We also explored different levels of treatment coverage of AIDS cases and incremental coverage of individuals in successively less advanced disease stages. Analyses were performed using the Runge-Kutta 4 integration method with a fixed time step of one fiftieth of a year.41
We performed uncertainty analyses to define the magnitude of variability in model output arising from uncertainty in model input parameter values. We performed sensitivity analyses to identify the relative contribution of specific input parameters to variability in model output.35-37
We performed time-dependent uncertainty analyses using Latin hypercube sampling (LHS)42 and 1000 simulation runs, as follows. First, we varied at random a set of 19 input parameters (see Table 1) to confirm that the results of our base-case scenario analysis were within the 95% prediction interval of the uncertainty analysis. Second, we varied only the input parameters that directly affected treatment outcome as follows: (1) treatment-mediated decreases in infectivity and disease progression were varied from 50% to 100%; (2) the treatment failure rate per year was varied from 12% to 23%, representing a median time to failure of 3 to 5 years;33,43-48 and (3) the sexual activity level of the individuals with AIDS on treatment ranged from 85% to 100% of that in the general population.49 We performed this uncertainty analysis for 2 scenarios: a conservative scenario with no increase in the sexual activity level in the general population and a sexual disinhibition scenario with sexual activity increasing from 0% to 20%. Third, we repeated the uncertainty analysis for the conservative scenario with treatment coverage of AIDS cases ranging from 10% to 100%.
A smaller effect of therapy is predicted with lower treatment coverage of AIDS cases. With 50% treatment coverage, the predicted decreases in cumulative new HIV-1 infections and AIDS deaths after 10 years were 15.6% and 32.7%, respectively. Treatment of at least 25% of AIDS cases was required to achieve approximately a 10% or greater decline in cumulative AIDS deaths at 10 years.
When all infected individuals were treated early in an epidemic (5% prevalence), the reduction in cumulative AIDS deaths was 67% after 10 years (data not shown). The residual AIDS mortality is attributable to treatment failure and subsequent disease progression. The decrease in cumulative new HIV-1 infections after 10 years rose to 78.1% with maximum treatment coverage. With 100% treatment coverage of individuals with CD4 counts <350 cells/mm3, there was a 45.1% reduction in HIV-1 incidence and a 41.2% reduction in cumulative new HIV-1 infections after 10 years. The corresponding figures after late (40% prevalence) treatment introduction were 28.2% and 36.5%, respectively. The disease stages for which treatment coverage was most influential in decreasing cumulative new HIV-1 infections were recent infection and chronic infection with a CD4 count >500 cells/mm3 (data not shown).
Uncertainty analysis of the sexual disinhibition scenario showed a progressive increase in cumulative new HIV-1 infections and HIV-1 prevalence. Specifically, after 10 years, the median change in cumulative new HIV-1 infections was +12% for early (see Fig. 4C), +2.7% for intermediate, and −0.5% for late treatment implementation, respectively. By contrast, cumulative deaths from AIDS declined over time, because individual benefit of treatment was still realized (see Fig. 4D).
When the conservative scenario was assumed, the decline in cumulative new HIV-1 infections was most sensitive to the effect of treatment on infectivity, whereas the decline in cumulative deaths from AIDS was most sensitive to the effect of treatment on disease progression. Uncertainty in the effect of treatment on infectivity explained more than 90% of the variance in the predicted decline in cumulative HIV-1 infections, regardless of the time of treatment introduction or duration of treatment use. Approximately 96% of the variance in predicted cumulative deaths from AIDS after 5 years was explained by uncertainty in the effect of treatment on disease progression.
Generalized sexual disinhibition emerged as a key factor negatively influencing the effect of antiretroviral therapy on new HIV-1 infections. At 15 years, uncertainty in sexual disinhibition explained 49.7% and 24% of the variance in decline of cumulative infections for early and late treatment introduction, respectively.
Simulations showed greater benefit with higher rates of treatment coverage of AIDS cases, at the individual and population levels, in terms of cumulative deaths from AIDS averted. Again, all levels of treatment coverage were superior to no treatment. The benefit in terms of reduction in cumulative new HIV-1 infections was modest, however, even when 100% of AIDS cases were treated according to WHO guidelines for resource-limited settings. This is likely because AIDS cases contribute a relatively small fraction of transmission compared with acute infection or other earlier stages of HIV-1 disease progression.
Our model predicted incremental reductions in cumulative new HIV-1 infections and deaths from AIDS when individuals at earlier stages were treated. In keeping with this, Auvert and colleagues55 predicted that over a 3-year period, the reduction in the annual risk of HIV-1 transmission using the WHO treatment strategy (CD4 count <200 cells/mm3) would be 12%, whereas that using the US Department of Health and Human Services (DHHS) 2001 through 2003 strategy (CD4 count <350 cells/mm3 or plasma HIV-1 RNA level >55,000 copies/mL)56 would be 72%. The median plasma HIV-1 RNA level in their population (South African township) was 55,750 copies/mL. For comparison, Gray et al,20 using a stochastic epidemic model, predicted that 100% treatment coverage by DHHS criteria (HIV-1 RNA level >55,000 copies/mL) in Rakai, Uganda would decrease HIV-1 incidence by 19%. Eligibility for treatment was based on serum HIV-1 RNA levels in Rakai, which has median values of 15,649 copies/mL in men and 9655 copies/mL in women.2 HIV-1 RNA levels in serum are known to be 30% to 80% lower than when measured in plasma.57,58 Thus, only 20% of the chronically infected individuals were eligible for treatment (HIV-1 RNA level >55,000 copies/mL) in the model by Gray et al.20 Furthermore, these authors assumed that treatment reduced viremia by only 27% to 44% and that cumulative 3-year treatment discontinuation was 55%. By comparison, our model predicted a 45.1% reduction in HIV-1 incidence and a 41.2% reduction in cumulative new HIV-1 infections after 10 years, with 100% treatment coverage of individuals with a CD4 count <350 cells/mm3 (DHHS 2004-2005 criteria53) and early (5% prevalence) treatment introduction. The corresponding figures after late (40% prevalence) treatment introduction were 28.2% and 36.5%, respectively. Thus, differences in the proportion of individuals eligible for treatment and in treatment efficacy are likely to account for the more favorable results in our model.
Our analyses identified persistent infectivity or disease progression on treatment and sexual disinhibition in the general population as key variables that could potentially undermine the beneficial effect of therapy. These findings are in general agreement with earlier models.12,17,20,23
Sub-Saharan Africa has 64% of the HIV-infected population of the world, totaling 23 million adults.62 Although a heterogeneous variety of epidemics are affecting this region,63 most of the Africans are not yet infected with HIV. Thus, effective treatment and prevention strategies are urgently needed. Our analyses indicate that roughly 3 to 7 million new HIV-1 infections and 5 to 9 million AIDS deaths could be averted in sub-Saharan Africa over the next 10 years if 100% of AIDS cases (approximately 4 million persons) are given treatment and sexual disinhibition is prevented. The corresponding estimates for South Africa alone are 0.4 to 1.2 million infections and 0.7 to 1.3 million AIDS deaths averted if 100% of AIDS cases (approximately 0.8 million persons) are treated. This enormous individual and public health benefit would require sustained access to antiretroviral therapy beyond the near-term goal of the WHO "3 by 5" program (3 million persons treated globally by 2005).64 In addition, the integration of treatment programs with voluntary counseling and testing services and prevention programs (eg, promotion of condom use and identification and treatment of sexually transmitted diseases) is key in controlling the spread of HIV-1.65
Others investigators have forecasted the effect of antiretroviral treatment rollout in resource-limited settings using simple mathematic models.21-25 Blower and colleagues23,24 used a model that included emergence of drug resistance but not the natural history of HIV-1 infection to predict that if 10% of the total HIV-infected population were treated (US global HIV/AIDS treatment strategy66), the decrease in HIV-1 incidence would be less than 5%24 after 10 years and that coverage of 10% to 50% would prevent 8.5% (range: 5.1%-31.5%) of cumulative new HIV-1 infections.23 Although these investigators assumed a 50% to 99% reduction in infectivity and a 1.5- to 3-fold increase in survival on treatment, they also assumed a 50% average increase in the sexual activity of the general population, which is much higher than the 10% increase we assumed for the sexual disinhibition scenario. Salomon et al25 used a more detailed model to predict the effect of treatment in sub-Saharan Africa.64 They simulated coverage of 50% of individuals with advanced disease by 2005 (WHO s 3 by 5 target), with extension to 80% by 2010 and beyond. Under different scenarios of sexual behavior and reduced transmission with treatment, their model predicted a −4% to +5% change in cumulative new adult HIV-1 infections and a 22% decline in cumulative adult deaths from AIDS through 2010.
The results of our base-case and conservative scenarios are more optimistic. This is because we simulated treatment according to WHO guidelines5 and our model incorporates recent data on HIV transmission29 and outcomes of treatment in sub-Saharan Africa.33 Furthermore, our model also stratified infected individuals according to disease progression,26-28,67 which made explicit the implementation and analysis of different treatment strategies. Despite differences in model structure and assumptions and scenarios of treatment use, our predictions confirm the general insights derived from earlier models, namely, that WHO s 3 by 5 program is likely to have a small public health benefit. Greater treatment coverage of the HIV-infected population is needed to achieve more substantial public health benefit.
The precise quantitative detail of our predictions will be affected by various refinements of the model structure. More complex stochastic models that incorporate, for example, sexual network structure, concurrent partnerships, age, and gender, together with complex treatment issues, can be easily formulated, and their properties can be analyzed using numeric methods. This is an important next step for the research reported in this article to increase the accuracy of model predictions. It is important to note, however, that the relatively simple structure used in the present analysis contains 55 parameters. With more complex models, parameter numbers spiral quickly as complexity increases, especially in connection with human sexual behavior. Because of limited knowledge, the problem of making parameter assignments is thus greatly exacerbated as complexity and realism are increased. Simple deterministic models can provide important and robust insights of a qualitative nature into the impact of control interventions for a wide variety of infectious agents.34 Our results must be interpreted with these caveats in mind, in terms of the relative merits of simplicity versus complexity.
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To model the effect of antiretroviral therapy within an HIV-1 epidemic, we classified the population into 4 groups: susceptible (X), infected and treatment naive (I), infected and on treatment (T), and infected and off treatment (F). After acquisition of HIV-1, each individual passed through 6 successive stages of infection, followed by death from AIDS. The stages included (1) acute infection, (2) recent infection, (3) chronic infection with a CD4 count >500 cells/mm3, (4) chronic infection with a CD4 count of 350 to 499 cells/mm3, (5) chronic infection with a CD4 count of 200 to 349 cells/mm3, and (6) AIDS (CD4 count <200 cells/mm3). Each stage was characterized by a stage-transition rate (disease progression) and a probability of transmission per sex act (infectivity). The model allowed for initiation and termination of treatment by infected individuals. The treatment discontinuation rate represented a composite of all causes of treatment failure (virologic failure and discontinuation for drug toxicity). Retreatment was not modeled, because second-line regimens are often not available in resource-limited settings.
The mathematic model is described by the following set of differential equations:
The recruitment of susceptible individuals to the sexually active class occurs at a constant net rate, μN0, where N 0 is the population size of sexually active adults in the absence of infection. Individuals leave the model after an average duration of sexual activity, 1/μ. Subscript s refers to the stage of HIV-1 infection based on the natural history of HIV-1 infection and CD4 cell count. For the 3 infected groups I, T, and F, the rates of progression through different stages of HIV-1 disease are represented by δs, ηs, and σs, respectively, with δ 6, η 6, and σ 6 being the respective death rates from AIDS. The per capita rates of treatment initiation and discontinuation are represented by α and κ, respectively. The rate of infection (incident HIV-1 cases per unit time) is the product of the number of susceptible individuals, X, and the risk of infection per susceptible individual, λ. Also known as the force of infection, λ is given by the following equation:34
Here, c is the average rate of sexual partner change. The subscripts s and τ represent the stage of infection and treatment status of the infected individuals, respectively. Y is the sum of all infected individuals (I, T, and F), and N is the total population. The probability of transmission of HIV-1 infection per partnership from an infected individual in disease stage s and treatment status τ to a susceptible individual is βsτ. The probability, βsτ, depends on the number of sex acts (a) and the probability of transmission per sex act (γsτ) and is given by the expression:
Women comprise half of the HIV-infected population worldwide and approximately 60% in Africa,62 where the efficiency of female-to-male transmission is similar to that for male-to-female transmission.2,3,68-70 We therefore assumed symmetry in sexual behavior and infectivity for both sexes.71 The stage-specific probabilities of transmission per heterosexual act used in our model were based on the rates of transmission per coital act estimated in a study of serodiscordant couples in Rakai, Uganda29 and the known heterogeneity in plasma HIV-1 RNA level among infected individuals.67
We assumed a dichotomous (and relatively pessimistic) treatment outcome: (1) treatment success with nil/reduced disease progression and transmission and (2) treatment failure with discontinuation of therapy and resumption of disease progression and transmission at pretreatment level. The rationale for this assumption is that antiretroviral therapy exerts its beneficial effects by suppressing HIV-1 replication,72 which delays disease progression73 and lowers infectivity,2,4,59,60 and that discontinuation of therapy leads to rebound of viremia to pretreatment levels,74 and to capture in a single composite rate (treatment failure at a rate κ) the different mechanisms of treatment failure over the entire simulation time period for individual patients. By assuming a median time to failure of 3 to 5 years with no retreatment and resumption of disease progression in, and transmission from, cases failing treatment, we tried to define a simple scenario for antiretroviral therapy. This obviated the need for assumptions regarding (1) the yet unknown long-term effects of antiretroviral therapy on survival and infectivity of infected individuals and (2) the rates of emergence of drug resistance, patient adherence in resource poor settings, temporary treatment discontinuations, and regimen changes.
The HIV-1 epidemic was initiated by allowing a single infected individual in the primary stage of HIV-1 infection at the beginning of simulation. Simulation was continued until a steady (endemic) stage was reached.