Reconstructing the incidence of human immunodeficiency virus (HIV) in Hong Kong by using data from HIV positive tests and diagnoses of acquired immune deficiency syndrome

Summary. The human immunodeficiency virus–acquired immune deficiency syndrome (HIV–AIDS) epidemic in Hong Kong has been under surveillance in the form of voluntary reporting since 1984. However, there has been little discussion or research on the reconstruction of the HIV incidence curve. This paper is the first to use a modified back-projection method to estimate the incidence of HIV in Hong Kong on the basis of the number of positive HIV tests only. The model proposed has several advantages over the original back-projection method based on AIDS data only. First, not all HIV-infected individuals will develop AIDS by the time of analysis, but some of them may undertake an HIV test; therefore, the HIV data set contains more information than the AIDS data set. Second, the HIV diagnosis curve usually has a smoother pattern than the AIDS diagnosis curve, as it is not affected by redefinition of AIDS. Third, the time to positive HIV diagnosis is unlikely to be affected by treatment effects, as it is unlikely that an individual receives medication before the diagnosis of HIV. Fourth, the induction period from HIV infection to the first HIV positive test is usually shorter than the incubation period which is from HIV infection to diagnosis of AIDS. With a shorter induction period, more information becomes available for estimating the HIV incidence curve. Finally, this method requires the number of positive HIV diagnoses only, which is readily available from HIV–AIDS surveillance systems in many countries. It is estimated that, in Hong Kong, the cumulative number of HIV infections during the period 1979–2000 is about 2600, whereas an estimate based only on AIDS data seems to give an underestimate.     

Human immunodeficiency virus, DNA and statistics

Understanding the causes and consequences of genetic variation in human immunodeficiency virus (HIV) is one of the most important tasks facing medical and evolutionary biologists alike. A powerful analytical tool which is available to those working in this field is the phylogenetic tree, which describes the evolutionary relationships of the sequences in a sample and the history of the mutational events which separate them. Although phylogenetic trees of HIV are becoming commonplace, their use can be improved by tailoring the underlying statistical models to the idiosyncrasies of viral biology. The design and refinement of phylogenetic analyses consequently represents an important practical use of statistical methods in HIV research.     

Semiparametric analysis of recurrent events: artificial censoring,truncation, pairwise estimation and inference

The analysis of recurrent failure time data from longitudinal studies can be complicated by the presence of dependent censoring. There has been a substantive literature that has developed based on an artificial censoring device. We explore in this article the connection between this class of methods with truncated data structures. In addition, a new procedure is developed for estimation and inference in a joint model for recurrent events and dependent censoring. Estimation proceeds using a mixed U-statistic based estimating function approach. New resampling-based methods for variance estimation and model checking are also described. The methods are illustrated by application to data from an HIV clinical trial as with a limited simulation study.     

A Bayesian approach for estimating antiviral efficacy in HIV dynamic models

The study of HIV dynamics is one of the most important developments in recent AIDS research. It has led to a new understanding of the pathogenesis of HIV infection. Although important findings in HIV dynamics have been published in prestigious scientific journals, the statistical methods for parameter estimation and model-fitting used in those papers appear surprisingly crude and have not been studied in more detail. For example, the unidentifiable parameters were simply imputed by mean estimates from previous studies, and important pharmacological/clinical factors were not considered in the modelling. In this paper, a viral dynamic model is developed to evaluate the effect of pharmacokinetic variation, drug resistance and adherence on antiviral responses. In the context of this model, we investigate a Bayesian modelling approach under a non-linear mixed-effects (NLME) model framework. In particular, our modelling strategy allows us to estimate time-varying antiviral efficacy of a regimen during the whole course of a treatment period by incorporating the information of drug exposure and drug susceptibility. Both simulated and real clinical data examples are given to illustrate the proposed approach. The Bayesian approach has great potential to be used in many aspects of viral dynamics modelling since it allow us to fit complex dynamic models and identify all the model parameters. Our results suggest that Bayesian approach for estimating parameters in HIV dynamic models is flexible and powerful.     

New approaches for censored longitudinal data in joint modelling of longitudinal and survival data,with application to HIV vaccine studies

In HIV vaccine studies, longitudinal immune response biomarker data are often left-censored due to lower limits of quantification of the employed immunological assays. The censoring information is important for predicting HIV infection, the failure event of interest. We propose two approaches to addressing left censoring in longitudinal data: one that makes no distributional assumptions for the censored data—treating left censored values as a “point mass” subgroup—and the other makes a distributional assumption for a subset of the censored data but not for the remaining subset. We develop these two approaches to handling censoring for joint modelling of longitudinal and survival data via a Cox proportional hazards model fit by h-likelihood. We evaluate the new methods via simulation and analyze an HIV vaccine trial data set, finding that longitudinal characteristics of the immune response biomarkers are highly associated with the risk of HIV infection.

     

Modelling human immunodeficiency virus infection and acquired immune deficiency syndrome cases in Scotland: data sources,prior information and Bayesian estimation

The paper describes the methodology developed to carry out predictions of the acquired immune deficiency syndrome (AIDS) epidemic in Scotland. Information on the human immunodeficiency virus (HIV) epidemic comes from formal case reports of AIDS cases and HIV positive tests, reports from surveillance schemes and from special studies. These sources of information, up to the end of 1994, are reviewed. Prior information on aspects of HIV disease is available from various published and unpublished sources. A simple model of the HIV epidemic in Scotland is proposed and the information is summarized in terms of this model. Bayesian methodology, using Markov chain Monte Carlo methods, is described and used to predict future cases of AIDS in Scotland and people who will be living with AIDS in the years 1995–1999.     

Estimation of Stratified Mark‐Specific Proportional Hazards Models with Missing Marks

Abstract. An objective of randomized placebo‐controlled preventive HIV vaccine efficacy trials is to assess the relationship between the vaccine effect to prevent infection and the genetic distance of the exposing HIV to the HIV strain represented in the vaccine construct. Motivated by this objective, recently a mark‐specific proportional hazards (PH) model with a continuum of competing risks has been studied, where the genetic distance of the transmitting strain is the continuous ‘mark’ defined and observable only in failures. A high percentage of genetic marks of interest may be missing for a variety of reasons, predominantly because rapid evolution of HIV sequences after transmission before a blood sample is drawn from which HIV sequences are measured. This research investigates the stratified mark‐specific PH model with missing marks where the baseline functions may vary with strata. We develop two consistent estimation approaches, the first based on the inverse probability weighted complete‐case (IPW) technique, and the second based on augmenting the IPW estimator by incorporating auxiliary information predictive of the mark. We investigate the asymptotic properties and finite‐sample performance of the two estimators, and show that the augmented IPW estimator, which satisfies a double robustness property, is more efficient.     

Human immunodeficiency virus risk: is it possible to dissuade people from having unsafe sex?

Summary. The cumulative number of human immunodeficiency virus (HIV) infections worldwide has reached 60 million in little over 30 years. HIV continues to spread despite a detailed understanding of the manner in which it spreads and measures which can prevent spread. Some governments have been highly successful in containing the spread of HIV through blood products and from mother to child and among injecting drug users. Lack of political will, lack of resources or challenges to widely accepted scientific evidence have held back similar interventions in other countries. It has proved much more difficult to reduce the sexual transmission of HIV in both high and low income countries. A wide range of strategies has been identified but it remains unclear which strategies deserve priority and what methods of promoting them have the greatest effect. There is ample evidence that awareness of HIV and changes in sexual behaviour have occurred widely but the penetration of information remains poor in some vulnerable groups especially adolescents and women in poorer countries. Further obstacles face those who have information about the risk. The subordinate position of women and a desire for large families are important obstacles to condom negotiation and use. Urbanization, poverty, conflict and declining public services all exacerbate unsafe sexual behaviour. We argue that so-called 'structural' interventions directed at these wider contexts of unsafe behaviour merit greater attention. Such approaches have the added benefit of being less susceptible to 'risk compensation' which has the potential to undermine strategies directed at reducing the transmission efficiency of HIV.     

Bayesian projection of the acquired immune deficiency syndrome epidemic

Short-term projections of the acquired immune deficiency syndrome (AIDS) epidemic in England and Wales have been regularly updated since the publication of the Cox report in 1988. The key approach for those updates has been the back-calculation method, which has been informally adapted to acknowledge various sources of uncertainty as well as to incorporate increasingly available information on the spread of the human immunodeficiency virus (HIV) in the population. We propose a Bayesian formulation of the back-calculation method which allows a formal treatment of uncertainty and the inclusion of extra information, within a single coherent composite model. Estimation of the variably dimensioned model is carried out by using reversible-jump Markov chain Monte Carlo methods. Application of the model to data for homosexual and bisexual males in England and Wales is presented, and the role of the various sources of information and model assumptions is appraised. Our results show a massive peak in HIV infections around 1983 and suggest that the incidence of AIDS has now reached a plateau, although there is still substantial uncertainty about the future.     

A multistate approach for estimating the incidence of human immunodeficiency virus by using data from a prevalent cohort study

Summary.  Cohort studies of individuals infected with the human immunodeficiency virus (HIV) provide useful information on the past pattern of HIV diagnoses, progression of the disease and use of antiretroviral therapy. We propose a new method for using individual data from an open prevalent cohort study to estimate the incidence of HIV, by jointly modelling the HIV diagnosis, the inclusion in the cohort and the progression of the disease in a Markov model framework. The estimation procedure involves the construction of a likelihood function which takes into account the probability of observing the total number of subjects who are enrolled in the cohort and the probabilities of passage through the stages of disease for each observed subject conditionally on being included in the cohort. The estimator of the HIV infection rate is defined as the function which maximizes a penalized likelihood, and the solution of this maximization problem is approximated on a basis of cubic M -splines. The method is illustrated by using cohort data from a hospital-based surveillance system of HIV infection in Aquitaine, a region of south-western France. A simulation study is performed to study the ability of the model to reconstruct the incidence of HIV from prevalent cohort data.     

Non-parametric Back-projection of HIV Positive Tests using Multinomial and Poisson Settings

Back-projection is a commonly used method in reconstructing HIV incidence. Instead of using AIDS incidence data in back-projection, this paper uses HIV positive tests data. Both multinomial and Poisson settings are used. The two settings give similar results when a parametric form or step function is assumed for the infection curve. However, this may not be true when the HIV infection in each year is characterized by a different parameter. This paper attempts to use simulation studies to compare these two settings by constructing various scenarios for the infection curve. Results show that both methods give approximately the same estimates of the number of HIV infections in the past, whilst the estimates for HIV infections in the recent past differ a lot. The multinomial setting always gives a levelling-off pattern for the recent past, while the Poisson setting is more sensitive to the change in the shape of the HIV infection curve. Nonetheless, the multinomial setting gives a relatively narrower point-wise probability interval. When the size of the epidemic is large, the narrow probability interval may be under-estimating the true underlying variation.     

Regression analysis for bivariate gap time with missing first gap time data

We consider ordered bivariate gap time while data on the first gap time are unobservable. This study is motivated by the HIV infection and AIDS study, where the initial HIV contracting time is unavailable, but the diagnosis times for HIV and AIDS are available. We are interested in studying the risk factors for the gap time between initial HIV contraction and HIV diagnosis, and gap time between HIV and AIDS diagnoses. Besides, the association between the two gap times is also of interest. Accordingly, in the data analysis we are faced with two-fold complexity, namely data on the first gap time is completely missing, and the second gap time is subject to induced informative censoring due to dependence between the two gap times. We propose a modeling framework for regression analysis of bivariate gap time under the complexity of the data. The estimating equations for the covariate effects on, as well as the association between, the two gap times are derived through maximum likelihood and suitable counting processes. Large sample properties of the resulting estimators are developed by martingale theory. Simulations are performed to examine the performance of the proposed analysis procedure. An application of data from the HIV and AIDS study mentioned above is reported for illustration.     

Cofactors and markers of disease progression in human immunodeficiency virus infection

The identification of factors which are related to human immunodeficiency virus (HIV) disease progression, either by a direct interaction with HIV to increase the rate of disease progression or by providing an indication of an infected individual's likely prognosis, can have great value when understanding HIV pathogenesis and in the development of novel therapeutic approaches. This paper describes the roles of the CD4 cell count and the viral load as markers of disease progression and discusses the recent findings on chemokine receptors in HIV infection. Our current knowledge on these factors is summarized and unresolved statistical issues are highlighted.     

Some state space models of hiv epidemic and its applications for the estimation of hiv infection and incubation

In this paper we have developed some state space models for the HIV epidemic for populations at risk for AIDS. By using these state space models, we have developed a general Bayesian procedure for estimating simultaneously the unknown parameters and the state variables. The unknown parameters include the immigration and recruitment rates, the death and retirement rates, the incidence of HIV infection ( and hence the HIV infection distribution ) and the incidence of HIV incubation ( and hence the HIV incubation distribution). The state variables are the numbers of susceptible people (S people), HIV-infected people (I people) and AIDS incidence over time. The basic approach is through multi-level Gibbs sampler combined with the weighted bootstrap method. We have applied the methods to the Swiss AIDS homosexual and IV drug data to estimate simultaneously the unknown parameters and the state variables. Our results show that in both populations, both the HIV infection and HIV incubation have multi-peaks indicating the mixture nature of these distributions. Our results have also shown that the estimates of the death and retirement rates for I people are greater than those of S people, suggesting that the infection by HIV may have increased the death and retirement rates of the individuals.     

STATISTICAL CHALLENGES OF AIDS: (The 1990 Knibbs Lecture, Canberra, 27 November 1990)

Issues that are central to the understanding and management of the HIV epidemic have generated numerous statistical challenges. This paper considers questions concerning the incubation period, the effects of treatments, pre diction of AIDS cases, the choice of surrogate end points for the assessment of treatments and design of strategies for screening blood samples. These issues give rise to a broad range of intriguing problems for statisticians. We describe some of these problems, how they have been tackled so far and what remains to be done. The discussion touches on topical statistical methods such as smoothing, bootstrapping, interval censoring and the ill-posed inverse problem, as well as asking fundamental questions for frequentist statistics.     

Influence of human immunodeficiency virus infection on neurological impairment: an analysis of longitudinal binary data with informative drop-out

A study to investigate the human immunodeficiency virus (HIV) status on the course of neurological impairment, conducted by the HIV Center at Columbia University, followed a cohort of HIV positive and negative gay men for 5 years and assessed the presence or absence of neurological impairment every 6 months. Almost half of the subjects dropped out before the end of the study for reasons that might have been related to the missing neurological data. We propose likelihood-based methods for analysing such binary longitudinal data under informative and non-informative drop-out. A transition model is assumed for the binary response, and several models for the drop-out processes are considered which are functions of the response variable (neurological impairment). The likelihood ratio test is used to compare models with informative and non-informative drop-out mechanisms. Using simulations, we investigate the percentage bias and mean-squared error (MSE) of the parameter estimates in the transition model under various assumptions for the drop-out. We find evidence for informative drop-out in the study, and we illustrate that the bias and MSE for the parameters of the transition model are not directly related to the observed drop-out or missing data rates. The effect of HIV status on the neurological impairment is found to be statistically significant under each of the models considered for the drop-out, although the regression coefficient may be biased in certain cases. The presence and relative magnitude of the bias depend on factors such as the probability of drop-out conditional on the presence of neurological impairment and the prevalence of neurological impairment in the population under study.     

A Bayesian approach in differential equation dynamic models incorporating clinical factors and covariates

A virologic marker, the number of HIV RNA copies or viral load, is currently used to evaluate antiretroviral (ARV) therapies in AIDS clinical trials. This marker can be used to assess the antiviral potency of therapies, but may be easily affected by clinical factors such as drug exposures and drug resistance as well as baseline characteristics during the long-term treatment evaluation process. HIV dynamic studies have significantly contributed to the understanding of HIV pathogenesis and ARV treatment strategies. Viral dynamic models can be formulated through differential equations, but there has been only limited development of statistical methodologies for estimating such models or assessing their agreement with observed data. This paper develops mechanism-based nonlinear differential equation models for characterizing long-term viral dynamics with ARV therapy. In this model we not only incorporate clinical factors (drug exposures, and susceptibility), but also baseline covariate (baseline viral load, CD4 count, weight, or age) into a function of treatment efficacy. A Bayesian nonlinear mixed-effects modeling approach is investigated with application to an AIDS clinical trial study. The effects of confounding interaction of clinical factors with covariate-based models are compared using the deviance information criteria (DIC), a Bayesian version of the classical deviance for model assessment, designed from complex hierarchical model settings. Relationships between baseline covariate combined with confounding clinical factors and drug efficacy are explored. In addition, we compared models incorporating each of four baseline covariates through DIC and some interesting findings are presented. Our results suggest that modeling HIV dynamics and virologic responses with consideration of time-varying clinical factors as well as baseline characteristics may play an important role in understanding HIV pathogenesis, designing new treatment strategies for long-term care of AIDS patients.     

The influence of behavioural heterogeneity on the population level effect of potential prophylactic human immunodeficiency virus type 1 vaccines

Theoretical results indicate that extensive coverage with low efficacy type 1 human immunodeficiency virus (HIV) vaccines could substantially reduce the incidence of HIV in developing countries. There is a non-linear relationship between effective vaccine coverage and HIV prevalence such that improved efficacy brings diminishing returns. The relative contribution of HIV-associated mortality and behavioural heterogeneity to this non-linear relationship is explored using deterministic mathematical models. If the duration of risk of acquiring HIV is long relative to the HIV incubation period then infection-associated mortality can generate the non-linear relationship. However, in its absence the same relationship results from behavioural heterogeneity. Models of HIV vaccination alongside other interventions generate qualitative results that suggest that targeted interventions lead to less redundancy in control efforts.     

Design of human immunodeficiency virus intervention trials in developing countries

Randomized controlled trials are recognized as the 'gold standard' for evaluating the effect of health interventions, yet few such trials of human immunodeficiency virus (HIV) preventive interventions have been conducted. We discuss the role of randomized trials in the evaluation of such interventions, and we review the strengths and weaknesses of this and other approaches. Randomization of clusters (groups of individuals) may sometimes be appropriate, and we discuss several issues in the design of such cluster-randomized trials, including sample size, the definition and size of clusters, matching and the role of base-line data. Finally we review some general issues in the design of HIV prevention trials, including the choice of the study population, trial end points and ethical issues. It is argued that randomized trials have an important role to play in the evolution of HIV control.     

Bayesian Experimental Design for Long-Term Longitudinal HIV Dynamic Studies

The study of HIV dynamics is one of the most important developments in recent AIDS research for understanding the pathogenesis of HIV-1 infection and antiviral treatment strategies. Currently a large number of AIDS clinical trials on HIV dynamics are in development worldwide. However, many design issues that arise from AIDS clinical trials have not been addressed. In this paper, we use a simulation-based approach to deal with design problems in Bayesian hierarchical nonlinear (mixed-effects) models. The underlying model characterizes the long-term viral dynamics with antiretroviral treatment where we directly incorporate drug susceptibility and exposure into a function of treatment efficacy. The Bayesian design method is investigated under the framework of hierarchical Bayesian (mixed-effects) models. We compare a finite number of feasible candidate designs numerically, which are currently used in AIDS clinical trials from different perspectives, and provide guidance on how a design might be chosen in practice.