## Abstract

The effect of antiretroviral therapy (ART) in preventing human immunodeficiency virus (HIV) infections and averting acquired immunodeficiency syndrome (AIDS) deaths in the San Francisco gay community over the next 10 years was predicted. A transmission model was coupled with a statistical approach that enabled inclusion of a high degree of uncertainty in the potential treatment effects of ART (in terms of infectivity and survival), increase in risky behavior, and rate of emergence of drug resistance. Increasing the usage of ART in San Francisco would decrease the AIDS death rate and could substantially reduce the incidence rate.

Currently, 30% of the San Francisco gay community are HIV-infected (1). About 50% of these HIV-infected men are taking combination ART (2); these three or more drug regimens include recently developed protease inhibitors, nonnucleoside reverse transcriptase inhibitors, or both. Part of the recent decrease in the San Francisco AIDS death rate (3) could be attributable to the effect of ART, as ART decreases disease progression rates (4). However, because treated individuals are likely to retain some degree of infectivity, it is possible that ART could lead to an increase in the infection rate (5). Furthermore, drug-resistant HIV strains (that are less responsive to therapy) have emerged (6), and risky behavior has begun to increase in San Francisco (7). Therefore, whether the epidemic-level effects of ART will be beneficial or detrimental is unclear.

To predict (with a degree of uncertainty) the effectiveness of ART in the San Francisco gay community, we developed and analyzed a mathematical model. Our model includes the potential effects of ART on the transmission dynamics of both drug-sensitive and drug-resistant HIV strains. It is specified by five ordinary differential equations (8) (Fig. 1) and allows for drug-resistant strains (that differ in their infectivity and disease progression rates from drug-sensitive strains) to emerge during treatment and to be sexually transmitted (6). Acquired resistance develops because of a variety of factors (8); we model the aggregate effect of all these factors by a single parameter *r*. We model the potential treatment effects of ART by assuming that ART [by reducing viral load (9)] increases average survival time and reduces infectivity, and that drug-resistant strains will be less responsive to therapy than drug-sensitive strains (6). Treatment (in our model) has three outcomes. A patient can respond to ART and remain as a nonprogressor for a specified amount of time, experience clinical failure and death without developing drug resistance (9), or virologically fail treatment and develop drug resistance (10). Individuals can go on and off ART, and drug-resistant infections can revert to drug-sensitive infections if the selective pressure of treatment is removed (11) (Fig. 1).

We predicted the effectiveness of a high usage of ART over the next 10 years in the San Francisco gay community by analyzing our model with time-dependent uncertainty analyses (12,13). Effectiveness was predicted in terms of the cumulative number of HIV infections prevented and the cumulative number of AIDS deaths averted (14). The San Francisco epidemic has been well studied, and the values of several of the parameters necessary for prediction are known (15); however, the values of other parameters are less certain. Hence, we conducted two uncertainty analyses (an optimistic and a pessimistic analysis) on the basis of different assumptions regarding the rate of increase in risky behavior and the rate of emergence of drug resistance. Both analyses included a high degree of uncertainty in the potential treatment effects of ART (on increasing survival and reducing infectivity). For the optimistic analysis we assumed that the rate of emergence of resistance would remain at a constant, fairly low value [only 10% of cases would acquire resistance per year (16)], and that risk behavior would not increase. For the pessimistic analysis we assumed that the rate of emergence of resistance could substantially increase [10 to 60% of cases could acquire resistance per year (17)], and that risk behavior could increase from almost no increase to a doubling (17).

For each uncertainty analysis we used our model and Latin hypercube sampling (LHS), a type of stratified Monte Carlo sampling (18); LHS has been described elsewhere (12). To make predictions, we assigned each uncertain parameter a probability density function (pdf); the pdf reflected either the uncertainty in the value of the parameter, or the degree to which the parameter could vary if it was being used as an “experimental variable” (12). We used the usage rate of ART (*F*
_{S}) as an “experimental variable” and predicted the effect of increasing usage rates that ranged from treating 50 to 90% of HIV-infected men; currently, only ∼50% of HIV-infected gay men in San Francisco take ART (2,19). We used a uniform pdf (range 0.5 to 0.9) to specify the uncertainty in *F*
_{S}, where*F*
_{S} represents the fraction of drug-sensitive cases treated with ART (20). We used LHS to sample 1000 values of *F*
_{S} from this pdf; hence, each of the 1000 predictions in each uncertainty analysis had a different ART usage rate (20).

By specifying pdfs, we included uncertainty in our estimates for the average duration of treatment, and the treatment effects of ART (in terms of survival and infectivity). Because ART has only been widely available for 2 to 3 years, it is not known how long patients will remain on ART. We assumed that from 1 to 25% (with a most likely value of 5%) of drug-sensitive patients would give up ART per year (21). Zidovudine monotherapy, by reducing viral load, decreases infectivity for both vertical and sexual transmission (22). Viral suppression with ART is 10- to 100-fold more effective than viral suppression with Zidovudine (9). Thus, to construct a pdf we assumed that ART could cause a 2- to 100-fold reduction in infectivity (23). Many patients on ART who have maintained a virological response have not clinically deteriorated, and clinical benefits of ART can exceed the duration of the virological response (24). However, the data on the survival effects of ART are from clinical studies with short-term follow-up. Hence, we included a high degree of uncertainty in estimating the potential longer term survival effect of ART. We assumed that the average survival time for ART patients who remain virologically suppressed could be 1.5 to 3 times greater than the average survival time for untreated drug-sensitive individuals (25).

Drug-resistant HIV strains have only recently emerged; thus, their behavior with respect to transmission and pathogenesis is relatively unknown. For our uncertainty analyses, we assumed that drug-resistant strains would be attenuated to some uncertain degree in comparison with drug-sensitive strains (26). We also included uncertainty in the degree to which individuals infected with drug-resistant strains would transmit drug-sensitive strains (27) and in the rate at which untreated drug-resistant infections would revert to drug-sensitive infections (27). Because drug-resistant strains are less responsive to therapy (6), we modeled a differential treatment response to ART (28). We assumed that ART would be less effective in decreasing disease progression rates and in reducing infectiousness in drug-resistant infections (28).

Our uncertainty predictions for the effectiveness of ART in terms of the cumulative number of AIDS deaths averted are shown in Fig. 2. Under optimistic assumptions the cumulative AIDS death rate was always reduced; hence, effectiveness was always positive (Fig. 2A). Under pessimistic assumptions, certain conditions led to an increase in the cumulative AIDS death rate over time; hence, effectiveness was sometimes negative (Fig. 2B). However, both analyses predicted that the most likely outcome would be a substantial reduction in the cumulative AIDS death rate (Fig. 2). After 10 years, 33% (median value under optimistic assumptions) (Fig. 2A) and 18% (median value under pessimistic assumptions) (Fig. 2B) of AIDS deaths had been averted.

The two uncertainty analyses generated very different predictions for the effectiveness of ART in terms of the cumulative number of HIV infections prevented. Under optimistic assumptions, effectiveness began at zero and increased with time; ART substantially reduced the number of new infections (Fig. 2C). After 10 years of ART, 40% (median value) of new HIV infections were prevented (Fig. 2C). Under pessimistic assumptions [because the levels of risky behavior rose by an average of 50% (17)], the incidence rate initially increased, and thus the effectiveness of ART began at a negative value (∼40% median value) (Fig. 2D). Over time effectiveness increased, as ART decreased the incidence rate. After 10 years, the effect of ART on decreasing the incidence rate finally balanced out the initial rise in the incidence rate (∼0% median value). If widespread usage of ART would directly cause significant increases in risk behavior, then our results imply that the net effect of ART on preventing new infections could be almost zero. However, if levels of risk behavior will increase independently of ART usage, then our results indicate that ART could prevent a substantial number of new infections.

To identify the key factors that determined the effectiveness of ART, we performed time-dependent sensitivity analyses (12,18). We used the 10 years of predicted data from the uncertainty analyses to calculate time-dependent sensitivity coefficients; a partial rank correlation coefficient (PRCC) was calculated annually for each parameter. Key factors that determined effectiveness (as identified by the value of their PRCC) are shown in Table 1 for years 1, 5, and 10. Not surprisingly, one of the two key factors that increased the effectiveness of ART (in terms of decreasing the death rate) was the degree to which ART increased survival (1/*v*
_{S}
^{T}) (Table 1). The longer ART-treated individuals survived, the lower the cumulative death rate; under optimistic assumptions, this effect was relatively constant over time. Under pessimistic assumptions, this effect waned over time as the increased levels of risk behavior that initially caused a rise in the incidence rate were translated into an increasing death rate. The most important key factor in increasing effectiveness was the usage rate of ART (*F*
_{S}) (Table 1). Increasing usage rates of ART significantly reduced the AIDS death rate and prevented a substantial number of new infections—under the optimistic assumptions and, perhaps surprisingly, even under the pessimistic assumptions. This conclusion is also shown by the unadjusted data generated by the uncertainty analyses (Fig. 3). Under optimistic or pessimistic assumptions, a high usage of ART substantially decreased the death rate (Fig. 3A) and prevented a significant number of new infections (Fig. 3B).

Not surprisingly, effectiveness decreased as both the infectiousness of treated drug-sensitive (β_{S}
^{T}) and drug-resistant patients (β_{R}
^{T}) increased (Table 1). Hence, if infectiousness was reduced (either by increasing condom usage in treated patients or by developing more effective drugs for viral suppression), the effectiveness of ART would substantially increase. High rates of emergence of drug-resistant strains would result in a fairly high prevalence of drug-resistant infections (29), but our sensitivity results revealed that even these high rates of emergence of resistance would not significantly affect either the death rate or the incidence rate (30). The most important key factor that decreased the effectiveness of ART (both in terms of the number of deaths averted and the number of infections prevented) was the degree of increase in risk behavior (Table 1). Even under pessimistic assumptions, a high usage of ART decreased the incidence rate (Fig. 3B); however, an increase in risky behavior of only 10% was enough to counterbalance the benefits of ART (Fig. 3C). Greater increases in risk behavior resulted in the incidence rate increasing, and hence effectiveness becoming negative (Fig. 3C).

Since 1996–97 (when ART became readily available), the AIDS death rate in San Francisco has decreased (3). Our predictions show that a decreased death rate is to be expected under both optimistic and pessimistic assumptions. However, our optimistic and pessimistic assumptions lead to very divergent incidence predictions. Our results show that the higher the usage of ART, the greater the number of infections that will be prevented (Fig. 3B). Because the current usage rate of ART in the San Francisco gay community is ∼50% (2, 19), the usage rate should be increased. Recently, increases in risky behavior in the gay community have been reported (7). Our results show that a high usage of ART could counterbalance the effect of increasing levels of risky behavior and prevent a substantial number of new infections. Our results imply that the incidence rate in San Francisco will first rise (to a level that will be determined by the degree of increase in risky behavior) and will then fall (to a level that will be determined by the degree of usage of ART). Significant efforts should be made to prevent risk behavior increasing because even small increases will overcome the effect of ART on reducing the incidence rate (Fig. 3C). To maximize the effectiveness of ART, 13 treatment programs should be combined with effective behavioral intervention programs.

Mathematical models can be used as health policy tools to guide public health decisions (31). However, a model is always an abstraction of reality and never a mirror of reality. Our model reflects current biomedical understanding; we applied Occam's razor to capture the essential processes of the transmission dynamics of drug-sensitive and drug-resistant strains in San Francisco. As biomedical knowledge accumulates our model can be made more complex, and as data accumulate we can reduce the uncertainty in our predictions. We have presented only short-term predictions because it is likely that more effective drugs and drug regimens will be developed over time; however, our model can also be used to evaluate the longer term consequences of ART (32). Here, we have used parameters that are specific for San Francisco, but our methodology can be applied to evaluate the impact of more widespread usage of ART in other HIV-infected communities.

↵* To whom correspondence should be addressed. E-mail: sally{at}itsa.ucsf.edu