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Planning of programs to prevent HIV infection or to treat infected persons requires close monitoring of the patterns and trends in the HIV epidemic.1 However, directly measured population-based incidence and prevalence data are not available in the United States. As a result of the delay between HIV infection and detection, surveillance for reported cases of HIV infection may be a poor reflection of the epidemiologic patterns and trends.2 Methods to supplement reported surveillance information have therefore been developed to meet the challenges of monitoring population-based incidence.
Advancing substantially over the past 2 decades, backcalculation techniques have addressed many challenges of estimating population-based HIV incidence and prevalence.3–11 More recently, the precision in estimates of recent incidence has been improved by the incorporation into modeling of information from HIV/non-AIDS cases.12–17 In addition, the inclusion of confidentially reported surveillance information on the first positive HIV test introduced the opportunity to study the likelihood of HIV detection.14–20
With these advancements, Aalen and colleagues16,17 developed a discrete-time Markov model to estimate HIV incident infections and detection based on reported HIV/AIDS case data in various European countries. Their nonparametric staged model is attractive to surveillance programs because the methodology is flexible for adaptation and statistically less complex than most current backcalculation techniques. Applying such a model to surveillance data in the United States requires an adaptation that incorporates the 1993 change in the Centers for Disease Control and Prevention (CDC) AIDS case definition.
The purpose of this study was to assess the functioning of a nonparametric Markov model for producing estimates of infection and detection rates in Louisiana while adjusting for the 1993 AIDS case definition. Additionally, we wanted to expand the use of the model for programmatic purposes by modeling demographic and risk subpopulations separately. This allowed us to examine how the epidemic may have evolved differently over time in each of these groups.
We applied a staged model to HIV/AIDS surveillance data from Louisiana to estimate the number of new HIV infections and stage-specific rates of detection over calendar quarters (3-month intervals). The model was based on the methodology developed by Aalen and colleagues, yet incorporated the 1993 change in AIDS case definition.
Overview of the Model
The model and its parameter descriptions are represented in Figure 1 and Table 1. The model assumes that infected cases progress through stages of disease until detection, after which their time until AIDS diagnosis is observed. The model estimates the number of newly infected cases, h(q), and stage-specific detection rates (α1-α4) over calendar time.
We assumed that progression is unidirectional using 4 pre-AIDS stages. Within 1 quarter-year, a proportion of cases in each stage (λ1–λ4) will progress to the next stage of disease. During the same unit of time, a proportion of cases in each stage will be detected (α1–α4).
Based on published studies of the natural history of disease progression,17,21–24 we defined the 4 progression stages as the ranges of time remaining until an AIDS diagnosis would be expected without treatment. The stages, the range of time remaining until an AIDS opportunistic infection diagnosis, the mean waiting time in quarters that individuals remain in the stage, and illustrative CD4 ranges were as follows: (1) Advanced HIV: 1–4 quarters until AIDS, waiting time of 4 quarters, <200 CD4 cells; (2) Intermediate II: 5–12 quarters until AIDS, waiting time of 8 quarters, 200–349 CD4 cells; (3) Intermediate I: 13–20 quarters until AIDS, waiting time of 8 quarters, 350–499 CD4 cells; and (4) Early stage: 21–42 quarters until AIDS, waiting time of 22 quarters, >500 CD4 cells. These assumptions in the model produce a mean time from infection to an AIDS opportunistic infection diagnosis of 10.5 years (42 quarters) without treatment. Cases diagnosed with AIDS based on immunologic criteria alone have a mean time of 9.75 years from infection to AIDS diagnosis. The observed time in quarters between HIV detection and AIDS diagnosis determines the stage of progression at the time of detection.
Affecting the rate of progression, a treatment effect (θ) slows disease progression among those in treatment. The proportion of detected cases entering into treatment is represented by ρ. Because one purpose of this study was to explore how well the model functioned before introducing more complex assumptions regarding advanced therapies, the treatment assumptions were limited to those relevant to the treatment effects and standards available through 1996.5,8,9,17,18,25–37
Expansion of the Model for the 1993 AIDS Case Definition Change
Expansion of the 1993 CDC AIDS case definition was incorporated into the model by stratifying all AIDS stages into 2 types. One type (“AIDS-IM”) represented those who meet the criteria for immunosuppression under the 1993 AIDS case definition, defined as having a detected CD4 count of under 200 CD4 cells/μL or a CD4% under 14%.38 Cases not meeting these criteria are considered as diagnosed based solely on the presence of an AIDS-defining opportunistic infection (“AIDS-OI”). In the model, tau (τ) represents the proportion of new AIDS-IM diagnoses in a given quarter and is derived from the observed data. For these cases, the transition rate (λ) was assumed to be 1, reflecting a mean waiting time of 3 months between entering the advanced HIV stage and having an AIDS-IM diagnosis.
Additional Methodologic Considerations
In this initial exploration of the model, we chose not to estimate confidence intervals based on an assumed probability distribution, but rather to explore the range and effects of several methodologic assumptions on model estimates. We performed sensitivity analyses by selecting plausible bounds for the parameters regarding length of the incubation period, treatment, and the assumption in addressing missing exposure information (see subsequently)—all of which may threaten the validity of the estimates. Unless specified, the ranges reported with the point estimates reflect the combined variations in the length of the incubation period for all estimates, as well as the choice of parameters for missing exposure information.
Model sensitivity to the length of the incubation period was assessed by adjusting each of the progression parameters (λ) by 0.005,17 resulting in a range of mean incubation periods between 9.8 and 11.4 years. To assess uncertainty in treatment initiation, we varied rho (ρ) from zero (representing no treatment of any cases) to double the original values, representing a yearly treatment initiation of 68%. We distributed cases with missing exposure information among exposure categories based on 2 independent sets of parameters described subsequently.
Pre-AIDS mortality was a concern primarily among injection drug users, who have increased mortality from overdoses, accidents, suicides, hepatitis C, and liver failure from other causes.39,40 Because pre-AIDS mortality in the data was low (ranging from 0.0% to 1.3% across all subgroups), transition to a pre-AIDS mortality state was dropped from this model. However, the low rates of pre-AIDS mortality in injection drug users may indicate underreporting of risk behaviors or deaths among this group.
The impact of sampling error was also assessed using bootstrap techniques. As a result of an extremely small resulting sampling error, bootstrapped sampling was not incorporated into the assessments of other uncertainties.
Method of Parameter Estimation
Using maximum likelihood estimation, we estimated numbers of new infections and stage-specific detection rates based on joint distributions of HIV detection and AIDS diagnosis dates (in quarter-years) for selected demographic and exposure subgroups. Because observed data on CD4 monitoring were not uniformly available, time from detection to AIDS diagnosis informed the progression stage at diagnosis. Equations for building the maximum likelihood function are described in the Appendix, which is available with the electronic version of this article.
Data were provided by the Louisiana HIV/AIDS Surveillance Program, which maintains active surveillance of HIV infection based on an extensive network of reporting sites in both public and private clinical and laboratory settings. The program initiated mandatory name-based reporting of all AIDS cases in 1984 and name-based reporting of HIV infection in 1993, including retrospective reporting. The HIV and AIDS status of each reported case is regularly updated with new and relevant information in a single database.
For each reported case, HIV detection was operationally defined as the diagnosis of HIV based on the earliest positive result of a reported confidential HIV test. We included all 14,887 adult and adolescent cases (13 years and older) for whom HIV was detected in Louisiana before the end of 1996 and reported through September 2000. We made our estimates only up to 1996 to assess the model's initial assumptions before the introduction of more effective therapies after 1996, which had complex and uncertain effects.
The distributions of various subgroups and of HIV/AIDS status are presented in Table 2 and 3, respectively. By the end of 1996, 63% were reported to have AIDS. Fifty-two of these were diagnosed with AIDS in a different state and reported to Louisiana; of those, 77% had developed AIDS by 1996. Studies of reporting delay for HIV surveillance data estimated that over 98% of Louisiana AIDS cases through 1996 were reported by the end of the third quarter of 2000 (Louisiana surveillance program, unpublished data, 2000); therefore, adjustment for reporting delay was unnecessary.
The larger demographic and exposure category subgroups were modeled separately. Using the CDC-defined exposure categories, subgroups included men who have sex with men (MSM), injection drug users (IDU), and high-risk heterosexuals (HRH). The combined exposure category of MSM/IDU was included within MSM. Demographic subgroups modeled were white men, black men, and women.
Exposure Category Redistribution Methods
In these data, 847 (9%) of AIDS cases and 1713 (31%) of HIV/non-AIDS cases had missing information regarding their possible exposure to HIV (commonly referred to as cases with “no identified risk”). To avoid underestimation of incident infections for recent years in specific exposure-category subgroups, we redistributed those with no identified risk by randomly assigning cases to an exposure category according to a set of weighted redistribution parameters that controlled for race, sex, and HIV/AIDS status. The redistribution parameters were based on the distribution of new HIV and AIDS cases with nonmissing exposure information in Louisiana during 1993–1996.
For the sensitivity analyses, we applied an alternate set of redistribution parameters. Developed by the CDC (CDC, unpublished data for southern states, 1997), these redistribution parameters were based on interviews with those with no identified risk who were questioned until they were either reclassified into an exposure category or closed because no further information could be obtained.41,42 Comparing the 2 sets of parameters, the alternate set gave greater weight to HRH and less to MSM, providing plausible bounds, particularly for these 2 exposure categories.
Estimates of Incident and Prevalent Cases
Figure 2 shows the estimated number of new infections for each exposure and demographic subgroup from 1979–1996. The estimated number of incident infections in the MSM and white men subgroups peaked in the early to mid-1980s and then dramatically decreased; it has increased somewhat over the more recent years. Newly infected cases steadily increased in HRH as well as in women. By 1993, incident cases were evenly distributed among the 3 major exposure groups. Accounting for the uncertainties in the incubation period and the exposure category redistribution methods, the range of quarterly estimates of incident cases during 1996 was 113–123 for MSM, 71–113 for IDU, and 103–138 for HRH. Among the demographic groups, the quarterly number of new infections in 1996 was similar in black men (122–134) and women (119–128); both of these groups had incident numbers that were more than double that in white men (55–58).
After removing the observed deaths, the estimated number of prevalent cases by the end of 1996 remained higher among MSM (range: 6607–7486) than IDU (3528–4514) and HRH (3103–4095). Among the demographic groups, black men (6331–6983) were highest, followed by white men (4234–4565) and women (3708–4249). Including the cases in subgroups not modeled, the estimate for the cumulative number of HIV/AIDS cases in 1996 was approximately 20,500 (19,200–22,200) of whom 14,600 (13,300–16,300) cases were alive.
Estimated Detection Rates
Figure 3 shows detection rates for HIV/AIDS converted to yearly aggregated rates for interpretability. There was a rapid rise in detection rates among previously undetected cases in all subgroups from 1988–1993, after which the rates tended to level off. By 1996, the model estimated that HIV or AIDS was diagnosed in 61% of the 14,600 estimated prevalent cases. Among the HIV/non-AIDS cases, only half had been detected, with an average time from infection to detection of 4 years. Although detection rates were highest among women and HRH, the ranges of rates were similar across subgroups when accounting for model uncertainties.
Effects of Uncertainty on Estimates
Variations in the incubation period ranged between -4% to 3% of the point estimates of new infections. The sensitivity to treatment initiation was also small, varying from -6% to +3%. Differences in the exposure category redistribution parameters had a much greater impact on incidence within exposure subgroups, producing ranges of 4% in the MSM model to 48% in the IDU model. Detection rates were affected only slightly in all sensitivity analyses.
We visually inspected the results to assess how well the models predicted observed trends. Stratified comparisons of the observed and expected cases helped to identify when assumptions may have been compromised. As seen in Figure 4, selected comparisons with the overall best and worst fit based on the goodness-of-fit tests demonstrate that the expected cases followed the trends in observed HIV detections and AIDS diagnoses.
Overall, the model produced plausible estimates of incident cases and detection rates that were consistent with other information available in Louisiana.43 Among MSM, the estimated number of new cases dropped substantially after the initial peak but may have increased again in recent years, consistent with MSM trends and related behaviors observed in other areas.44,45 The epidemic among HRH was increasing, resulting in equal distribution among the 3 main exposure categories by 1996; however, given the hierarchical classification of cases with multiple major potential exposure routes into MSM and IDU categories, rather than HRH, heterosexual transmission may be underestimated. The direction of incident cases in IDU during recent years was less clear as a result of the uncertainty of exposure category redistribution. The results by demographic subgroup confirm a shift from an epidemic predominantly in white men to one in blacks and in women.
The point estimates of rates of HIV detection, among those with HIV infection, were slightly lower among MSM and IDU and slightly higher among HRH compared with results obtained in other countries.14–20 However, the ranges of estimates were plausible given the various methodologies and the geographic diversity in other studies of detection. The result for cumulative detection was almost identical to the CDC estimate that 61% of all living persons with HIV have been confidentially tested by 1996.2
Based on the ranges of estimates while accounting for uncertainty, there were no substantial differences in the rates of HIV detection across subgroups. It was noted, however, that women and HRH had slightly higher point estimates of detection rates, which was consistent with the implementation of guidelines for HIV screening in pregnant women.46 Across time, the increases in detection until 1993 in all subgroups was somewhat artificial as a result of a combination of true changes in individual test-seeking behaviors, improved access to testing and services, and improvements in documentation, reporting, and data quality—particularly with the initiation of retrospective HIV reporting in 1993.
Varying the incubation period within ranges found in published studies did not produce substantial changes in estimates of incident cases, although a larger range is certainly plausible. Among the exposure-category subgroups, varying the parameters used to redistribute cases of unknown exposure category had much greater impact on estimates of incident cases, particularly for the most recent years. The small effect of uncertainty in the treatment initiation parameter reduced concern for bias introduced by variations in access to care across subgroups.
This model is designed to provide information about new infections and HIV detection. The assumption of unidirectional progression to AIDS (ie, persons do not return to previous stages of progression) is acceptable for the purpose of the model because the total time from detection to diagnosis (rather than stage or CD4 monitoring) contributes to the fitting of the model. In addition, the progression parameters applied in this model were derived from published studies of observed progression using unidirectional models in which the mean waiting times would have accounted for cases that return to previous states or progress to AIDS from earlier stages. The use of these broadly defined stages permitted substantial true variation in progression without compromising the reliability of the estimates.
The validity of model assumptions relating to the 1993 AIDS case definition, however, warrants discussion. Although numerous graphic comparisons suggested good overall model fit, the greatest divergence between observed and expected frequencies occurred when AIDS cases previously detected with HIV were stratified by the diagnosis of immunosuppression or an opportunistic infection. With the introduction of the 1993 AIDS definition and the resulting increased use of CD4 measurement, AIDS cases that were diagnosed based on an opportunistic infection may have been increasingly characterized by cases that develop an opportunistic infection before reaching immunosuppression. This likely violated that model's assumption that all opportunistic infections occur after the true (but undetected) CD4 count falls below 200. Of the 15% to 25% of cases that were classified as having an “AIDS-OI” diagnosis, some may have been infected in a slightly more recent quarter than the model estimates.
Although it is well documented that age has an effect on the length of the incubation period and response to treatment,8,26,34,47 the effects of age would likely not influence the conclusions regarding trends within the subgroups because the effects of age are represented by the variations in the length of the incubation period.
The model assumed that all infected individuals eventually would be detected with HIV or diagnosed with AIDS. Although collection methods for surveillance have increased the quality and completeness of the current and retrospective HIV/AIDS case data over time, the potential for infected individuals to be unreported or lost to follow up introduce challenges to estimation and interpretation. Reasons for underreporting or loss to follow up include anonymous testing, in-state residence while seeking out-of-state testing, unreported out-of-state relocation, unreported death, and underreporting from the private sector. Depending on the magnitude of underreporting, data quality, and the potential for changing characteristics over time of each of these challenges, the magnitude of the estimates and the peak of the epidemic curve may be biased in either direction.
Implications for Future Use of the Model
As more effective treatments with earlier initiation prolong the period of HIV infection without progression to AIDS, and as the use of these treatments change over time and among subgroups, the range of uncertainty in estimates produced by this model or similar models is expected to increase. More informed population-based estimates of the uptake of treatment at specific stages of progression and the long-term effects of newer treatments would be required. The flexibility of the model allows the user to adjust the treatment parameters and add new dimensions such as different levels of treatment effectiveness.
In addition, users who apply more recent data should consider pre-AIDS mortality,48 reporting delays, multiple transitions to AIDS, different shapes of the incubation distribution, and factors related to the incubation period such as age8,26,34,47 and exposure to opportunistic infections diagnosed at higher CD4 counts among specific groups.47–52 Methods for incorporating age at infection or diagnosis into the model should also be explored because they may produce temporal challenges. In addition, the method of redistribution of persons of unknown exposure category should be reassessed as a result of an increasingly prolonged period of HIV infection without progression to AIDS; longer HIV status in the absence of AIDS increasingly represents better access to care rather than shorter times since infection when compared with AIDS cases.
HIV/AIDS surveillance programs could benefit from exploring the use of discrete Markov modeling because precision in estimating infection can be improved when recent HIV testing data is incorporated.12 Surveillance programs that focus on abstracting the earliest HIV detection date may derive greater benefit in precision and interpretation, particularly in jurisdictions with shorter average times from infection to detection.12 Changes in the model's estimated detection rates as a result of individual test-seeking behaviors are indistinguishable from the artificial increases caused by reporting and improved data quality; nonetheless, further exploration of differences in detection over time or between subgroups may be useful for monitoring the validity of assumptions in other methodologies for estimating population-based incidence such as the serologic testing algorithm for recent HIV seroconversion.53
Adapted for surveillance programs in the United States, Markov models that integrate HIV detection data with AIDS case data can provide estimates of new infections, detection rates, and the impacts of various uncertainties. The flexibility of the discrete-time model for adapting parameter assumptions and model design makes estimation and sensitivity analyses substantially more manageable given the resources for such estimation within surveillance programs.
We thank the staff of the Louisiana HIV/AIDS Surveillance Program for their valuable contributions to this study and Odd Aalen for generously sharing details for the estimation procedure.
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