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JAIDS Journal of Acquired Immune Deficiency Syndromes:
doi: 10.1097/01.qai.0000194234.31078.bf
Epidemiology and Social Science

Potential Impact of Antiretroviral Therapy on HIV-1 Transmission and AIDS Mortality in Resource-Limited Settings

Abbas, Ume L. MD*; Anderson, Roy M. PhD†; Mellors, John W. MD*

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From the *Division of Infectious Diseases, School of Medicine, University of Pittsburgh, Pittsburgh, PA; and †Department of Infectious Disease Epidemiology, Imperial College, Faculty of Medicine, University of London, London, United Kingdom.

Received for publication May 10, 2005; accepted October 21, 2005.

R.M. Anderson acknowledges grant support from the Wellcome Trust. J.W. Mellors acknowledges support from the Bristol Myers Squibb Research Foundation and grants from the National Institute of Allergy and Infectious Diseases (U01AI38858) and from the National Cancer Institute (Science Applications International Corporation (SAIC) contract 20XS190A).

Reprints: Ume L. Abbas, Division of Infectious Diseases, University of Pittsburgh School of Medicine, Falk Medical Building, Suite 3-A, 3601 Fifth Avenue, Pittsburgh, PA 15213 (e-mail: abbasu@dom.pitt.edu).

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Abstract

Objective: To estimate the potential impact of antiretroviral therapy on the heterosexual spread of HIV-1 infection and AIDS mortality in resource-limited settings.

Methods: A mathematic model of HIV-1 disease progression and transmission was used to assess epidemiologic outcomes under different scenarios of antiretroviral therapy, including implementation of World Health Organization guidelines.

Results: Implementing antiretroviral therapy at 5% HIV-1 prevalence and administering it to 100% of AIDS cases are predicted to decrease new HIV-1 infections and cumulative deaths from AIDS after 10 years by 11.2% (inter-quartile range [IQR]: 1.8%-21.4%) and 33.4% (IQR: 26%-42.8%), respectively. Later implementation of therapy at endemic equilibrium (40% prevalence) is predicted to be less effective, decreasing new HIV-1 infections and cumulative deaths from AIDS by 10.5% (IQR: 2.6%-19.3%) and 27.6% (IQR: 20.8%-36.8%), respectively. Therapy is predicted to benefit the infected individual and the uninfected community by decreasing transmission and AIDS deaths. The community benefit is greater than the individual benefit after 25 years of treatment and increases with the proportion of AIDS cases treated.

Conclusions: Antiretroviral therapy is predicted to have individual and public health benefits that increase with time and the proportion of infected persons treated. The impact of therapy is greater when introduced earlier in an epidemic, but the benefit can be lost by residual infectivity or disease progression on treatment and by sexual disinhibition of the general population.

The benefit of antiretroviral treatment of HIV-infected individuals is undisputed.1 Because the level of viremia correlates with the likelihood of sexual transmission of HIV-1,2,3 suppression of viremia by treatment is also likely to reduce HIV-1 spread.4 Although guidelines for treatment of HIV-1 infection in resource-limited settings have been issued,5 the approach that would yield maximum individual and public health benefit is not defined.

Several studies have explored the potential effect of antiretroviral therapy on HIV-1 epidemics using mathematic models.6-25 Most of these models, however, have focused on monotherapy and on epidemics in developed nations among men who have sex with men (MSM). Much less attention has been directed to the impact of antiretroviral therapy on heterosexual epidemics in resource-limited settings.19-25 We therefore developed a transmission model of the HIV-1 epidemic to estimate the individual and public health benefit of antiretroviral therapy in resource-limited settings.

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METHODS

Model Structure

We used a deterministic system of 20 differential equations to represent the natural history of HIV-1 infection, heterosexual transmission of HIV-1, and impact of antiretroviral therapy (Fig. 1). A staged progression model, based on CD4 cell decline,26 was used to make explicit the timing of treatment initiation. The model parameter set was chosen to mimic an epidemic in a sub-Saharan African nation reaching an endemic prevalence of 40% in the sexually active population 15 to 49 years of age. Parameter assignments were made from recent literature on HIV-1 disease progression,27,28 infectivity,29 sexual behavior,30-32 and response to antiretroviral therapy33 in resource-limited settings. Model parameters are shown in Table 1, and model equations and details are provided in the Appendix.

Figure 1
Figure 1
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Table 1
Table 1
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Model Analysis

We analyzed the model numerically34 and by time-dependent uncertainty and sensitivity analyses.35-37 We determined the following epidemiologic outcomes: HIV-1 prevalence, HIV-1 incidence, cumulative new HIV-1 infections, and cumulative deaths from AIDS. To measure the impact of treatment, we made comparisons between the treated and untreated epidemics at each simulation time step and calculated the percent change in outcomes.

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Numeric Analyses
Base-Case Scenario

We simulated the effect of antiretroviral therapy using the World Health Organization (WHO) criteria5 for treatment initiation in resource-limited settings (CD4 count <200 cells/mm3) at 3 different phases of the epidemic, defined by HIV-1 prevalence levels of 5% (early), 20% (intermediate), and 40% (late/endemic equilibrium). At these epidemic phases, AIDS cases comprised 12%, 13%, and 18% of the total HIV-1-infected population, respectively. Treatment was administered to 100% of AIDS cases. We assumed that (1) the sexual activity level (number of sexual partners and total sexual acts per year) of AIDS cases was 70% of that for the general population29,38,39 and was not changed by treatment; (2) 60% of the individuals initiating treatment would maintain viral suppression for 3 years,33 corresponding to an average treatment failure rate of 17% per year (and a median time to failure of 4 years)40; and (3) disease progression was stopped and infectivity reduced to nil during virus suppression.

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

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Individual Versus Population Benefit

We determined the population-level benefit of antiretroviral therapy and contrasted this with benefit exclusive to infected individuals by assuming 2 scenarios. For scenario I, we assumed that the decrease in infectivity on treatment was 100%. For scenario II, we assumed that the decrease in infectivity on treatment was 0%. For each scenario, we varied the proportion of AIDS cases on treatment from 10% to 100%. Other assumptions were as described for the base-case scenario. We calculated the benefit of antiretroviral therapy as the percent change in cumulative AIDS deaths. The difference between the untreated epidemic and scenario II defined the individual benefit (direct effect attributable to improved patient survival), whereas the difference between scenario II and scenario I defined the population benefit (effect attributable to reduced transmission).

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Uncertainty and Sensitivity Analyses

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

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Uncertainty Analyses

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%.

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Sensitivity Analyses

After confirming monotonicity37 and rank transformation of data,50 we performed multivariate regression and derived the standardized rank regression coefficients.51

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RESULTS

Base-Case Scenario

When antiretroviral therapy was implemented early in the epidemic (at 5% prevalence) and administered to 100% of AIDS cases, the predicted decrease in cumulative new HIV-1 infections was 25.8% after 5 years, 29.4% after 10 years, and 32% after 15 years (Fig. 2A). The predicted decrease in cumulative deaths from AIDS was 68% after 5 years, 58.4% after 10 years, and 56.1% after 15 years (see Fig. 2B). After 15 years, HIV-1 incidence had declined by 35.6% but prevalence had only declined by 4.4% because of longer survival of treated individuals.

Figure 2
Figure 2
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Table 2 quantifies the predicted impact of antiretroviral therapy for South Africa and sub-Saharan Africa as a whole, where HIV-1 prevalence in adults is 18.5% and 7.4%, respectively. In South Africa, up to 1.2 million new HIV-1 infections and 1.3 million AIDS deaths could be averted over 10 years by treating 100% of AIDS cases. For sub-Saharan Africa, the projections are 6.9 million new HIV-1 infections and 9.3 million AIDS deaths averted.

Table 2
Table 2
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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).

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Population Versus Individual Benefit

Figure 3 shows the predicted individual and population benefit after 5 to 25 years of antiretroviral therapy for 10% to 100% of AIDS cases as the relative contribution to decline in cumulative AIDS deaths. A population benefit was observed at all times after treatment implementation, although the benefit was small initially. For example, 5 years after early implementation of therapy, the population benefit accounted for only approximately 1% of the decrease in AIDS deaths. After 25 years, however, the population benefit accounted for 70.6%, 55.4%, and 52.3% of the decrease in AIDS deaths after early, intermediate, and late implementation of therapy, respectively (see Fig. 3). Thus, the population benefit exceeded the individual benefit of therapy after 25 years.

Figure 3
Figure 3
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Analysis of Uncertainty

Figures 4A and B show the uncertainty analysis of the effect of treating 100% of AIDS cases under the conservative scenario (no generalized sexual disinhibition). After 10 years, the decline in cumulative deaths from AIDS was always positive and had a median value of 33.4% for early (see Fig. 4B), 30.7% for intermediate, and 27.6% for late treatment implementation. The decline in cumulative new HIV-1 infections was also largely positive, although a few simulation runs predicted an increase after the first few years of treatment because of increased sexual activity and extended life of HIV-1-infected individuals. The median decline in HIV-1 infections after 10 years was 11.2% for early (see Fig. 4A), 10.7% for intermediate, and 10.5% for late treatment implementation, respectively.

Figure 4
Figure 4
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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).

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Sensitivity Analysis

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.

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DISCUSSION

Antiretroviral therapy unequivocally benefits HIV-infected individuals by reducing HIV-related morbidity and mortality.1 The public health benefits of therapy are uncertain, however, and optimal treatment strategies are debated.5,52,53 Using a simple but well-parameterized deterministic model of HIV-1 spread, we found that antiretroviral therapy benefits the infected individual by decreasing AIDS deaths and the uninfected community by producing a net reduction in transmission (population benefit). The population benefit became greater than the individual benefit for a major part of the simulated epidemic. Larger individual and population benefits were observed when treatment was introduced early into the epidemic, before peak infection rate and prevalence were reached. Our findings are in line with the more general rule for control of infectious disease epidemics-early intervention has a greater impact than later efforts.54 It is important to note, however, that late implementation of treatment was only modestly inferior to early treatment and considerably better than no treatment, because there was a persistent and sustained decrease in cumulative deaths from AIDS at early and late intervention times.

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.

A decrease in the HIV-1 infection rate after widespread availability of potent antiretroviral therapy has been empirically observed. Porco and colleagues59 attributed the decreasing HIV-1 seroincidence in a cohort of MSM in San Francisco to a 60% decline in per partnership probability of HIV-1 transmission after widespread use of antiretroviral therapy. Six years after providing free antiretroviral therapy, the rate of HIV-1 transmission in Taiwan decreased by 53%.60 Similarly, heterosexual transmission of HIV-1 decreased by 80% in a Spanish cohort of 393 steady couples.4 These observations support reduced infectivity of individuals on antiretroviral therapy, as we have modeled. Additional studies are being conducted to define the impact of antiretroviral therapy on infectivity better.61

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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ACKNOWLEDGMENTS

U.L. Abbas is grateful to Bob Eberlein, Bard Ermentrout, and Michael McKay for helpful discussions and constructive suggestions and to Debbie Bradshaw and Leigh Johnson for providing the 2004 HIV/AIDS estimates for South Africa.

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APPENDIX

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:

Equation (Uncited)
Equation (Uncited)
Image Tools

The recruitment of susceptible individuals to the sexually active class occurs at a constant net rate, μN0, where N0 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 δ6, η6, and σ6 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

Equation (Uncited)
Equation (Uncited)
Image Tools

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 β. The probability, β, depends on the number of sex acts (a) and the probability of transmission per sex act (γsτ) and is given by the expression:

Equation (Uncited)
Equation (Uncited)
Image Tools

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.

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Keywords:

antiretroviral therapy; HIV-1 epidemic; mathematic model; transmission; developing countries; public health

© 2006 Lippincott Williams & Wilkins, Inc.

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