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

New challenges for mathematical and statistical modeling of HIV and hepatitis C virus in injecting drug users

Injecting drug users are not only driving blood-borne transmission of HIV and hepatitis C virus but also likely drive sexual transmission of HIV in large parts of the world. Mathematical and statistical modeling can provide important insights in these epidemiological processes and on the potential impact of interventions but have been little used to date. This review aims to discuss the potential areas of application of modeling in the field of viral infections in injecting drug users. After reviewing key examples of published modeling work on HIV and hepatitis C virus in injecting drug users, we discuss recent developments in the epidemiology, diagnosis, treatment and prevention of these infections. In particular, new methods for the diagnosis of early HIV infection, new antivirals for a more effective treatment of HIV, hepatitis B and hepatitis C virus infections, new concepts in design and surveillance of interventions for drug users and increasing possibilities of molecular typing of pathogens are changing the questions and decisions for public health policy makers who deal with drug-related infectious diseases. Research including mathematical modeling is needed to understand the impact of new diagnostic tools, new treatment options and combined intervention strategies on the epidemiology of viral infections in injecting drug users. Methodological advances in mathematical modeling are required to adequately approach some of the ensuing research questions. Modeling has much to offer for solving urgent policy questions, but current levels of funding in modeling research are insufficient and need to be scaled up to make better use of these possibilities.

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

"This paper considers questions concerning the incubation period [of HIV infections], the effects of treatments, prediction 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." The geographical scope is worldwide, with some data for selected developed countries used to illustrate the models.

Estimation and interpretation of models of absolute risk from epidemiologic data, including family-based studies

Absolute risk is the chance that a person with given risk factors and free of the disease of interest at age a will be diagnosed with that disease in the interval (a, a + tau]. Absolute risk is sometimes called cumulative incidence. Absolute risk is a "crude" risk because it is reduced by the chance that the person will die of competing causes of death before developing the disease of interest. Cohort studies admit flexibility in modeling absolute risk, either by allowing covariates to affect the cause-specific relative hazards or to affect the absolute risk itself. An advantage of cause-specific relative risk models is that various data sources can be used to fit the required components. For example, case-control data can be used to estimate relative risk and attributable risk, and these can be combined with registry data on age-specific composite hazard rates for the disease of interest and with national data on competing hazards of mortality to estimate absolute risk. Family-based designs, such as the kin-cohort design and collections of pedigrees with multiple affected individuals can be used to estimate the genotype-specific hazard of disease. Such analyses must be adjusted for ascertainment, and failure to take into account residual familial risk, such as might be induced by unmeasured genetic variants or by unmeasured behavioral or environmental exposures that are correlated within families, can lead to overestimates of mutation-specific absolute risk in the general population.

Complementary approach of data analysis and modeling to estimate the pattern of the BSE epidemic: the example of France

Clinical surveillance was the only way to detect bovine spongiform encephalopathy (BSE) until July 2000 in France. From the 103 cases identified as such between 1991 and June 2000, we used a back-calculation method to reconstruct the longitudinal trend of BSE infections. Between July 1987 and June 1997, an estimated 51,300 (CI =[24,300-84,700]) cattle were infected in France. The comprehensive surveillance of BSE with rapid tests, set up in France since 2001 at the abattoir and fallen plant, allowed study of the relative exposure of the successive birth cohorts with nonconditional logistic regression models adjusted for possible confounding variables. The results were in agreement with those of the back-calculation model, estimating a decrease of the BSE exposure from the birth cohort July 1995-June 1996 that matched with the decrease of the infection after June 1996. In view of the long incubation period of BSE, it is not possible to precisely assess the impact of any control measure before several years. Modeling was therefore used to estimate prospectively the efficiency of the ban of meat and bone meal extended to all farm species in November 2000. Using parameters about age at infection and incubation time estimated earlier, we assessed the minimum time to first detection if infections still occurred. We have waited up to June 2007 to know if less than 100 infections occurred among French cattle during the 6 months following January 2001.

BSE risk assessment as a basis for updating French screening policy

The current French bovine spongiform encephalopathy (BSE) surveillance system, based on rapid testing of all cattle over 24 months of age and on clinical diagnosis, detects all clinical cases and some preclinical cases of BSE. Several indicators point to a marked shrinkage of the French BSE epidemic in recent years, owing to risk reduction measures. Meat and bone meal, the only known vector of the BSE agent, was banned in feed for all farmed species in November 2000. Thus the surveillance system may be relaxed. The objective of this risk assessment study was to provide information for decisionmakers on the minimum age at which healthy and high-risk cattle now need to be screened with rapid tests. For this purpose, we used the back-calculation method to project the course of the BSE epidemic. We examined the predicted patterns of the number and age distribution of cases of BSE that would be detected by the different existing surveillance streams. Various theoretical sensitivities of rapid tests were explored. Assuming that feed-borne sources of infection no longer exist, and that BSE does not occur spontaneously, our models suggest that it would have been possible to raise the minimum age for rapid tests to 66 months in early 2006, whereas theoretical reasoning, based on the assumption that the total meat and bone meal ban was effective in November 2001, suggests that this age cutoff could only be raised to 48 months in early 2006. These results only apply to cattle born and bred in France. If the situation remains unchanged, the age cutoff could be raised incrementally each year.

Assessment of intervention measures for the 2003 SARS epidemic in Taiwan by use of a back-projection method

OBJECTIVES

To reconstruct the infection curve for the 2003 severe acute respiratory syndrome (SARS) epidemic in Taiwan and to ascertain the temporal changes in the daily number of infections that occurred during the course of the outbreak.

METHOD

Back-projection method.

RESULTS

The peaks of the epidemic correspond well with the occurrence of major infection clusters in the hospitals. The overall downward trend of the infection curve after early May corresponds well to the date (May 10) when changes in the review and classification procedure were implemented by the SARS Prevention and Extrication Committee.

CONCLUSION

The major infection control measures taken by the Taiwanese government over the course of the SARS epidemic, particularly those regarding infection control in hospitals, played a crucial role in containing the outbreak.

Estimating incidence of the French BSE infection using a joint analysis of both asymptomatic and clinical BSE surveillance data

Bovine Spongiform Encephalopathy (BSE) clinical surveillance data were the main source of information to perform back-calculation of BSE infection incidence. Since 2001, systematic BSE screening tests enhanced the clinical surveillance and allowed to detect some preclinical, i.e. asymptomatic, cases of BSE. We propose a method to incorporate additional information provided by screening tests. It was the first time that a back-calculation model was developed for a full BSE clinical surveillance. In the spirit, our approach resembles what it was done in the Acquired Immune Deficiency Syndrome (AIDS) epidemic to incorporate the Human Immunodeficiency Virus (HIV) diagnosis. Nevertheless, in the BSE epidemic, we had to consider different surveillance systems, their peculiarity, and the phenomenon of communicating vessels between these surveillance systems. In addition, both the preclinical sensitivity of tests and the status of BSE cases, asymptomatic or clinical, were not precisely known. We applied the model to the French BSE epidemic in order to obtain an updated estimate of the incidence of BSE infection. Our back-calculation model fitted very well the observed data of each surveillance system. We detected a lengthening of the incubation period and estimated that the number of infections was very small in the late 1990s and zero in July 2001.

Infectiousness of smallpox relative to disease age: estimates based on transmission network and incubation period

This study investigated the infectiousness of smallpox relative to the onset of fever using a likelihood-based estimation procedure based on the observed transmission network (n=223) and on the distribution of the incubation period (n=379). Who-infected-whom information enabled us to back-calculate the infectiousness by disease age, employing a step function model for infectiousness. Frequency of secondary transmissions was highest between 3 and 6 days after onset of fever, yielding an expected daily frequency of 20.6% (95% CI 15.1-26.4) of the total number of secondary transmissions, which is consistent with previous observations. The estimated cumulative frequency suggests that 91.1% of secondary transmissions occurred up to 9 days after onset of fever. The proposed method appeared to be useful for diseases with an acute course of illness, where transmission was not hampered by depletion of susceptible contacts.

Including pre-AIDS mortality in back-calculation model to estimate HIV prevalence in France, 2000

Since the advent of highly active antiretroviral therapy (HAART) the lengthening of AIDS incubation time has led to a decrease of AIDS incidence and mortality, and to the increase of the proportion of pre-AIDS mortality. The objective was to develop an extension of the back-calculation model by including pre-AIDS mortality and to estimate HIV prevalence in France. Our previous back-calculation model was modified to take into account the probability of survival for HIV-infected individual using the relative risk to die at different period for an HIV-infected person versus the general population (psi). AIDS cases reported to InVS (Institut de Veille Sanitaire) until March 2003 were back-calculated to estimate HIV incidence until December 2000. AIDS deaths occurring until December 2000 were used to obtain HIV prevalence in 2000. Plausible intervals were calculated taking into account uncertainties on AIDS incubation time. Taking into account pre-AIDS mortality increased the goodness-of-fit of the model to the data. The relative risk, psi, was estimated as 3 for homo-bisexual men, haemophiliacs and transfused cases, 10 for intravenous drug users, and 4 for heterosexual cases, with no difference over period. HIV prevalence in 2000 was estimated as 88,200 (with a plausible interval of 52,300-168,000), versus 78,200 when mortality was not considered. Pre-AIDS mortality estimates show the amount of this mortality during the era of HAART but also evidenced its existence before HAART. Taking into account pre-AIDS mortality of HIV-infected person in the back-calculation model increased the estimated HIV prevalence.

Estimation of the age-dependent risk of infection to BSE of dairy cattle in Great Britain

An important aspect of the bovine spongiform encephalopathy (BSE) epidemic has been an apparent age-dependent risk of infection, with younger cattle being more likely to become infected than older cattle. Our objective was to determine the age-dependent risk of infection of dairy cattle. We first reviewed unpublished data on the feeding patterns of proprietary concentrates for dairy-replacement cattle. These data showed that autumn- and spring-born cattle would receive different feeding patterns of proprietary concentrates, and so age-dependent risk of infection profiles were obtained separately for autumn- and spring-born cattle. We used back-calculation methods to analyse BSE-epidemic data collected in Great Britain between 1984 and 1996. Dairy cattle were most at risk in the first 6 months of life; adult cattle were at relatively low risk of infection. Between 6 and 24 months of age, risk profiles reflected feeding patterns of proprietary concentrates in each of the autumn- and spring-born cohorts.

Bayesian back-calculation using a multi-state model with application to HIV

Back-calculation is a method of obtaining estimates of the number of infections of a disease over time. Data on an endpoint of the disease, together with knowledge of the time from infection to endpoint, allows reconstruction of the incidence of infection. The technique has had much success when applied to the HIV epidemic, using incidence of AIDS diagnoses to inform past HIV infections. In recent years, the period from infection to AIDS has changed considerably due to new regimes of anti-viral therapies. This has led to attempts to use incidence of first positive HIV test as an alternative basis for back-calculation. Developing on earlier work, this paper explores the feasibility of a multi-state formulation of the back-calculation method that models the disease and diagnosis processes and uses HIV diagnoses as an endpoint. Estimation is carried out in a Bayesian framework, which naturally allows incorporation of external information to inform the diagnosis probabilities. The idea is illustrated on data from the HIV epidemic in homosexuals in England and Wales.

Modelling studies on bovine spongiform encephalopathy occurrence to assist in the review of the over 30 months rule in Great Britain

The objective of this study was to contribute to a risk assessment to review the over 30 months (OTM) scheme in cattle, whereby all cattle aged over 30 months slaughtered in the UK are removed from the human food chain. We use back-calculation methods to estimate the impact of changes to the OTM rule, by using passive and active surveillance data collected between 1 July 2001 and 30 June 2002. There are two types of change considered: increasing the age limit and allowing animals born after a certain date into the food chain. Results indicate that under the OTM rule less than 1 animal in the last 12 months of the incubation period would enter the food chain in 2003. The birth date changes considered and small changes to the upper age limit would increase this number by a relatively small amount.

Assessment of the risk posed by bovine spongiform encephalopathy in cattle in Great Britain and the impact of potential changes to current control measures

We extended an existing back-calculation model to analyse data on reported clinical cases of bovine spongiform encephalopathy (BSE), data from random testing of healthy animals slaughtered in abattoirs and testing data from animals reported as sick or dying on the farm. Extensive analysis of demographic data was also undertaken. We estimated past and current BSE infection prevalences in the cattle population and the degree of case under-ascertainment resulting from excess mortality in cattle near to disease onset. Ongoing levels of human exposure to BSE infectivity were also estimated, together with the effect on these of a range of possible exposure-reduction strategies that might replace the current rule banning tissue from cattle over 30 months (OTM) of age from the human food supply. While any policy change that allows a wider age range of animals into the human food supply will increase levels of human exposure to infectivity, the risk posed by such increases is small by comparison with historical exposure levels. Making the pessimistic assumption that there will be 5000 deaths during the variant Creutzfeldt-Jakob disease (vCJD) epidemic in total, our analysis indicates that replacement of the OTM rule with testing would result in 0.04 additional vCJD deaths over the next 60 years. However, there is substantial (more than 40-fold) uncertainty surrounding this estimate, the sources of which are discussed.

Extending backcalculation to analyse BSE data

We review the origins of backcalculation (or back projection) methods developed for the analysis of AIDS (acquired immunodeficiency syndrome) incidence data. These techniques have been used extensively for >15 years to deconvolute clinical case incidence, given knowledge of the incubation period distribution, to obtain estimates of past HIV (human immunodeficiency virus) infection incidence and short-term predictions of future AIDS incidence. Adaptations required for the analysis of bovine spongiform encephalopathy (BSE) incidence included: stratification of BSE incidence by age as well as birth cohort; allowance for incomplete survival between infection and the onset of clinical signs of disease; and decomposition of the age- and time-related infection incidence into a time-dependent feed risk component and an age-dependent exposure/susceptibility function. The most recent methodological developments focus on the incorporation of data from clinically unaffected cattle screened using recently developed tests for preclinical BSE infection. Backcalculation-based predictions of future BSE incidence obtained since 1996 are examined. Finally, future directions of epidemiological analysis of BSE epidemics are discussed taking into account ongoing developments in the science of BSE and possible changes in BSE-related policies.

Implications of BSE infection screening data for the scale of the British BSE epidemic and current European infection levels

The incidence of confirmed clinical cases of bovine spongiform encephalopathy (BSE) in Great Britain continues to decline, but the recent discovery of cases in previously unaffected countries (including Israel, Japan, Poland, Slovenia and Spain) has heightened concerns that BSE transmission was more intense and widespread than previously thought. We use back-calculation methods to undertake an integrated analysis of data on infection prevalence in apparently healthy cattle and the incidence of confirmed clinical disease. The results indicate substantial underascertainment of clinical cases over the course of the British epidemic, and consequently that two- to fourfold more animals were infected than previously estimated. Upper bounds on the predicted size of the new variant Creutzfeldt-Jakob Disease (vCJD) epidemic are unaffected, as the prediction methods employed fit to observed vCJD mortality data, and are not sensitive to estimates of the absolute magnitude of past human exposure to BSE-infected cattle, only to relative changes in exposure through time. We also estimate the per-head incidence of infection in cattle born between 1993 and 1997 in other European Union countries, using data on the testing of apparently healthy cattle slaughtered for consumption. Infection incidence for cattle born after mid-1996 was highest in Greece, Italy and Belgium, with Spain and The Netherlands having intermediate levels, and estimates for Great Britain, Germany and France being comparably low.

Multinomial analysis of smoothed HIV back-calculation models incorporating uncertainty in the AIDS incidence

Back-calculation models, developed to reconstruct the past trend of human immunodeficiency virus (HIV) and to project future acquired immunodeficiency syndrome incidence (AIDS), are usually and unrealistically based on the assumption that the observed AIDS counts are independently distributed according to a Poisson process. In contrast, we argue that a multinomial framework is more suitable to this situation, leading to a natural covariance structure. The ill-conditioned nature of the problem is solved by modelling the HIV parameters according to a cubic spline function to reduce the dimensionality of the parameter space and obtain smoother parameter estimates. We applied a regression spline technique which yields to a computationally stable basis incorporating the incubation period in the new design matrix. We directly incorporate the reporting delay distribution in the AIDS incidence data, leading to a more complex formulation of the variance and covariance model that is adapted to the iteratively reweighted least square (IRLS) algorithm. In this case we obtain more accurate estimates of the standard error of the HIV incidence, especially in the most recent time. Our model, which uses a cubic spline reparameterization based on a multinomial probability distribution, is applied to the AIDS epidemic data in Italy.

Mathematical biology and medical statistics: contributions to the understanding of AIDS epidemiology

Some of the many ways in which mathematical biology and statistics have been used in investigating the acquired immunodeficiency syndrome (AIDS) epidemic are reviewed. Aspects of the spread of the disease via social and sexual networks are discussed. The different kinds of data involved are critically compared. Some studies of the incubation period are briefly reviewed and some comments made on the role of adherence to therapy.

Advances in medical statistics arising from the AIDS epidemic

Many statisticians have contributed to studies of the HIV epidemic and progression to AIDS. They have developed new statistical methodology, where needed, to address HIV-related issues. The transfer of methods from one area to another often involves a substantial delay. This paper points to methods that were developed in the HIV context and have either already found applications in other areas of medical research or have the potential for such applications, with the hope that this will promote a speedier transfer of the research methods. Among the new tools that HIV studies have placed firmly into the pool of statistical methods for medical research are the methods of back-calculation, methods for the analysis of retrospective ascertainment data and methods of analysis for the combined data from clinical trials and associated longitudinal studies. Notions that have been stimulated substantially are use of surrogate endpoints in clinical trials and screening blood products by the use of pooled serum samples. Research activity in many other areas has been boosted substantially through contributions motivated by HIV/AIDS studies. Noteworthy examples are analyses for doubly-censored lifetime data and methods for assessing vaccines for transmissible diseases.

Estimating HIV incidence using dates of both HIV and AIDS diagnoses

Knowledge of HIV incidence is important to formulate sensible strategies aimed at controlling the HIV/AIDS epidemic. Back-projection is one of the methods for reconstructing the HIV incidence curve from AIDS incidence data. However, because of the low risk of developing AIDS during the first few years after infection, precise estimates of HIV incidence for the recent past are unlikely if we use AIDS incidence data only. As a result there have been recent attempts to use, not only the date of AIDS diagnosis, but also to use the date of their first positive HIV test. The objective of this paper is to incorporate into back-projection the additional information provided by those individuals who have tested HIV positive but have not yet developed AIDS. This adds information on a very large number of other individuals, and provides the hope that the precision of back-projection is improved considerably. The date of a positive HIV test or an AIDS diagnosis of an individual, whichever comes first, is used in a generalized convolution equation for the purpose of back-projection. The method is illustrated by an application to Australian HIV and AIDS data. Study results show that dramatic improvement in precision is gained for estimates of HIV incidence in recent years when both HIV and AIDS diagnosis dates are used on all individuals.

Uncertainty of AIDS incubation time and its effects on back-calculation estimates

Incubation time is the period from the onset of HIV infection to AIDS. The distribution of the incubation time is one of the main parameters of the back-calculation method for the estimation of incidence of HIV infection. Because of the long and variable incubation time, the assessment of its distribution is uncertain and this uncertainty spreads through the back-calculation method and affects the estimation of the precision of incidence of HIV infection. We propose a method to investigate the sensitivity of the estimates to variations of the incubation times, with particular regard to the covariate AGE in the modelling of the incubation period, making use of the parametric bootstrap. An application to the HIV epidemic in Italy is presented. The amplification of the uncertainty of the HIV incidence estimates resulting from the implementation of our proposed method tends to concentrate around the earlier periods of the epidemic, corresponding to the right tail of the incubation time distribution, which is very sensitive to small perturbations.

Using AIDS mortality data to reconstruct HIV/AIDS epidemics

The back-calculation methods used to reconstruct HIV epidemics and to predict future AIDS cases use data on the number of AIDS reported cases and information about the time from HIV infection to AIDS diagnosis. In this paper we modify a back-calculation method, presented in Verdecchia and Mariotto, to estimate HIV incidence from AIDS mortality data rather than the AIDS registered cases. In this application the AIDS incubation time is replaced by the time from HIV infection to AIDS death. Since AIDS is an irreversible condition leading to death within a few years, mortality data is expected to be approximately as informative as AIDS data. The main application of this method would be in countries where an AIDS registry is not available or where the registration of AIDS cases is not consistent, for example, due to changes in the definition of AIDS. Application to the Italian AIDS mortality data shows that the number of AIDS cases reconstructed from this method is very close to the registered number of cases. Also, the two estimates of HIV incidence and prevalence obtained from back-calculation on both AIDS mortality data and the number of AIDS reported cases were very similar, although there was more variability in the estimates obtained from mortality data.

Is the AIDS incubation time changing? A back-calculation approach

Contradictory literature was recently published on possible changes in AIDS incubation time over the period 1978-1994. The purpose of this work was to test if a change in incubation time (shortening or lengthening) was observed in France, either globally or in specific transmission groups (homosexual-bisexual men, heterosexual subjects), using a back-calculation approach. An age dependent TSI model (time since infection), which took into account a temporary treatment effect and allowed us to test for a change in the incubation time, was applied to the French AIDS cases (Réseau National de Santé Publique). The EM algorithm was used to maximize the likelihood and the best model was selected on the basis of the likelihood ratio statistic. The analysis on all AIDS cases indicated a shortening of the AIDS incubation time estimated to begin in 1983 (95 per cent CI 1982-1984). This shortening of incubation time was also apparent when analysis was restricted to homosexual-bisexual men and to heterosexual subjects. This shortening corresponded to a median incubation time of 9.6 years (95 per cent CI 8.1-10.5) for people infected at 30 years of age in 1983, versus 12.7 years for people infected at 30 years of age before the change.

A hierarchical Bayesian approach to age-specific back-calculation of cancer incidence rates

We propose a Bayesian hierarchical model to estimate age-specific cancer incidence per year from age-specific cancer mortality. The model is based upon the empirical Bayesian approach of Liao and Brookmeyer (1995) and extends that model by consideration of the dependence on age. The incident cases per year are considered as observations from a discrete-time stochastic process following an autoregressive structure within a Poisson regression model. The model assumes that the survival probability among those with cancer is known. We have investigated the sensitivity of the model to the choice of this distribution and have found that this is the most sensitive part of the model. By comparison the predictions of the model are relatively robust to changes in other key areas, such as the number of years an incident case contributes before death, assumptions about parameter equality for identification and the initial prior distributions. The proposed methodology has been investigated using lung cancer mortality data from Scotland. Parameter estimates were obtained through Markov chain Monte Carlo methods, implemented using BUGS.

BSE in Northern Ireland: epidemiological patterns past, present and future

By 30 January 1998, there had been 170,259 confirmed cases of BSE in Great Britain (GB), 1766 confirmed cases in Northern Ireland (NI) (2 January 1998), and 276 confirmed cases in the Republic of Ireland (31 January 1998). Analysis of the epidemiological patterns in the NI epidemic reveals significant clustering of cases in herds and counties. The observed clustering of cases within herds results in lower per capita incidence of BSE in previously unaffected herds, providing support for the introduction of a certified herd scheme in NI. By fitting a backcalculation model to the case data, we can estimate the number of animals infected with the aetiological agent of BSE and project the number of future cases. We predict that the epidemic will decline rapidly, with approximately 99 cases (95% confidence interval 30,504) occurring in the five year period 1997-2001.

The epidemiology of BSE in cattle herds in Great Britain. II. Model construction and analysis of transmission dynamics

Mathematical model that describe the key processes determining the pattern of the bovine spongiform encephalopathy (BSE) epidemic in British cattle are derived that allow for infection from feed as well as maternal and direct horizontal transmission. Heterogeneous susceptibility classes are also incorporated into the analysis. Maximum likelihood methods are used to estimate parameters and to obtain confidence intervals from available experimental and epidemiological data. A comprehensive sensitivity analysis of all model parameters and distributional assumptions is presented. Additional validation is provided by fitting the model to independent data collected in Northern Ireland. Model estimates and predictions based on BSE case data for Great Britain and Northern Ireland, together with their implications, are reviewed, and future research priorities discussed.

Uses of the EM algorithm in the analysis of data on HIV/AIDS and other infectious diseases

The analysis of data on infectious diseases is a natural setting for applications of the EM algorithm, because the infection process is only partially observable. Difficulties in determining the expectation at the E step have been side-stepped by adopting pragmatic models which reflect only part of the mechanism that generates the data. In the HIV/AIDS context the EM algorithm has helped in the reconstruction of the unobserved HIV infection curve, the so-called backprojection problem, as well as in the estimation of the distribution for the incubation period until AIDS, in estimating the infectivity of HIV in partnerships and in estimating parameters describing the decline in the immune system. There is a need for smooth estimates of functions in these applications, suggesting the use of the EMS algorithm or use of the EM algorithm to maximize a penalized likelihood. For data on other infectious diseases the application of the EM algorithm has so far been restricted to analyses of data on the size of outbreaks in a sample of households.

Transmission dynamics and epidemiology of BSE in British cattle

A comprehensive analysis of the bovine spongiform encephalopathy (BSE) epidemic in cattle in Great Britain assesses past, present and future patterns in the incidence of infection and disease, and allows a critical appraisal of different culling policies for eradication of the disease.

A meta-analysis of estimates of the AIDS incubation distribution

Information from 12 studies is combined to estimate the AIDS incubation distribution with greater precision than is possible from a single study. The analysis uses a hierarchy of parametric models based on a four-parameter generalized F distribution. This general model contains four standard two-parameter distributions as special cases. The cases are the Weibull, gamma, log-logistic, lognormal distributions. These four special cases subsume three distinct asymptotic hazard behaviors. As time increases beyond the median of approximately 10 years, the hazard can increase to infinity (Weibull), can plateau at some constant level (gamma), or can decrease to zero (log-logistic and lognormal). The Weibull, gamma and 'log-logistic distributions' which represent the three distinct asymptotic hazard behaviors, all fit the data as well as the generalized F distribution at the 25 percent significance level. Hence, we conclude that incubation data is still too limited to ascertain the specific hazard assumption that should be utilized in studies of the AIDS epidemic. Accordingly, efforts to model the AIDS epidemic (e.g., back-calculation approaches) should allow the incubation distribution to take several forms to adequately represent HIV estimation uncertainty. It is recommended that, at a minimum, the specific Weibull, gamma and log-logistic distributions estimated in this meta-analysis should all be used in modeling the AIDS epidemic, to reflect this uncertainty.

Computing back-calculation estimates of AIDS epidemic

The method of back-calculation estimates the number of HIV infections from AIDS incidence data and projects future AIDS incidence. We explore a conditional likelihood approach for computing estimates of the number of HIV infections and the parameters in the epidemic density. This method is asymptotically equivalent to the usual likelihood method. The asymptotic normal distribution of the estimates facilitates the computation of confidence intervals. We compute standard deviations for the estimates of HIV incidence and project AIDS incidence from the underlying multinomial distributions. We illustrate the methods with applications to AIDS data in the United States.

A back-calculation method to estimate the age and period HIV infection intensity, considering the susceptible population

An extension of the back-calculation method to reconstruct the past history of HIV infections is presented. In this method the HIV infection intensity, as a function of age and calendar time, is assumed to belong to a family of parametric functions and is back-calculated from the observed AIDS incidence cases, knowledge of the incubation period and of the susceptible population, the final model being chosen as the one that gives the best fit. The model allows for the interaction between age and calendar time. The application to the main risk groups in Italy, male IVDUs and homosexuals, showed that age and the susceptible population were important in estimating the epidemic among IVDUs; interaction between age and calendar time was found to be significant, showing that among young IVDUs the risk of infection decreased faster than among older IVDUs. Sensitivity of the estimates to the assumption of population and choice of the infection function is investigated.

Estimating the human immunodeficiency virus infection curve of intravenous drug users in Lombardia, Italy

We present a model to estimate the infection curve of the human immunodeficiency virus in intravenous drug users in Lombardia. We based estimates on AIDS incidence data, according to a backcalculation model accounting for 'therapy' and changes in the surveillance definition of AIDS.

A direct approach for correcting AIDS incidence: variance formula and comparison with other methods

We present a direct approach for correcting the acquired immunodeficiency syndrome (AIDS) incidence data for reporting delays, based on a non-parametric method for the analysis of right truncated data. We show that the proposed method when applied for grouped data is equivalent to three other published methods. We give a simple formula for the variance of the estimated AIDS incidence. Both estimator and variance are assessed in a simulation study. It is important for the estimation of AIDS incidence in the last quarter of the period under consideration to use month rather than quarter or half-year as the time unit for the analyses. The method is illustrated using data from the United States Centers for Disease Control.

Seasonal and other short-term influences on United States AIDS incidence

This paper models monthly AIDS diagnosis counts in terms of smooth secular trend, calendar month effects, and the number of workdays per month. A parameterization of month effects allows separation of true seasonal effects from a linear trend over the calendar year and an arbitrary June effect. There is strong evidence for seasonal patterns, other calendar month effects, and workday effects. Examination of subgroups defined by reporting delay, initial diagnosis, risk group, and region shows evidence for seasonal patterns in some diagnosis groups, for effects due to imputed diagnosis dates, and for effects due to patients' choices of when to seek diagnosis.

Using time of first positive HIV test and other auxiliary data in back-projection of AIDS incidence

Estimation of HIV incidence by the method of back-projection typically uses data on the time of diagnosis of AIDS cases, together with known information about the incubation distribution of AIDS. This paper discusses back-projection using auxiliary data on AIDS cases, particularly the time of first positive HIV test. We discuss the possibility that certain types of auxiliary data, including time of first positive test, can be useful in back-projection because they provide extra information about the incubation period of AIDS cases. Under a back-projection model, theoretical efficiency calculations are given comparing back-projection with and without the time of first positive HIV test of AIDS cases. These calculations suggest that such data have the potential to significantly improve HIV incidence estimates, particularly in the recent past. Smoothed non-parametric estimates of both HIV incidence and time-dependent testing rates are described. These can be obtained using the EM algorithm, in conjunction with a smoothing step or a penalized likelihood. The benefit of these methods in practice needs to be assessed as such data become available.

Back-projection of German AIDS data using information on dates of tests

The paper presents an application of back-projection methods to the reported AIDS data which are collected by the AIDS Center in the Federal Office of Health, Berlin. The analysis is based on all data reported up to 31 December 1992. The data are broken down by major risk groups. Correction for reporting delays takes into account temporal changes in their distribution. The paper applies the EMS algorithm for the estimation of HIV incidence. The incubation period is modelled according to a convolution of several exponential distributions which describe time dependent phenomena like change of case definition and therapy effects. For each case it is reported whether an antibody test has been performed before diagnosis, and if yes, at what time this test was performed. The modelling of the incubation distribution takes into account the transition from a state 'not yet tested' into a state 'tested'. Only tested individuals in three pre-AIDS states are eligible for treatment. The model allows us to estimate not only the current total HIV prevalence but also a breakdown into those that are not yet tested, those that are tested but not yet treated and those that are under treatment. The results depend on the assumptions about the effect of treatment and on the degree of smoothing applied in the EMS algorithm.

Simultaneous back-projection of AIDS incidence data for two or more groups

Back-projection of AIDS incidence data is useful for estimating characteristics of the HIV epidemic curve and forms a basis for projections of the AIDS epidemic curve. Its application to subgroups of the population is limited by its imprecision for groups with a small number of cases. Back-projection can be made more precise by pooling data from different groups and linking their HIV infection intensities by a model. Here we propose a method based on proportional infection intensities and study its performance with simulations and applications to AIDS in different States of Australia and haemophiliacs in the U.S.A. This method of simultaneous back-projection is shown to reduce substantially the width of confidence intervals for HIV infection intensities and for total numbers infected.

Estimates of disease incidence in women based on antenatal or neonatal seroprevalence data: HIV in New York City

Piecewise constant incidence models were developed to estimate the force of infection in women from age- and time-specific antenatal or neonatal seroprevalence data. Differential inclusion of infected women in sero-surveys compared to uninfected women was taken into account, with respect to both changes in inclusion rate following infection, and changes in relative inclusion rate over calendar time. These models were applied to anonymous HIV seroprevalence data collected from neonates born to black and Hispanic women in New York City 1988-1992, with incidence and fertility parameters estimated by maximum likelihood. Estimates of inclusion rate parameters accorded well with what is known about the natural history of HIV. The data could not distinguish between additive and multiplicative combination of the effects of age and time on incidence. Incidence was strongly dependent on age with the highest incidence in women aged 20-34 years. There was strong evidence that incidence had been falling in Hispanic women since 1982-1984. The results illustrate the extent to which trends in incidence over time may be confounded by changes in the relative inclusion rate of infected and uninfected women.

[Short-term prediction of cases of AIDS in Catalonia (1991-1994)]

Predicting AIDS incidence is a useful strategy for health service planning and for the design of preventive and control programmes. The different predictive mathematical models of AIDS can be classified in simple and complex, according to the assumptions used. The present study presents the prediction of the minimum AIDS incidence in Catalonia for the period 1991-94 by means of three simple mathematical models. The models that have been compared are the projection method of Chin and Lwanga, the retroprojection method of Brookmeyer and Gail, and the extrapolation method of Cox and Medley. Baseline information about AIDS cases comes from the AIDS Registry of the "Generalitat" of Catalonia. In a comparable manner, the three methods point out an increase of AIDS incidence until 1994, varying from 1160 to 1733 new cases in that year depending on the method used. We conclude that it will be necessary to increase the provision of health, social and preventive resources against AIDS in Catalonia in the coming years.

Relationships between incidence and mortality in non-reversible diseases

The estimation of the incidence of chronic diseases from mortality and survival rates is shown to be attainable, under some simplifying hypotheses, through the solution of a convolution equation. Exact solutions are derived for the cases in which survival times of incident cases are exponential or gamma distributed. Applications to breast cancer, HIV infection and stomach cancer incidence in the Italian population are presented.

Assessing the extent of the Australian HIV epidemic from AIDS surveillance data

Current knowledge about human immunodeficiency virus (HIV) disease is used to assess past and future trends in Australian HIV/AIDS (acquired immune deficiency syndrome) incidence, focusing on the precision with which such assessments can be made. The statistical method of back-projection is applied to reconstruct the past pattern of HIV incidence from surveillance data on AIDS incidence to June 1992. The results indicate that HIV incidence rose rapidly in the early 1980s to peak in 1983-1984, followed by a sharp decline. This finding is insensitive to plausible variations from the assumptions made, and is consistent with both success in preventive strategies and high levels of infection in a subgroup having a high probability of exposure. Cumulative HIV incidence to the end of 1987 is estimated with a 90 per cent confidence interval from 9,350 to 10,350. Estimation of the cumulative HIV incidence to June 1992 is less precise, with a 90 per cent confidence interval of 12,900 to 17,800. After adjustment for underreporting the incidence could be as high as 22,000, but only if recent infection rates, which cannot be quantified accurately, were very high. Based on data to June 1992, the estimated trend in AIDS incidence indicates 680 new cases in 1993, rising gradually to 695 in 1995. The estimated rate of increase in AIDS incidence over the recent past and near future is significantly less than that observed earlier in the epidemic. This is a consequence of both the earlier peak in HIV incidence and the effect of therapy.

An empirical Bayes approach to the estimation of the incidence curve of HIV infection

A new approach is proposed for estimating the incidence curve of HIV infection and obtaining short term prediction of AIDS incidence. It is based on the method of back calculation which utilizes the fact that AIDS incidence is generated from HIV infection incidence by convolution with the incubation period distribution, but avoids the difficulties associated with approximating the infection incidence curve by a class of step functions. Instead, the infection incidence is modelled as a simple stochastic epidemic process which ensures smoothness of the estimate. We first derive the distribution of AIDS incidence knowing infection incidence. The best linear estimators of infection incidence as well as future AIDS incidence are given; a smoothed past AIDS incidence is also obtained. The parameters of the stochastic infection process can be estimated by maximum likelihood using a normal approximation to the marginal distribution of the AIDS incidence. This approach is applied to AIDS incidence data in the United States up to mid 1987. We find that the epidemic started in mid 1976, peaked at the end of 1983 and dropped afterwards.