Censoring in an epidemic with an application to hemophilia-associated AIDS

In epidemiologic studies of infectious diseases, the times of infection may be known only up to an interval. A two-stage parametric regression model is proposed for the analysis of cohort studies during an epidemic in which the exact times of infection cannot be ascertained. The methods permit joint estimation of the effects of covariates both on the risk of infection and the risk of progression to clinical disease once infected. The methodology is applied to a cohort of hemophiliacs who were at risk of infection with the AIDS virus. It was found that hemophiliacs with severe Type A hemophilia were at highest risk of infection, and the risk of infection increased sharply in the early 1980s. Hemophiliacs who were over the age of 20 at infection were at higher risk of progression to AIDS than hemophiliacs who were under age 20. The estimate of the cumulative probability of developing AIDS within t years of infection (the incubation period distribution) for hemophiliacs over age 20 was 1 - exp(-.0021t2.516). Since follow-up in this cohort was restricted to about 10 years from infection, estimates of the incubation period distribution beyond 10 years depend on model extrapolation and should be interpreted cautiously.     

Estimation of the Distribution of Infection Times Using Longitudinal Serological Markers of HIV: Implications for the Estimation of HIV Incidence

Summary In the last decade, interest has been focused on human immunodeficiency virus (HIV) antibody assays and testing strategies that could distinguish recent infections from established infection in a single serum sample. Incidence estimates are obtained by using the relationship between prevalence, incidence, and duration of recent infection (window period). However, recent works demonstrated limitations of this approach due to the use of an estimated mean “window period.” We propose an alternative approach that consists in estimating the distribution of infection times based on serological marker values at the moment when the infection is first discovered. We propose a model based on the repeated measurements of virological markers of seroconversion for the marker trajectory. The parameters of the model are estimated using data from a cohort of HIV‐infected patients enrolled during primary infection. This model can be used for estimating the distribution of infection times for newly HIV diagnosed subjects reported in a HIV surveillance system. An approach is proposed for estimating HIV incidence from these results.     

Estimation of the incidence of HIV infection

The aim of the method of 'back projection' is to provide estimates of the number of new infections with the human immunodeficiency virus (HIV) as a function of time, by using the numbers of diagnoses of the acquired immune deficiency syndrome (AIDS) together with information on the distribution of the incubation period between infection and diagnosis. Here, the method is investigated with particular reference to cases of HIV infection and AIDS in the United Kingdom.     

A modeled time-varying density function for the incubation period of AIDS

Building on the Weibull distribution, we develop a modeled time-varying density function of the incubation time between exposure to HIV infection and full-blown AIDS. This approach leads to a series of cohort-specific density functions that take into account the increasing impact of new therapies such as zidovudine (AZT). The resulting modeled density functions are studied in detail, particularly with regard to their modes and medians. The mode is sensitive to changes in the period incubation time distribution, with even a possibility of a bimodal distribution for certain combinations of the parameters that determine the rate at which the period median incubation time changes. An important substantive result is that when a period median incubation period slowly increases to some leveling off value, say m(x _c ), then it is surprisingly early on that cohorts of infected individuals have a median incubation period very close to that ultimate value m(x _c ).     

Estimation of the incubation period of influenza A (H1N1-2009) among imported cases: addressing censoring using outbreak data at the origin of importation

Empirical estimates of the incubation period of influenza A (H1N1-2009) have been limited. We estimated the incubation period among confirmed imported cases who traveled to Japan from Hawaii during the early phase of the 2009 pandemic (n=72). We addressed censoring and employed an infection-age structured argument to explicitly model the daily frequency of illness onset after departure. We assumed uniform and exponential distributions for the frequency of exposure in Hawaii, and the hazard rate of infection for the latter assumption was retrieved, in Hawaii, from local outbreak data. The maximum likelihood estimates of the median incubation period range from 1.43 to 1.64 days according to different modeling assumptions, consistent with a published estimate based on a New York school outbreak. The likelihood values of the different modeling assumptions do not differ greatly from each other, although models with the exponential assumption yield slightly shorter incubation periods than those with the uniform exposure assumption. Differences between our proposed approach and a published method for doubly interval-censored analysis highlight the importance of accounting for the dependence of the frequency of exposure on the survival function of incubating individuals among imported cases. A truncation of the density function of the incubation period due to an absence of illness onset during the exposure period also needs to be considered. When the data generating process is similar to that among imported cases, and when the incubation period is close to or shorter than the length of exposure, accounting for these aspects is critical for long exposure times.     

Epidemiological and statistical aspects of the AIDS epidemic: introduction

In 1988, a government working party studied estimates of incidence and prevalence of numbers of acquired immunodeficiency syndrome (AIDS) cases. They investigated a series of epidemiological, statistical and mathematical problems associated with predicting trends in incidences of AIDS. This paper introduces a series of papers that give a fuller and more technical exposition of the appendixes of that working party report. The papers provide a brief background to the current state of knowledge on the epidemiology of the infection and the disease; a deterministic model for human immunodeficiency virus (HIV) transmission in the male homosexual community in England and Wales is introduced. Back-projection methods are studied in two papers, following the distribution of the incubation period of the disease. The concept of minimum size of the epidemic is introduced. Mathematical functions to describe the spread of HIV infection are refined by using past trends in the incidence of AIDS to estimate values for some parameters. Survival times for AIDS patients from the point of diagnosis are considered and evidence for changes in male homosexual sexual behaviour is studied; lag-time from the point of diagnosis to the report of the case is also examined. There is a comparative analysis of the AIDS epidemic in various European countries. The incubation period of HIV in patients with haemophilia A and B infections and the problems associated with making predictions for different at-risk groups or small subgroups based on geographical area are discussed. Reasons for fluctuation between the number of reported cases from month to month are provided.     

Monitoring the trend of the transmission rate of vertically acquired HIV infection: a simple method applied to Italian data.

The objective of this study was to develop and validate a method for estimating and monitoring over time the transmission rate of vertically acquired HIV infection at the population level. We estimated the annual number of children born to HIV-infected women in Italy in 1991-1994 by multiplying the seroprevalence rates, provided by Anonymous Unlinked HIV Serosurveys among Italian Newborns, by the annual number of births, provided by the Italian National Institute of Statistics. The number of HIV-infected children was estimated by applying a simplified back-calculation method to the incident cases of vertically acquired AIDS reported to the AIDS surveillance registry, using seven different estimates of the distribution of the incubation period identified through a literature search. The annual vertical transmission rates were estimated by dividing the estimated number of children with vertically acquired HIV infection by the estimated number of births to an HIV-infected mother. Depending on the chosen distribution of the incubation period, the estimated transmission rate for the four-year period ranges from 0.10 to 0.30. Five of the seven incubation distributions provided a rate falling within the very narrow interval 0.18-0.20. The method provided estimates of vertical transmission rates consistent with those of longitudinal studies performed in European countries. The method presented here could be useful for monitoring the impact of interventions aimed at reducing HIV vertical transmission rate.     

The correlation between infectivity and incubation period of measles, estimated from households with two cases

The generation time of an infectious disease is the time between infection of a primary case and infection of a secondary case by the primary case. Its distribution plays a key role in understanding the dynamics of infectious diseases in populations, e.g. in estimating the basic reproduction number. Moreover, the generation time and incubation period distributions together characterize the effectiveness of control by isolation and quarantine. In modelling studies, a relation between the two is often not made specific, but a correlation is biologically plausible. However, it is difficult to establish such correlation, because of the unobservable nature of infection events. We have quantified a joint distribution of generation time and incubation period by a novel estimation method for household data with two susceptible individuals, consisting of time intervals between disease onsets of two measles cases. We used two such datasets, and a separate incubation period dataset. Results indicate that the mean incubation period and the generation time of measles are positively correlated, and that both lie in the range of 11-12 days, suggesting that infectiousness of measles cases increases significantly around the time of symptom onset. The correlation between times from infection to secondary transmission and to symptom onset could critically affect the predicted effectiveness of isolation and quarantine.     

Population modeling of influenza A/H1N1 virus kinetics and symptom dynamics

Influenza virus kinetics (VK) is used as a surrogate of infectiousness, while the natural history of influenza is described by symptom dynamics (SD). We used an original virus kinetics/symptom dynamics (VKSD) model to characterize human influenza virus infection and illness, based on a population approach. We combined structural equations and a statistical model to describe intra- and interindividual variability. The structural equations described influenza based on the target epithelial cells, the virus, the innate host response, and systemic symptoms. The model was fitted to individual VK and SD data obtained from 44 volunteers experimentally challenged with influenza A/H1N1 virus. Infection and illness parameters were calculated from best-fitted model estimates. We predicted that the cytokine level and NK cell activity would peak at days 2.2 and 4.2 after inoculation, respectively. Infectiousness, measured as the area under the VK curve above a viral titer threshold, lasted between 7.0 and 1.3 days and was 15 times lower in participants without systemic symptoms than in those with systemic symptoms (P < 0.001). The latent period, defined as the time between inoculation and infectiousness, varied from 0.7 to 1.9 days. The incubation period, defined as the time from inoculation to first symptoms, varied from 1.0 to 2.4 days. Our approach extends previous work by including the innate response and providing realistic estimates of infection and illness parameters, taking into account the strong interindividual variability. This approach could help to optimize studies of influenza VK and SD and to predict the effect of antivirals on infectiousness and symptoms.     

Trans-shell infection by pathogenic micro-organisms reduces the shelf life of non-incubated bird's eggs: a constraint on the onset of incubation?

Many birds initiate incubation before clutch completion, which results in asynchronous hatching. The ensuing within-brood size disparity often places later-hatched nestlings at a developmental disadvantage, but the functional significance of the timing of the onset of incubation is poorly understood. Early incubation may serve to maintain the viability of early-laid eggs, which declines over time owing to the putative effects of ambient temperature. An unexplored risk to egg viability is trans-shell infection by micro-organisms. We experimentally investigated the rate and magnitude of microbial trans-shell infection of the egg, and the relative effects of ambient temperature and micro-organisms on hatching success. We show that infection of egg contents is prevalent and occurs within the time required to lay a clutch. The probability of infection depends on the climatic conditions, the exposure period and the phylogenetic composition of the eggshell microbiota. We also demonstrate that microbial infection and ambient temperature act independently to reduce egg viability considerably. Our results suggest that these two factors could affect the onset of avian incubation in a wide range of environments.     

Epidemiological determinants of the pattern and magnitude of the vCJD epidemic in Great Britain.

Understanding the epidemiology and aetiology of new-variant Creutzfeldt-Jakob (vCJD) disease in humans has become increasingly important given the scientific evidence linking it to bovine spongiform encephalopathy (BSE) in cattle and hence the wide exposure of the population of Great Britain (GB) to potentially infectious tissue. The recent analysis undertaken to determine the risk to the population from dorsal route ganglia illustrated the danger in presenting point estimates rather than ranges of scenarios in the face of uncertainty. We present a mathematical template that relates the past pattern of the BSE epidemic in cattle to the future course of any vCJD epidemic in humans, and use extensive scenario analysis to explore the wide range of possible outcomes given the uncertainty in epidemiological determinants. We demonstrate that the average number of humans infected by one infectious bovine and the incubation period distribution are the two epidemiological factors that have the greatest impact on epidemic size and duration. Using the time-series of the BSE epidemic and the cases seen to date, we show that the minimum length of the incubation period is approximately nine years, and that at least 20% of the cases diagnosed to date were exposed prior to 1986. We also demonstrate that the current age distribution of vCJD cases can only arise if younger people were either exposed to a greater extent, more susceptible to infection, or have shorter incubation periods. Extensive scenario analyses show that given the information currently available, the very high degree of uncertainty in the future size of the epidemic will remain for the next 3-5 years. Furthermore, we demonstrate that this uncertainty is unlikely to be reduced by mass screening for late-stage infection.     

Non-parametric estimation of the age-at-onset distribution from a cross-sectional sample

We propose and study a simple and innovative non-parametric approach to estimate the age-of-onset distribution for a disease from a cross-sectional sample of the population that includes individuals with prevalent disease. First, we estimate the joint distribution of two event times, the age of disease onset and the survival time after disease onset. We accommodate that individuals had to be alive at the time of the study by conditioning on their survival until the age at sampling. We propose a computationally efficient expectation–maximization (EM) algorithm and derive the asymptotic properties of the resulting estimates. From these joint probabilities we then obtain non-parametric estimates of the age-at-onset distribution by marginalizing over the survival time after disease onset to death. The method accommodates categorical covariates and can be used to obtain unbiased estimates of the covariate distribution in the source population. We show in simulations that our method performs well in finite samples even under large amounts of truncation for prevalent cases. We apply the proposed method to data from female participants in the Washington Ashkenazi Study to estimate the age-at-onset distribution of breast cancer associated with carrying BRCA1 or BRCA2 mutations.     

Estimation of the size of the vCJD epidemic

Estimates of the final size of the variant Creuzfeldt-Jakob Disease epidemic have been made by fitting theoretical curves of the incubation period distribution to the histogram of observed annual deaths to 2002, using various assumptions of the mean and standard deviation of this distribution, and also of the efficacy of the Specified Bovine Offals ban of 1989. Unless the mean incubation time is greater than 15 to 20 years the estimates lie in the low hundreds to about a thousand, and the most likely situation, of a mean between 11 and 15 years, gives estimates of about 150 to 500 deaths. Numbers above a few thousands would only occur if the mean incubation period is of the order of 25 to 30 years and reasons are adduced to indicate this is very unlikely. These numbers are not greatly increased if the ban was poorly observed. This method of analysis may be applicable to other situations where a cause that is limited in space and time is expected to have late effects.     

Determining disease prevalence from incidence and survival using simulation techniques

ObjectivesWe present a new method for determining prevalence estimates together with estimates of their precision, from incidence and survival data using Monte-Carlo simulation techniques. The algorithm also provides for the incidence process to be marked with the values of subject level covariates, facilitating calculation of the distribution of these variables in prevalent cases.MethodsDisease incidence is modelled as a marked stochastic process and simulations are made from this process. For each simulated incident case, the probability of remaining in the prevalent sub-population is calculated from bootstrapped survival curves. This algorithm is used to determine the distribution of prevalence estimates and of the ancillary data associated with the marks of the incidence process. This is then used to determine prevalence estimates and estimates of the precision of these estimates, together with estimates of the distribution of ancillary variables in the prevalent sub-population. This technique is illustrated by determining the prevalence of acute myeloid leukaemia from data held in the Haematological Malignancy Research Network (HMRN). In addition, the precision of these estimates is determined and the age distribution of prevalent cases diagnosed within twenty years of the prevalence index date is calculated.ConclusionDetermining prevalence estimates by using Monte-Carlo simulation techniques provides a means of calculation more flexible that traditional techniques. In addition to automatically providing precision estimates for the prevalence estimates, the distribution of any measured subject level variables can be calculated for the prevalent sub-population. Temporal changes in incidence and in survival offer no difficulties for the method.     

Weight calibration to improve the efficiency of pure risk estimates from case-control samples nested in a cohort

Cohort studies provide information on relative hazards and pure risks of disease. For rare outcomes, large cohorts are needed to have sufficient numbers of events, making it costly to obtain covariate information on all cohort members. We focus on nested case-control designs that are used to estimate relative hazard in the Cox regression model. In 1997, Langholz and Borgan showed that pure risk can also be estimated from nested case-control data. However, these approaches do not take advantage of some covariates that may be available on all cohort members. Researchers have used weight calibration to increase the efficiency of relative hazard estimates from case-cohort studies and nested cased-control studies. Our objective is to extend weight calibration approaches to nested case-control designs to improve precision of estimates of relative hazards and pure risks. We show that calibrating sample weights additionally against follow-up times multiplied by relative hazards during the risk projection period improves estimates of pure risk. Efficiency improvements for relative hazards for variables that are available on the entire cohort also contribute to improved efficiency for pure risks. We develop explicit variance formulas for the weight-calibrated estimates. Simulations show how much precision is improved by calibration and confirm the validity of inference based on asymptotic normality. Examples are provided using data from the American Association of Retired Persons Diet and Health Cohort Study.     

Impact of the Infection Period Distribution on the Epidemic Spread in a Metapopulation Model

Epidemic models usually rely on the assumption of exponentially distributed sojourn times in infectious states. This is sometimes an acceptable approximation, but it is generally not realistic and it may influence the epidemic dynamics as it has already been shown in one population. Here, we explore the consequences of choosing constant or gamma-distributed infectious periods in a metapopulation context. For two coupled populations, we show that the probability of generating no secondary infections is the largest for most parameter values if the infectious period follows an exponential distribution, and we identify special cases where, inversely, the infection is more prone to extinction in early phases for constant infection durations. The impact of the infection duration distribution on the epidemic dynamics of many connected populations is studied by simulation and sensitivity analysis, taking into account the potential interactions with other factors. The analysis based on the average nonextinct epidemic trajectories shows that their sensitivity to the assumption on the infectious period distribution mostly depends on , the mean infection duration and the network structure. This study shows that the effect of assuming exponential distribution for infection periods instead of more realistic distributions varies with respect to the output of interest and to other factors. Ultimately it highlights the risk of misleading recommendations based on modelling results when models including exponential infection durations are used for practical purposes.     

Huntington disease in Georgia: age at onset.

Age at onset of motor symptoms was collected on 611 persons affected with Huntington disease (HD) among 3,201 persons "at risk" in 108 kindreds. Life-table estimates correcting for truncated intervals of observation (censoring) produced a median age at onset 5 years older than the observed mean. Risk estimates of HD onset for persons at risk, as calculated by life-table methods, were significantly higher for older ages than were estimates based on the observed distribution of onsets. Age-specific incidence was found to be highest at age 35-64 years, a considerably older age interval than suggested by previous estimates. The offspring of affected males had significantly younger onset than did offspring of affected females, and a trend suggesting and excess of paternal descent among juvenile-onset cases was present. Life-table analysis is contrasted with analyses of (a) the observed distribution of age at onset and (b) remote cohorts age 63 or older at the time of data collection. The implications for risk prediction, genetic counseling, and genetic analysis of HD are discussed.     

Determination of the spread of HIV from the AIDS incidence history

The relation between the incidence of HIV in the general population, the number of AIDS cases, and the incubation period for the disease is examined. The number of AIDS cases can be expressed in terms of a convolution integral over the incubation period distribution and the temporal history of HIV incidence. In order to determine the level of HIV incidence it is necessary to invert the convolution. In this manner, it is possible to determine the spread of HIV up to the present time from knowledge of the AIDS incidence history and the incubation period. We describe the inversion of the convolution in terms of a Laplace transform technique that is applicable for any given incubation period distribution. Substantial simplifications in the technique are found in the case of an Erlang distribution for the probability density. The spread of HIV infections in the United States is charted through 1988 using AIDS incidence data that are corrected for both the revised AIDS case definition and reporting time delays. The results are consistent with current estimates of the HIV incidence in the United States and show no evidence of saturation in the rate of new infections. Indeed, the rate of new infections still appears to be climbing as of that date. While the technique is unable to predict the future course of the epidemic, it may provide a useful benchmark for comparison with mathematical models of the epidemic. The techniques are conceptually applicable to diseases other than AIDS.