A Method for Obtaining Short-Term Projections and Lower Bounds on the Size of the AIDS Epidemic

AIDS: trends, predictions, controversy

One of the persistent controversies that surfaces in the media about AIDS is whether the heterosexual population is at risk. The latest projections provide an emphatic affirmative.

Population estimates of intravenous drug users and HIV infection in Los Angeles County

This study applies multiple-capture models to drug treatment data and the synthetic estimation method to arrestee data to provide estimates of the number of IVDUs in Los Angeles County in 1989. Based on the 5% HIV-prevalence rate currently found in IVDUs in Los Angeles, it is estimated that there could be as many as 9,500 HIV-infected IVDUs. The estimates of IVDUs are generally higher than those obtained by back-calculation methods which often undercount IVDU-related AIDS cases and do not consider deaths for causes other than HIV infection.

[The prediction of the minimum incidence of AIDS in Spain for the period 1992-1995]

In order to estimate minimum AIDS incidence in Spain between 1992 and 1995, annual AIDS incidence up to 1991 has been obtained from the June 1992 update of the National Register. Correction was made for reporting delays in cases diagnosed since July 1989, in order to run a subsequent back calculation on all cases and for each separate mode of transmission. It was assumed that no new HIV infections would appear after 1991. Since 1981 to 1995, more than 38,000 AIDS cases will have been diagnosed. Minimum AIDS incidence as forecast exhibits a rise between 1992 and 1995 for the total number of cases and for all categories of transmission, except for recipients of blood and blood products, and children of mothers at risk. Real incidence will probably prove higher than estimated owing to the effect of new infections which may arise, and to the underreporting of cases.

Use of backcalculation for estimation of the probability of progression from HIV infection to AIDS

Backcalculation has been used to estimate the rate of past HIV infection and to predict future AIDS incidence. In this study we examine another use: estimating the probability of progression from HIV infection to AIDS as a function of time from infection. Given observed AIDS incidence data, the technique of backcalculation estimates the most likely number of persons infected with HIV in the past. Assumptions about probability of progression from HIV infection to AIDS are necessary. By varying these assumptions and examining the resulting goodness of fit to the AIDS incidence data, we can theoretically estimate parameters of progression. We report on implementation of this method and examine its practical utility in deciding among four competing progression models specified on a priori grounds. The four specific models comprise three Weibull distributions with medians of 8, 10, and 12 years, respectively, and one model that begins as a Weibull with 8 year median but where the hazard is level after 3.5 years. To employ asymptotic maximum likelihood methods, we define a two parameter family of progression models that includes all four a priori models. One parameter sets the scale for an initial Weibull progression (the shape parameter being fixed for all models), and the other specifies a levelling point after which the hazard remains constant. AIDS incidence data from Canada's national surveillance system provided the empiric data for this evaluation. First we corrected these data for reporting delay by Poisson modelling of the delay distribution. We used three parametric families of infection curves: step-function, log-logistic, and logistic. The results support the hypothesis of an early levelling of the hazard function. When we fixed the scale parameter to that of the Weibull curve with 8 year median, the maximum likelihood estimate of the levelling point was 2.7 years, and a clearly superior fit was produced compared to a pure Weibull progression with the same scale parameter (likelihood ratio chi-square of 10.6 on 1 degree of freedom, p = 0.001). The maximum was indistinguishable in fit from the levelling point of 3.5 years hypothesized in advance (chi-square = 0.30, d.f. = 1, p = 0.58). Backcalculation, however, could not determine the Weibull scale parameter itself because the likelihood was quite flat as a function of this parameter. We conclude that one must determine the parameters governing the initial shape of the hazard function from other kinds of data.(ABSTRACT TRUNCATED AT 400 WORDS)

AIDS: the statistical basis for public health

The backcalculation method has been extensively used in AIDS modelling and forecasting. Knowledge of reported AIDS cases, information on the time between HIV infection and onset of AIDS, and assumptions on the rate at which infections occurs, can be used to reconstruct the past history of the HIV epidemic, as well as to provide short term predictions of AIDS incidence. Uncertainty in the three components of the backcalculation method and the increasingly available information on HIV prevalence must be taken into account in order to provide realistic projections. In this paper we discuss ways of acknowledging uncertainty and suggest a Bayesian formulation of the backcalculation idea as a means of combining into a single model both random and systematic variation as well as prior information.

An analysis of AIDS incidence data by clustering trends

AIDS incidence trends vary greatly among geographic areas in the United States. We define clusters of areas within which AIDS incidence trends are similar, as areas within a cluster may have similar human immunodeficiency virus epidemic patterns and thus may lead to similar prevention/intervention strategies. Methods of exploratory data analysis are used to define such clusters from reported quarterly AIDS incidence to December 1990 (adjusted for estimated reporting delays) in homosexual and bisexual men not using intravenous drugs in 39 metropolitan statistical areas (MSAs) in the United States. After smoothing AIDS incidence in each MSA, we define groups from cluster analysis based on a measure of similarity between pairs of MSAs. A log-linear model gives estimates of the scale factors and the common trend for the MSAs in each group. Alternative metrics and simulated data suggest that the clustering is fairly robust to variations in AIDS incidence data. The resulting clusters separate MSAs with different trends, for example, MSAs in which AIDS incidence shows signs of reaching a plateau are separated from MSAs in which incidence continues to increase rapidly.

The epidemic dynamics of HIV-1 in Italy: modelling the interaction between intravenous drug users and heterosexual population

Data gathered through the Latium HIV Surveillance System were used in conjunction with a compartmental mathematical model to describe the transmission dynamics of HIV-1 in Italy. In the Latium region, as in the rest of Italy, fewer than 1 in 5 cumulative cases of AIDS are attributable to male homosexual transmission, while 55-60 per cent of the cases have been observed among intravenous drug users (IVDUs). Moreover, the number of non-drug-using heterosexual cases is increasing (14 per cent of cumulative cases). Anonymous notification of positive HIV-1 tests, mandatory in Latium since 1985, were used to produce a time series of new HIV-1 diagnoses; just over 400 new cases of infection per quarter were diagnosed from mid 1989 to mid 1990, with no evidence of increasing incidence. A minimum of 6009 and a maximum of 10,000 individuals with HIV-1 as of the end of 1989 were estimated, 80 per cent of adult cases occurring among IVDUs. The model included two main subgroups: IVDUs and non-drug-using heterosexuals, both with behavioural heterogeneities. Sharing of needles among IVDUs and heterosexual contact were considered as possible ways of transmitting HIV-1. A mathematical framework was developed to reproduce different mixing patterns within and between subgroups. A scenario analysis of the model showed incident cases of HIV-1 among IVDUs peaking early, then declining dramatically and stabilising at low values, with a stable-state prevalence higher than 0.75. Heterosexual interaction with IVDUs resulted in a significant, but non-self-sustaining, virus spread in the general population, affecting females more than males. The extent of this spread is associated with the assortativeness of the sexual mixing pattern adopted. The qualitative features of the Italian epidemic are well represented by the model, which highlights the role of IVDUs as an infection reservoir. However, the need emerges for more accurate information on the key parameters influencing the transmission dynamics of HIV-1.

Estimating HIV prevalence and projecting AIDS incidence in the United States: a model that accounts for therapy and changes in the surveillance definition of AIDS

The AIDS incubation distribution is changing in calendar time because of treatment and changes in the surveillance definition of AIDS. To obtain reliable estimates of HIV prevalence and projections of AIDS incidence in the 1990s using the method of backcalculation, we constructed an appropriate incubation distribution for each calendar date of infection. We parameterized the impact of treatment on the incubation distribution by specifying the relative hazard for AIDS in treated versus untreated people as a function of duration of HIV infection. To account for trends in the incubation distribution, we modelled the prevalence of treatment, the distribution of treatment onset times, and the impact of the revision of the AIDS surveillance definition in 1987. We selected and evaluated backcalculation models based on consistency with external information. We defined a 'plausible range' of estimates that took into account uncertainty about the natural incubation distribution and treatment efficacy, as well as bootstrap assessment of stochastic error. Using these methods, we projected that national United States AIDS incidence will plateau during 1991-1994 at over 50,000 cases per year. Projections exhibited substantial systematic uncertainty, and we calculated a plausible range for AIDS incidence in 1994 of 42,300 to 70,700 cases. An estimated 628,000 to 988,000 cumulative HIV infections occurred as of 1 January 1991. After accounting for AIDS mortality, we estimated that 484,000 to 844,000 people were living with HIV infection on 1 January 1991. Favourable trends in HIV incidence appeared in gay men and intravenous drug users. Plausible ranges for our estimates overlapped with those from a 'stage model' approach to incorporating treatment effects in backcalculations. Our approach, however, tended to yield smaller estimates of epidemic size, mainly because the parameters used with the stage model implied that more treatment was in use and that treatment was more effective than in our model.

Estimating the stage-specific numbers of HIV infection using a Markov model and back-calculation

The back-calculation method has been used to estimate the number of HIV infections from AIDS incidence data in a particular population. We present an extension of back calculation that provides estimates of the numbers of HIV infectives in different stages of infection. We model the staging process with a time-dependent Markov process that partitions the HIV infectious period into the following progressive stages and/or substages: stage 1, infected but antibody negative; substages 2-3; antibody positive but asymptomatic; substages 4-6, pre-AIDS symptoms and/or abnormal haematologic indicator, stage 7, clinical AIDS. We also model an eight stage, decreased due to AIDS. The model allows for time-dependent treatment effects that slow the rate of progression in substages 4-7. We use the estimated AIDS incubation period distribution for the Markov model in back calculation from AIDS incidence data to estimate the total number of HIV infections and the parameters of the infection probability distribution. We then use these estimates in the Markov model to estimate the stage-specific numbers of HIV infections over the course of the epidemic in the population under study. Example calculations employ data for epidemic in San Francisco City, Clinic Cohort.

A bibliography and comments on the use of statistical models in epidemiology in the 1980s

This paper reviews developments in statistical modelling in epidemiology in the 1980's, with emphasis on cohort and case-control studies. The central roles of the logistic and proportional hazard models are highlighted, and it is shown how these models lead to a deeper understanding of classical designs and methods of analysis as well as to efficient new designs and analytical procedures. The important area of model misspecification is discussed, including the problems of omitted latent structure, mis-modelling of available measurements, missing data and errors in measurements. Various designs motivated by the logistic model are illustrated numerically, and designs based on the proportional hazards model are discussed, as are papers on sample size determination. There are brief introductions to the literature on other topics, including attributable risk, disease clustering, family studies and genetics, analysis of disease incidence data, infectious disease, longitudinal data, screening and miscellaneous related topics in statistics. An extensive bibliography is indexed according to the outline of the paper.

A method of non-parametric back-projection and its application to AIDS data

The method of back-projection has been used to estimate the unobserved past incidence of infection with the human immunodeficiency virus (HIV) and to obtain projections of future AIDS incidence. Here a new approach to back-projection, which avoids parametric assumptions about the form of the HIV infection intensity, is described. This approach gives the data greater opportunity to determine the shape of the estimated intensity function. The method is based on a modification of an EM algorithm for maximum likelihood estimation that incorporates smoothing of the estimated parameters. It is easy to implement on a computer because the computations are based on explicit formulae. The method is illustrated with applications to AIDS data from Australia, U.S.A. and Japanese haemophiliacs.

Mean square error of estimates of HIV prevalence and short-term AIDS projections derived by backcalculation

We simulated multinomial AIDS incidence counts from 27 'representative' AIDS epidemics that spanned a period corresponding to previous applications of backcalculation (1 January 1977 to 1 July 1987) and assessed mean square error for several back-calculated estimators of HIV prevalence and short-term AIDS projections. Estimators were based on flexible model selection procedures that chose the best-fitting non-negatively constrained model of the infection curve from a family of possible step-function models. Selection of the best-fitting model from a family of four-step models each with a long last step of width of 4 or 4.5 years offered a favourable tradeoff between bias and variance when compared with selection from families of models with three steps or from families with a short last step. Five-step models performed as well as four-step models. Three-step models had substantially larger mean square error in some epidemic situations. Percentage root mean square error (PRMSE) for estimates of cumulative HIV prevalence as of 1 January 1985 was less than 14 per cent over a range of hypothetical epidemics of N = 50,000 infected individuals. PRMSE for short-term projections was less than 18 per cent. Estimates of cumulative HIV prevalence as of 1 July 1987 were substantially more uncertain and had a PRMSE of 33 per cent in the unfavourable case of a rapidly rising HIV epidemic. Estimates of cumulative HIV prevalence as of 1 July 1987 were positively biased in HIV epidemics with a rapidly decreasing recent HIV incidence rate and negatively biased in rapidly increasing HIV epidemics. Despite these uncertainties, we obtained useful estimates even for HIV epidemics with as few as 5000 infected individuals.

Reconstruction and future trends of the AIDS epidemic in the United States

There has been considerable uncertainty in estimates of past and current human immunodeficiency virus (HIV) infection rates in the United States. Statistical estimates of historical infection rates can be obtained from acquired immunodeficiency syndrome (AIDS) incidence data and the incubation period. However, this approach is subject to a number of sources of uncertainty and two other approaches, epidemic models of HIV transmission and surveys of HIV prevalence, are used to corroborate and refine the statistical estimates. Analyses suggest the HIV infection rate in the United States grew rapidly in the early 1980s, peaked in the mid-1980s, and subsequently declined markedly. Due both to the decline in the underlying infection rate and to the development of effective therapies that may delay AIDS diagnosis, overall AIDS incidence may plateau during the next 5 years. However, the number of individuals with advanced HIV disease without a diagnosis of AIDS who could potentially benefit from therapy is expected to increase 40% by 1995 as infected individuals progress to more advanced stages of HIV disease. Thus, although the overall HIV infection rate has declined, the demands on the U.S. health care system for treatment and care of HIV-infected individuals remain enormous.

Some statistical issues in studies of the epidemiology of AIDS

Analysis of studies of the epidemiology and natural history of infection with the Human Immunodeficiency Virus and subsequent onset of AIDS are complicated by many statistical issues. Several such problems are associated with the nature of data collection which is often incomplete. Here we briefly survey some of the statistical methods that have been developed to meet the needs of analysis of AIDS data. In particular, we consider projection of the number of future cases, and estimation and identification of two key epidemiological unknowns, namely the properties of the incubation distribution and those of the infectivity associated with transmission.

Projections of the HIV/AIDS epidemic for homosexual/bisexual men in France, the Federal Republic of Germany and the United Kingdom

Projections of the spread of the acquired immunodeficiency syndrome (AIDS) and of its etiologic agent, the human immunodeficiency virus (HIV), are presented for homosexual/bisexual men in the three European countries with the largest caseloads. The results suggest that the HIV epidemic for French, German, and British homosexual/bisexual men has peaked around 1985 and declined rapidly thereafter. By the end of the century, and for a median incubation period of AIDS equal to 8 years, the total numbers infected in these groups are predicted to be about 31,200, 10,400 and 9,800, respectively. (These estimates more than double if the median incubation period is 12 years). In all cases the annual incidence of AIDS will reach its maximum in the early to mid-1990s. However, the AIDS epidemic will be protracted because of the long incubation period.

Estimation of haemophilia-associated AIDS incidence in Japan using individual dates of diagnosis

This paper presents a procedure for obtaining short-term projections and lower bounds on the size of the acquired immunodeficiency syndrome (AIDS) epidemic. The method is similar to that proposed by Brookmeyer and Gail but adapted to the situation where individual dates of AIDS diagnosis are available. It gives results that are not dependent on the choice of time intervals by contrast to the procedure proposed by Brookmeyer and Gail which was based on the tabulated number of AIDS cases. Applications are illustrated with haemophilia-associated AIDS incidence in Japan where only 45 cases have been diagnosed by 31 December 1987.

Incubation period of AIDS in San Francisco

In a closed population, the distribution of AIDS diagnoses over time is the convolution of the distributions of human immunodeficiency virus (HIV) infections and the incubation period. This has motivated estimates of the infection distribution, assuming known diagnosis and incubation distributions, but the usefulness of this method is limited by uncertainty about incubation. The large amount of information on the distribution of HIV infections in San Francisco's gay community suggests the opposite approach--estimating the incubation distribution, assuming known infection and diagnosis distributions. A non-parametric implementation of this strategy produced an estimate with a median at 9.8 years, increasing hazard rates, and less uncertainty than previous estimates.

Models for the HIV infection and AIDS epidemic in the United States

Statistical models of the HIV infection epidemic in the U.S. which account for the observed incidence of AIDS cases in the years 1978-1987 are considered. The models assume a known distribution of times from infection to AIDS. The best model estimates that there were approximately 563,000 to 1,110,000 individuals infected in the U.S. in April 1987. These estimates do not take into account underreporting of AIDS cases. The sensitivity of the conclusions to the model's assumptions is ascertained by investigating a variety of parametric models for the infection epidemic, a variety of likely distributions for the time from infection to AIDS, and some plausible alternatives for the history of AIDS cases in the U.S.. It is concluded that there is too much uncertainty in the data and the models to be able to give highly accurate predictions of the number of people currently infected in the U.S., however, the results from the best fitting models suggest that there are less than the 1 to 1.5 million infected as estimated by the Centers for Disease Control. A Bayesian scheme is suggested for incorporating the uncertainty in the models.

Statistical methods for short-term projections of AIDS incidence

Short-term projections of AIDS incidence are critical for assessing future health care needs. This paper focuses on the method of back-calculation for obtaining short-term projections. The approach consists of back-calculating from AIDS incidence data through use of the incubation period distribution to obtain estimates of the numbers previously infected. The numbers previously infected are then projected forward to obtain short-term projections. An approach is suggested for accounting for new infections in short-term projections of AIDS incidence. Back-calculation requires accurate AIDS incidence data. A method which is computationally easy to implement is proposed for estimating the distribution of the delays in reporting AIDS cases. It was found that the reporting delay distribution in the United States varies by geographic region of diagnosis. Back-calculation also requires a reliable estimate of the incubation period distribution. Statistical issues associated with estimating the incubation period distribution are considered. The methods are applied to obtain short-term projections of AIDS incidence in the United States. The projected cumulative AIDS incidence in the U.S. by the end of 1992 was 287,100 under the assumption that there are no new infections after 1 July 1987, and 330,600 under the assumption that the infection rate remains constant. These projections do not account for the new broadened AIDS surveillance definitions or the underreporting of AIDS cases to the Centers for Disease Control.