness of partially effective vaccines also depends on the prevailing level of HIV infectious spread. Therefore, a second criterion is introduced through a third summary measure, the hazard index (ξ), to describe the effectiveness of a vaccine in substantially reducing HIV incidence. The qualitative features of the impact are delineated by studying 4 distinct scenarios of HIV vaccination. Accordingly, our work delineates the link between vaccine efficacies and the impact of vaccination at the population level and provides the tools for vaccine developers to assess the utility and effectiveness of a given imperfect vaccine straightforwardly and rapidly.

From the *Statistical Center for HIV/AIDS Research and Prevention, Fred Hutchinson Cancer Research Center, Seattle, WA; †Center for Studies in Demography and Ecology, University of Washington, Seattle, WA; ‡Department of Infectious Diseases Epidemiology, Faculty of Medicine, Imperial College London, London, United Kingdom; and the §Program of Biostatistics and Biomathematics, Fred Hutchinson Cancer Research Center and Department of Biostatistics, School of Public Health and Community Medicine, University of Washington, Seattle, WA.

Received for publication September 8, 2006; accepted April 24, 2007.

Research supported by the University of Washington Center for AIDS Research, a National Institutes of Health-funded program (P30 AI 27757).

Reprints: Laith J. Abu-Raddad, PhD, Statistical Center for HIV/AIDS Research and Prevention, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, LE-400, PO Box 19024, Seattle, WA 98109 (e-mail: laith@scharp.org).

HIV/AIDS continues to be among the leading causes of global morbidity and mortality, particularly in sub-Saharan Africa, where it is the primary cause of death.^{1} Progressively larger efforts are being devoted to vaccine development.^{2} Nevertheless, these efforts continue to face formidable scientific challenges in developing a vaccine that protects against established infection, as is evident by the trials on nonhuman primate models.^{3-5} Accordingly, interest is rising in vaccines that may only have an impact on progression to disease and might be effective in controlling HIV spread by reducing infectivity in infected vaccinees.^{6-8} In part, this consideration is motivated by the historical success of several vaccines that have been shown to prevent disease but not infection, such as those for measles and polio.^{9,10} Such an outcome necessitates a study of the potential public health implications of partially effective vaccines. Mathematical modeling provides a useful tool to this end.

There has been a series of studies that attempted to investigate the impact of HIV vaccination at the population level.^{6,7,11-23} Despite this theoretic progress, there is still widespread confusion about the potential effectiveness of HIV vaccination. The confusion stems from the lack of clear and well-defined solutions to key issues surrounding population implications of vaccination. Moreover, different models provided contradictory predictions, which further distorted the conclusions of these studies. One example is the contentious issue of perversity after vaccine administration.^{24,25} A recent publication^{17} has stressed population-level perversity, but this assertion is contradicted by other approaches.^{19,20} Such issues need to be clarified, because any skepticism over the role of first-generation vaccines may contribute to excessively stringent licensure guidelines.

This article attempts to provide lucid and explicit answers to several questions about the impact of vaccination. Here, we address 3 key questions. The first is concerned with the precise link between the biologic mechanisms of action and vaccine efficacies as measured in clinical trials and the impact of vaccination at the population level. Vaccine efficacies refer to the effects of reducing susceptibility to infection *VES*, reducing infectiousness of those who become infected *VEI*, and slowing clinical disease progression *VEP*.^{26,27} The relevancy of this question is tied to the ongoing efforts of optimizing and augmenting trial designs to measure different vaccine efficacies.^{28-30} Because first-generation AIDS vaccines are likely to be licensed for delivery as a result of their disease-modifying aspects, and despite their minimal protection against infection,^{31} any vaccination policy needs to evaluate effectiveness by taking into account the different biologic effects of the vaccine and their imprint on HIV epidemiology.

The second question is concerned with defining a criterion for the utility of a given vaccine. Although it is relatively straightforward to define beneficial vaccine effects at the individual level in terms of reduction in susceptibility or infectivity and retardation of disease progression, the criterion for a favorable vaccine impact at the population level is not forthright. For example, one can define a favorable vaccine as one that reduces HIV prevalence in a given community. Such a criterion would overlook the utility of a vaccine that decreases the HIV incidence rate, although increasing HIV prevalence, as a consequence of slower disease progression to AIDS, however. Therefore, we introduce a criterion for the utility of the vaccine at the population level by characterizing the impact as definitely beneficial, partially beneficial, or perverse. The criterion is then explicitly expressed mathematically by deriving the summary measures, namely, vaccine utility (ϕ) and vaccine infection fitness (ψ), that characterize the utility of the vaccine at the population level once its individual-level efficacies are determined. In this fashion, we provide tools for vaccine developers and the broader HIV scientific community to assess the impact of imperfect HIV vaccines straightforwardly and rapidly.

The third question is concerned with identifying the conditions under which a vaccine leads to a substantial reduction in HIV infection incidence. Having a beneficial vaccine does not automatically imply that the vaccine is considerably effective in controlling HIV spread. Several beneficial HIV interventions proved to be of limited effectiveness, at least in some settings. Although there is little doubt that behavioral interventions are effective in principle, as evidenced by many successful behavioral interventions,^{32,33} several randomized behavioral intervention studies found no or limited effectiveness in reducing the acquisition of HIV or other sexually transmitted infections (STIs), despite the reduction in sexual risk behavior.^{34-37} The same is true for STI treatment interventions that largely failed in 2 major studies,^{38-40} although they were successful in an earlier study.^{41}

One explanation for the contrasting outcome of beneficial interventions may be the dynamics of competing risks. As we show elsewhere in this article, the utility of partially effective vaccines does not materialize as effectiveness in substantially reducing infection incidence unless the vaccine increases the waiting time to the next exposure to HIV to the extent that it is longer than the sexual activity lifespan. We introduce a measure, the hazard index, which sets the criterion for effectiveness of vaccination as an intervention. Further, we demonstrate how the poor impact of a given imperfect vaccine may not faithfully reflect the utility of the vaccine but mainly exposes the necessity of synergistic approaches that combine vaccination with other interventions. Finally, we delineate several qualitative features of the impact by studying 4 distinct scenarios of HIV vaccination. These scenarios are closely tied to the coming analysis and interpretation of the vaccines that are in or going to phase 2b trials.^{42}

## METHODS

### Vaccine Efficacy Measures and Related Properties

We first define the vaccine efficacy measures that characterize the biologic mechanisms of action of the vaccine and form part of the parameter input of the mathematical model. The first is *VES*, the classic vaccine efficacy of reducing susceptibility to infection as determined by phase 3 prophylactic vaccine trials and defined as the percent reduction in the rate of infection attributable to vaccination. The second is *VEI*, which measures the vaccine's efficacy of reducing HIV infectivity for those who had been vaccinated before the time of acquiring the infection. Here, *VEI* is defined as the percent reduction in risk of secondary transmission attack rate resulting from exposure to an infected vaccinee compared with exposure to an unvaccinated infected individual (see Appendix).^{43,44} The *VEI* can be measured in vaccine trials designed specifically for this purpose^{45} or in partner-augmented phase 3 prophylactic vaccine trials.^{28-30} Because *VEI* is likely to reflect the biologic mechanism of vaccine-induced reduction in HIV viral load,^{3-5} *VEI* can also be derived using viral load as a surrogate.^{27,46,47} Additionally, *VEI* can be estimated by contact tracing in community-based vaccination programs^{47} or using incidence data of vaccinated and control groups from trials conducted in multiple populations even without sexual contact information.^{48}

It is possible to express *VEI* mathematically in terms of the reduction in viral load by using the functional relation between HIV transmission probability per coital act^{49} and the logarithmic (base 10) change in viral load attributable to vaccination (see Appendix). It is generally believed that a candidate HIV vaccine that reduces viral load by 1 to 2 logs for at least 2 years might be suitable for licensure considerations.^{27,49a} A reduction of 1 log in viral load would correspond to *VEI* ≈ 60%, whereas a reduction of 2 log would correspond to *VEI* ≈ 80%.

The third efficacy is *VEP*, defined here as the percent reduction in the transition rate from HIV infection to AIDS. It represents the vaccine's effect of slowing HIV clinical disease progression in those who were vaccinated before the time of acquiring the infection. The biologic mechanism reflects observed vaccine outcomes in primate models,^{3,4,50} the potent prognostic value of the viral load set point for the progression to AIDS,^{51-57} and the established relation between viral load and disease progression in macaques.^{58,59} Therefore, *VEP* and *VEI* are biologically closely related. The *VEP* can be estimated using the endpoints of efficacy trials as surrogates for the rate of disease progression. The surrogates can include viral titers in various tissue compartments, such as blood plasma, lymph nodes, and genital secretions, and CD4 cell concentration counts.^{47}

In addition to the biologic effects (*VES*, *VEI*, and *VEP*), a vaccine may induce behavioral effects attributable to changes in the perception of risk among those vaccinated or even among the general population after vaccine administration. Vaccination can induce behavioral disinhibition if those vaccinated perceive vaccination as a protection from infection, but vaccine administration may also induce behavioral inhibition if the vaccine is distributed in conjunction with a behavioral intervention program. Behavioral disinhibition has been observed in a study of participants in phase 1 and phase 2 HIV vaccine safety and immunogenicity trials,^{60} although no such increase in risk has been observed in a phase 3 vaccine trial.^{61,62} In a similar vein, risk increase has been observed after the availability of highly active antiretroviral therapy (HAART),^{63-67} although there is also evidence that the administration of HAART has decreased risk behavior.^{68} To quantify this effect, one can define a behavioral vaccine efficacy^{69} and measure it in a unblinded vaccine efficacy trial or in a 3-armed efficacy trial that includes a unblinded arm.^{70} Community randomized clinical trials provide another setting for measuring this effect. It must be noted, however, that defining behavioral efficacies is fraught with complexity, because the changes in risk behavior may vary across populations, such as vaccinated versus nonvaccinated, and can take different forms, such as changes in partnership formation and frequency of coital acts as well changes in the use of preventive measures (eg, condoms). In this work, we considered a change in risk behavior after vaccine administration among the vaccinated relative to the nonvaccinated population in addition to a change in risk behavior among the nonvaccinated population after vaccine administration.

Vaccine efficacies may wane with time not only because of the emergence of escape mutants^{71} but as a result of the reduced antigenic exposure to the virus.^{72} As we show here, and as has been reported earlier,^{7,73} the durability of the efficacies plays an integral role in the impact of the vaccine at the population level. Note that the effect of waning of vaccine-induced immunity after infection is implicitly factored in the model in *VEP*, because it represents the vaccine-induced improvement of the waiting time from seroconversion to AIDS. In presence of waning, *VEP* would be reduced.

Although there are promising HIV vaccine candidates under development,^{74} there is not yet any vaccine with determined efficacy measures that we can use to assess the impact of vaccination. Therefore, we explore the impact in a spectrum of settings that spans plausible ranges for these efficacies.

### Model Structure

The vaccine properties discussed so far provide the mechanisms that influence the impact at the population level. We used an adaptation of the Anderson-Hanson model^{6} to translate what these properties imply at the population level. The formalism is delineated in the Appendix, along with parameter definitions and values. The model includes 2 epidemiologically distinct infected classes depending on vaccination status and assumes an adolescent vaccination policy. It is a culmination of several closely related efforts that started with the work of Anderson et al^{18} and continued in a series of influential papers by Blower and his colleagues,^{11-17} who progressively incorporated more complexity into the model, such as waning of vaccine immunity, changes in infectivity and sexual risk behavior, and slower clinical disease progression to AIDS.

## RESULTS

### Summary Measures of Vaccine Utility at the Population Level

We consider the following criterion to assess the utility of vaccination at the population level. We define a vaccine to be definitely beneficial if it reduces endemic equilibrium values for disease prevalence, incidence, incidence rate (number of incident infections per susceptible individual per year or hazard rate of infection), and disease mortality compared with these values in the absence of vaccination. We define a vaccine to be partially beneficial if at least one but not all of these equilibrium values are reduced after vaccination. Finally, we define a vaccine to be perverse if none of these equilibrium values are reduced and at least one of them increases after vaccination.

The next logical step is to investigate whether there is a simple mathematical measure that can determine if a vaccine is definitely beneficial, partially beneficial, or perverse. Knowing that the best measure of the strength of infectious spread is the basic reproductive number (number of secondary infections that an index case would generate on introduction into an infection-free population^{75}), it is reasonable to postulate that a vaccine lowering the reproductive number would also define a definitely beneficial vaccine. Indeed, this is the case. We find that the vaccine impact measure introduced by McLean and Blower^{15} to quantify the vaccine efficacy in reducing the reproductive number is a single measure that can distinguish a definitely beneficial vaccine from a partially beneficial or perverse one. Because we use this measure to express the criterion for vaccine utility, we label it here as vaccine utility. It is expressed in our formalism as follows:

Here, *r* is the measure of risk behavior change of the vaccinated relative to the unvaccinated in the partially vaccinated population, *g* is the measure of risk behavior change of the nonvaccinated after vaccine administration relative to that before vaccine demonstration, *f* is the fraction of susceptibles entering the sexually active population who are vaccinated, *Dvp* is the average duration of vaccine protection, *T* is the average duration from seroconversion to AIDS, and *L* is the average duration of the sexual activity lifespan. It bears notice that ϕ dependence on *VES* and *VEI* is symmetric, which highlights how *VEI* can be just as important as *VES* in determining the utility of vaccination.

The ϕ provides a useful measure to assess the utility of a vaccine rapidly, given its efficacy measures. We found that if ϕ > 0, the vaccine is definitely beneficial and that if ϕ ≤ 0, the vaccine is partially beneficial or perverse, because the vaccine would lead to at least 1 unfavorable outcome in the epidemiologic measures that define utility according to the previous criterion of utility. For example, a vaccine may increase HIV prevalence, although simultaneously decreasing incidence, incidence rate, and AIDS mortality. According to our criterion, this is a partially beneficial vaccine; however, it still can be considered favorable overall as an intervention measure, because the increase in prevalence may simply reflect the slower disease progression.

Another useful summary measure is vaccinee infection fitness (ψ)

which compares the number of secondary infections that an infected vaccinee would cause in a partially vaccinated but infection-free population with the number of secondary infections that an infected nonvaccinated individual would cause in the same partially vaccinated but infection-free population (see Appendix). This measure can be seen as a generalization of the immunologically naive susceptible measure introduced by Halloran et al^{76} and the fitness ratio measure introduced by Smith and Blower.^{17} The ψ underscores the heterogeneity in transmission that vaccination introduces in a partially vaccinated population and is independent of *VES* and *Dvp* because these variables affect preinfection mechanisms.

The ψ is convenient from a public health perspective because it captures the effect of the vaccine on secondary transmissions and can help to determine whether additional intervention efforts should be predominantly aimed at nonvaccinated or vaccinated infected individuals. Note that even if a vaccinated infected causes more secondary cases than an unvaccinated infected (eg, ψ > 1), this does not necessarily imply that the vaccine is detrimental, because vaccination can also reduce susceptibility to infection (ie, ψ > 1 but ϕ > 0). The following expression provides a useful link between these two measures:

### Summary Measure of Vaccine Effectiveness at the Population Level

The utility of the vaccine, which depends on vaccine properties and changes in sexual risk behavior after vaccination, does not alone guarantee the effectiveness of a vaccine in substantially reducing HIV infection incidence. The effectiveness depends on vaccine properties and the prevailing risk of HIV acquisition in a given population. To provide a criterion for the effectiveness, we define a measure labeled the hazard index (see Appendix) as follows:

This measure has the simple interpretation as the average number of HIV infectious exposures per lifetime that a typical susceptible experiences in a given population group. Although ξ, unlike ϕ and ψ, is not strictly a vaccine impact measure, vaccination changes the waiting time before the next HIV exposure, thereby implicitly changing ξ. The waiting time before the next exposure may vary across groups within the population, because different subpopulations experience different incidence rates. There are 2 hazard indices in our formalism corresponding to the unvaccinated and vaccinated subpopulations, respectively.

As opposed to vaccines that provide sterilizing immunity, imperfect vaccines provide only partial protection against infection (or transmission). Therefore, their mechanism of action is basically to increase the waiting time before the next infectious exposure to HIV instead of preventing HIV acquisition. We find that a beneficial vaccine becomes an effective vaccine only when it increases the waiting time before the next exposure so that it is longer than the sexually active lifespan (ξ < 1). Otherwise, if the waiting time is substantially less than the sexual lifespan, vaccinated individuals are still likely to acquire HIV because of the repeated exposures to the virus. Figure 1 illustrates this assertion, where HIV incidence is calculated as a function of ξ for each of the unvaccinated and vaccinated subpopulations. The population here consists of 100,000 adults, half of whom are vaccinated with a partially effective protection that varies from *VES* = 100% at ξ = 0.0 to *VES* = 0% at ξ = 4.5 (ie, no vaccination).

It is evident how HIV incidence changes slowly as we lower ξ (by increasing *VES*) until we reach the boundary region at ξ ≈ 1. HIV incidence then diminishes rapidly with the decrease in ξ. The vaccine is most effective in reducing HIV incidence only when it reduces the average number of HIV infectious exposures a person experiences to <1 per lifetime, implying that most people would never encounter an exposure to HIV. The vaccine may be successful in substantially reducing ξ, for example, from ξ = 4 to ξ = 2, but this large reduction translates into a more modest impact on HIV incidence.

The different vaccine efficacies dictate, in part, the value of ξ for any given population. The ξ can be reduced by increasing *VES* or *VEI* and decreasing sexual risk behavior, but it is increased with increasing *VEP* and increasing sexual risk behavior. The failure of a definitely beneficial vaccine to reduce ξ to <1 does not strictly construe a “failure” of vaccination as an intervention but indicates the need for other interventions, such as sexual risk behavior reduction, to be incorporated synergistically with vaccination. First-generation vaccines may not provide the “silver bullet” to control HIV, but they do take us a long way toward this end. Any vaccination policy needs to optimize the intervention package so that ξ is substantially <1.

### Qualitative Features of the Impact of HIV Vaccination

The interplay between the different vaccine efficacies and the changes in sexual risk behavior after vaccination and the prevailing hazard rate of HIV infection in a given population leads to an intricate picture for the population level impact. We examine this picture by discussing 4 potential vaccination scenarios at the endemic equilibrium.

We start first (Fig. 2) with an optimistic scenario in which we have a high-level vaccination coverage of *f* = 80%, a limited protection against infection of *VES* = 30%, a high infectivity reduction of *VEI* = 70% (corresponding to a 1.4-log reduction in HIV-1 viral load), and a substantial decrease in the rate of progression to AIDS of *VEP* = 50%. These properties embody potential characteristics of first-generation AIDS vaccines that may be licensed despite the minimal protection against infection.^{31} We also assume a modest increase in risk behavior of those vaccinated relative to those unvaccinated (*r* = 10%) but assume here, and in all subsequent results, that the unvaccinated population maintains the same level of risk behavior as that before vaccine administration *g* = 0%.

Figure 2 illustrates how such a vaccine is effective at controlling the spread of the disease. In Figure 2A, the vaccination reproductive number (*R*_{0V}), vaccine utility (ϕ), vaccine infection fitness (ψ), and hazard indices (ξ) of the unvaccinated and vaccinated groups are shown as a function of vaccine duration of protection. Clearly, ϕ > 0 confirms that this vaccine is a definitely beneficial vaccine, irrespective of the duration of protection. The utility, as expressed by ϕ, increases with the duration of protection that can be achieved by a vaccine with long-lasting immunity or with repeated vaccination, such as for influenza. The *R*_{0V} is greater than the classic endemicity threshold at *R*_{0V} = 1, except when the duration of protection is longer than 35 years, indicating how it is difficult to eradicate HIV/AIDS without a substantially long period of vaccine protection. Noteworthy here is that the classic criterion of *R*_{0V} < 1 for disease eradication^{75} may not be sufficient in certain parameter regimens if backward bifurcation is present.^{21,77-81} The ψ = 54% reflects that the positive impact of the vaccine is largely attributable to infected vaccinees generating almost 50% fewer secondary infections compared with nonvaccinated infected persons. The ξ < 1 for the unvaccinated and vaccinated groups for any duration of protection exceeding 2 years testifies to the effectiveness of this vaccine in substantially reducing HIV infectious spread.

Figure 2B displays the population size, prevalence, incidence rate (hazard rate of infection) among unvaccinated and vaccinated, incidence, and AIDS population (proxy for AIDS mortality) relative to their values in the absence of vaccination and also as a function of vaccine duration of protection. It is manifest how the 3 summary measures ϕ, ψ, and ξ have expressed the utility and effectiveness of the vaccine, as is visible in such epidemiologic measures as incidence and prevalence in Figure 2B. This highlights the usefulness of these simple mathematical expressions as powerful tools to characterize the impact of vaccination once vaccine efficacies are determined. It is also evident in the figure how the vaccine is effective in rebalancing the demographics that were under strain because of AIDS mortality.

One salient feature of the impact is the slower reduction in HIV incidence and AIDS mortality (and prevalence to some extent) compared with incidence rate. This aspect arises, in part, because the slower progression and reduced transmission allow the population size to grow, thereby feeding a larger pool of susceptibles to the system to be infected. It is not surprising then that the incidence rate is a much better proxy for the impact of the vaccine than the overall incidence, prevalence, or disease mortality. This re-emphasizes the importance of taking this aspect of HIV epidemiology into consideration after vaccine administration, because slow declines in some epidemiologic measures may mask the benefits of the vaccine as an intervention.

Next, we explore a pessimistic scenario of a detrimental vaccine at the population level, as shown in Figure 3. The vaccine properties here are a high-level vaccination coverage of *f* = 80%, a limited protection against infection of *VES* = 30%, a meager infectivity reduction of *VEI* = 30% (corresponding to a 0.4-log reduction in HIV-1 viral load), a modest decrease in the rate of progression to AIDS of *VEP* = 30%, and a substantial increase in risk behavior after vaccination of *r* = 50%. Despite the protective effects of the vaccine, and as suggested by ϕ < 0, ψ > 1, and ξ > 1, the increase in the HIV infectious period (ie, slower disease progression) and the behavioral disinhibition after vaccination have undermined the impact of the vaccine to the extent that the vaccine is perverse from a public health point of view. The prevalence, incidence rate, incidence, and AIDS cases have all increased in the presence of vaccination (ϕ < 0). Additionally, the ψ > 1 confirms that the perversity is largely attributable to infected vaccinees being more efficient transmitters of the virus than the unvaccinated infected population. Finally, although the vaccine is certainly perverse from a population point of view, it is still beneficial from an individual perspective because it slows disease progression and confers protection, although limited, against infection. This outcome emphasizes the potential complexity of forming a public health policy once a vaccine is available. Similar issues are at the center of the recent debate over antiretroviral therapy.^{82-84}

The presence of a negative impact of a vaccine on some epidemiologic measures does not automatically imply that the vaccine is perverse overall (Fig. 4). Here, *f* = 80%, *VES* = 30%, *VEI* = 60% (corresponding to a 1.1-log reduction in HIV-1 viral load), *VEP* = 60%, and *r* = 50%. Despite the presence of adverse effects (ϕ < 0) and the increase in *R*_{0V}, the vaccine could be deemed beneficial because it reduces, albeit marginally, the incidence rate among the nonvaccinated and improves the toll of HIV on disease mortality and population size, even though the incidence rate among the vaccinated has increased as well as the prevalence. Again, this mixed outcome is captured by our fitness measure ψ, which reflects the heterogeneity in disease spread between the vaccinated and nonvaccinated infected. Infected vaccinees cause more secondary infections (ψ > 1), in addition to being exposed to a higher hazard of infection because of their behavioral disinhibition.

This result hints at a potential conceptual transformation in the fight against HIV after vaccine introduction from an emphasis on reducing infection spread to that of managing and improving the prognosis of infection. Yet, the greater number of subjects living with HIV but not AIDS may be a concern, particularly in resource-poor settings, because public health systems may find it difficult to deal with a larger pool of HIV persons who are more prone to such coinfections as malaria and tuberculosis.^{85}

Behavioral inhibition is a possibility after vaccine administration if vaccines are administered as part of a comprehensive program that includes prevention education.^{86} Although some may doubt the feasibility of such programs, the success of Uganda in checking HIV spread and reducing incidence suggests otherwise.^{33,38} Therefore, we present a fourth scenario in Figure 5, where we assume the same coverage and biologic efficacies as those of the mixed impact vaccine in Figure 4 (*f* = 80%, *VES* = 30%, *VEI* = 60%, and *VEP* = 60%) but assume a reduction of 35% in risk behavior among vaccinees (*r* = −35%). This value for risk reduction is our estimate of the average effective reduction that was achieved in Uganda starting from the late 1980s. It has been derived by modeling the Uganda epidemic using an HIV transmission model^{87} with 2 risk groups and 3 HIV stages to fit the observed decline in incidence and prevalence in a longitudinal cohort study in rural southwest Uganda.^{88} The approximate estimate accords well with repeated behavioral surveys from 1989 through 2002, which found a 2-year delay in the onset of sexual activity, a 9% decrease in casual sex, and an increase in ever use of condoms among women from 1% to 16% and among men from 16% to 40% by 2000.^{89,90} The estimate is also in line with earlier modeling estimates that indicated a 50% reduction in the proportion of men having one-off sexual contacts with female sex workers and a 25% reduction in partner change rates in the whole population.^{33,91}

The synergy of vaccination and prevention programs is striking, as can be seen in Figure 5. The same biologic vaccine that rendered a mixed impact with behavioral disinhibition is now rendering a robust beneficial impact (ϕ > 0). It is even possible to contain the disease with a long duration of vaccine protection (>24 years). The vaccine alone was not able to reduce the number of exposures per lifetime to <1 (ξ < 1), but administered synergistically with behavioral inhibition, this was done easily, leading to an effective vaccination policy (ξ ≪ 1).

## CONCLUSIONS

We explored the population-level implications of administering imperfect prophylactic HIV vaccines and provided analytic insights into the dynamics using a mathematical model that incorporates the major potential vaccine effects. Although our study provides general features of the impact, it should be seen as a building block toward a more comprehensive assessment of vaccine impact. Other aspects that need to be explored are the differential effect of the vaccine on HIV stages, the differential effect across sexual risk groups, the role of antiretroviral treatment, and changes in the virus ecologic landscape after vaccine introduction.

Our work highlights how a vaccine that reduces viral load by 1 to 2 logs might have a substantial benefit at the population level. Such an effect could be obtained by the current adenovirus-vectored vaccine candidates that induce cellular immune responses. An ongoing efficacy trial is testing a candidate vaccine consisting of a mixture of 3 adenovirus serotype-5 vectors, each encoding the HIV-1 *gag*, *pol*, or *nef* gene, as vaccine antigens.^{42} A parallel efficacy trial of the same candidate vaccine has just started in South Africa. Moreover, an efficacy trial testing a DNA prime plus adenovirus serotype-5 boost regimen is planned for 2007, where each component encodes HIV-1 *gag* and *pol* genes (the DNA component also contains HIV-1 *nef*) and 3 envelope gp140 molecules of subtypes A, B, and C. We should know the efficacy of these vaccines within few years. Based on the work presented here, there is room for optimism that first-generation vaccines, despite having limited efficacy against infection acquisition, may still be effective, particularly if administered synergistically with a prevention program.

## ACKNOWLEDGMENTS

The authors thank Peter B. Gilbert and M. Elizabeth Halloran for their informative comments and suggestions on a draft version of this manuscript.

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### APPENDIX

### Model Structure

We use an adaptation of the Anderson-Hanson model^{6} to study the impact of vaccine administration. We elected to use this model, as opposed to more complex vaccine or HIV epidemiology models,^{19,21,22,92,93} because our goal is to provide simple and widely accessible measures that can be used as tools to distinguish qualitatively and rapidly the impacts of potential vaccines among the many candidate products for vaccine development. The model consists of 5 differential equations

that represent 5 epidemiologically distinct populations. The *S* population stands for susceptible individuals who have no immune protection, and the *V* population represents those whose vaccine immunity is biologically effective. The HIV-infected subjects who have not yet progressed to AIDS are divided into the 2 groups *Y* and *W*. The *Y* group is the population of those who acquired the infection at a time they possessed no vaccine-induced immunity, whereas the *W* population stands for those infected despite being effectively vaccinated. Finally, *A* is the population of those living with AIDS. A schematic diagram of the model is shown in Figure A1, and further mathematical analyses of this type of model can be found elsewhere.^{6,11-18,21} Elbasha and Gumel,^{21} in particular, provide detailed analyses of the mathematical structure, including the impact of sexual risk behavior among AIDS patients.^{94} Further details of this model and its analysis are available on request to the corresponding author.

Population demographics are incorporated through a constant rate at which susceptibles enter the sexually active population (Υ) and a constant rate of removal from this population (μ). Vaccine administration and effects are expressed through the vaccine efficacies (*VES*, *VEI*, and *VEP*), the fraction of susceptibles entering the sexually active population (ie, adolescents) who are vaccinated (*f*), the risk behavior change of the vaccinated relative to the unvaccinated in the partially vaccinated population (*r*), the risk behavior change of the nonvaccinated after vaccine administration relative to that before vaccine demonstration (*g*), and the rate at which vaccine immunity wanes (γ). Note that the transition distributions in this model are implicitly assumed to be exponential. Therefore, *Dvp* = 1/γ is the average duration of vaccine protection, *T* = 1/ω is the average duration from seroconversion to AIDS, *G* = 1/α is the average duration from AIDS to death, and *L* = 1/μ is the average duration of the sexual activity lifespan. Finally, λ is the HIV force of infection (hazard rate) given by the following:

Here, *Nsex* is the sexually active population size, ρ is the average new partner acquisition rate in the population before vaccine administration, and *z* is the HIV transmission probability per partnership. Note that *g* must be a function of *f*, that is, *g* = *u*(*f*), where *u*(0) = 0. Moreover, the incidence rate (incidence per capita of susceptible population or hazard rate of infection) among the fully susceptible population is given by λ, and that among the vaccinated susceptible population is given by (1 − *VES*) (1 + *r*)λ.

We have not explicitly included waning of vaccine immunity from the *Y* population to the *W* population because this effect is implicitly incorporated through the model parameterization and structure. From empiric studies in nonhuman primate models,^{3-5,50,58,59} the biologic effect of an HIV vaccine is to change the virologic set point, thereby putting the vaccinated in a different path to disease progression and viremia levels considering the strong relation between the virologic set point and progression to AIDS and viral load level in the chronic stage.^{51-57} Therefore, the effect of vaccine-induced immunity is to set the trajectory for the following HIV pathogenesis, which is represented in the model by 2 distinct progression profiles based on vaccination status. The risk of emergence of escape mutants^{72} at a population level shortens the average time from seroconversion to AIDS, thereby reducing the value of *VEP* for those vaccinated.

### Model Parameters

The model parameters and their assumed numeric values are listed in Table A1. For the demographics, we assume, for the sake of simplicity, a stationary population size in absence of AIDS mortality (eg, Υ = μ*N*_{0}).^{75} This assumption facilitates the disentanglement of epidemiologic effects from demographic effects. The sexual activity lifespan is set at 35 years to represent the age group from 15 to 49 years that is typically used to characterize the sexually active population.^{95}

The average HIV transmission probability per coital act over all HIV stages and the average frequency of coital acts are derived from an analysis of the measurements of Wawer et al.^{96} The average HIV transmission probability per partnership is derived using the binomial (Bernoulli) model *z* = 1 − (1 − *P*)τ_{p}*n*, where *p* is the transmission probability per coital act, τ_{p} is the duration of partnership, and *n* is the frequency of coital acts.

The sexual behavior input is informed by a study in Kisumu, Kenya, which is roughly representative of that measured across sub-Saharan Africa.^{97-100} The average new sexual partner acquisition rate is set at 2.0 per year as a representative value considering the following measurements of sexual behavior in Kisumu: (1) the mean number of nonspousal partners (excluding contact with sex workers) of 1.67 for men and of 1.23 for women during the past 12 months,^{99} (2) the average number of nonspousal partnerships (excluding contact with sex workers) for men of 701 per 1000 men-years,^{98} and (3) the average number of male client contacts with sex workers of 960 per 1000 men-years.^{98,100}

We have chosen the representative value of 6 months for the average duration of sexual partnerships. This estimate reflects the midrange value between the long duration and high coital frequency of spousal and nonspousal partnerships, excluding contacts with sex workers (median nonspousal partnership is 11 months in Kisumu),^{98} and the variable but generally short duration and low coital frequency partnerships with sex workers.^{100} The estimate also reflects the comparable number of nonspousal partnerships, excluding contacts with sex workers (701 per 1000 men-years in Kisumu)^{98} and the number of contacts with sex workers (960 per 1000 men-years in Kisumu).^{98,100}

### Model Output

We examine the impact of vaccination at the endemic equilibrium. Equilibrium analysis is useful because it disentangles the interaction-strength effects from the temporal effects. Moreover, the calculations at the equilibrium provide an effective proxy of the cumulative impact of the vaccine on HIV epidemiology as the HIV epidemic bursts, passes through its different stages, and eventually saturates. We found that the results of the equilibrium analyses presented here are similar to those we performed at specific time points in the epidemic (eg, HIV prevalence peak) or over a fixed period of time. Noteworthy is that the endemic equilibrium needs approximately 2 centuries to be reached in the present model, because risk behavior is uniform. Inclusion of heterogeneity and mixing in risk behavior shortens this duration to just a few decades.^{87}

The HIV prevalence and incidence at equilibrium are derived using a fixed-point approach.^{101} To this end, it is useful to recast Equation A1 in the matrix form,

as follows:

At equilibrium,

resulting in

which provides analytical expressions for each population variable in the problem in terms of λ and model parameters. Explicitly, this yields the following expressions at equilibrium:

Substituting the expressions for *Y* and *W*, along with that for the sexually active population,

in Equation A2 results in the force of infection being provided in the form of

where *h*(λ) is a lengthy nonlinear function of λ. Although Equation A9 is difficult to solve analytically, one can show that there is only 1 real positive (biologically meaningful) solution to this equation and that this solution is given by the following:

Detailed proof of these statements follows along the lines discussed by Abu-Raddad and Ferguson.^{101} Equation A10 establishes a numeric solution for λ using successive approximations with rapid convergence to the exact solution. Using this solution, one can easily calculate all population variables and epidemiologic measures at the endemic equilibrium.

### Definition of Vaccine's Efficacy of Reducing Infectivity (*VEI*) and Its Relation to HIV-1 Viral Load Level in Vaccinees

*VEI* is defined as follows:

where *SAR*_{1} is the secondary attack rate from an infected vaccinated person to his or her unvaccinated sexual partner, whereas *SAR*_{0} is that of an infected nonvaccinated person to his or her unvaccinated partner.

*VEI* can be linked mathematically to the change in HIV-1 viral load level in infected vaccinees relative to infected nonvaccinated individuals. This is done using the measured relation between HIV-1 transmission probability per coital act and HIV-1 viral load.^{49} Explicitly,

Here, *x* is the log (base 10) reduction in viral load level attributable to vaccination.

### Measures of HIV Vaccination Impact at the Population Level

The basic reproductive number in absence of vaccination is given by the following:

whereas that in a partially vaccinated population, the vaccination reproductive number, is given for the previous model by the following:

Both of these expressions have been derived by forming the next-generation matrix and then finding the dominant eigen value of this matrix along the lines described elsewhere.^{102,103} This is a method for deriving the reproductive number, and other methods, such as the Jacobian matrix, can lead to different results in some circumstances.^{103} Of particular note is that the classic requirement of having *R*_{0V} < 1 for disease eradication^{75} may not be sufficient in certain parameter regimens because of the possibility of backward bifurcation in multiple group models, including the model we study here.^{21,77-81}

Vaccine impact is defined by McLean and Blower^{15} as follows:

but is labeled in our work as vaccine utility. Its expression in Equation 1 is derived by substituting Equations A13 and A14 in Equation A15.

The vaccinee infection fitness is defined by the following:

This formula compares the number of secondary infections that an infected vaccinee would cause in a partially vaccinated but infection-free population

with the number of secondary infections that an infected nonvaccinated individual would cause in the same partially vaccinated but infection-free population

The latter 2 expressions are derived using the elements of the second-generation matrix,^{102} which describes the number of secondary infections for a given index HIV case whether this index infection is vaccinated (*RW*) or not (*RY*).

We define the hazard index in terms of the hazard rate of infection (force of infection) as follows:

Hence, ξ among the unvaccinated population is given by the following:

and that among the vaccinated population is given by the following:

Because the transition distributions in this model are implicitly assumed to be exponential, ξ can be expressed in terms of the waiting time to next HIV exposure (Equation 4). Note that because the hazard rate can vary with time, ξ varies throughout the epidemic.

### Multivariate Sensitivity Analysis

We have incorporated a multivariate sensitivity analysis to examine the impact of uncertainty in the most relevant parameters for the impact of a potentially efficacious vaccine at the individual level: *VES*, *VEI*, *VEP*, and *r*. We did so for the results of Figure 2 at a vaccine duration of protection of 5 years by Monte Carlo sampling from the specified ranges of uncertainty for each parameter using the uniform distribution for 1000 runs of the model. The ranges of variation are [0.2, 0.4] for *VES*, [0.5, 0.9] for *VEI*, [0.3, 0.7] for *VEP*, and [0.0, 0.2] for *r*. Figures 2 through 5 serve as sensitivity analyses for the vaccine impact versus the vaccine's duration of protection.

The results of the multivariate sensitivity analysis are shown in Figure A2. The ψ and ξ_{vaccinated} show the highest sensitivity to the variations in these parameters, whereas vaccine utility and vaccination reproductive number show the least sensitivity. All other variables show intermediate sensitivity. The high sensitivity of ψ and ξ_{vaccinated} is attributable to these variables depending primarily on effects arising from the vaccinated population, whereas the other variables are affected by both the unvaccinated and vaccinated populations. Cited Here...

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**Keywords:**

HIV; infectious disease; mathematical model; vaccines; vaccine efficacy