Unobserved Cases Research Articles

Evolution of the BSE epidemic in Catalonia (1990–2015) based on a stochastic model

A stochastic model was used to estimate the number of bovine spongiform encephalopathy (BSE) cases that may have occurred in Catalonia (north-east Spain) from January 1990 to December 2015. The model allowed the evolution of the unobserved cases to be estimated differentiating the population into two subgroups: (1) livestock slaughtered for human consumption without clinical signs and (2) cattle at increased risk (including fallen stock, clinical suspects, emergency slaughtered cattle and cattle with clinical signs at ante mortem inspection). If the first bovine affected by BSE had been introduced into the region in 1986, the model predicted that during the period of study the surveillance system would detect a median of 197 outbreaks (2.5-97.5 percentiles: 157-245). The estimated median number of outbreaks that may have appeared before the establishment of the monitoring program was 80 (2.5-97.5 percentiles: 35-152), and the number of cases would have peaked in 2001.

Estimating the variance of cancer prevalence from population-based registries

Cancer prevalence is the proportion of people in a population diagnosed with cancer in the past and still alive. One way to estimate prevalence is via population-based registries, where data on diagnosis and life status of all incidence cases occurring in the covered population are collected. In this paper, a method to estimate the complete prevalence and its variance from population-based registries is presented. In order to obtain unbiased estimates of the complete prevalence, its calculation can be thought as made by three steps. Step 1 counts the incidence cases diagnosed during the period of registration and still alive. Step 2 estimates the expected number of survivors among cases lost to follow-up. Step 3 estimates the complete prevalence by taking into account cases diagnosed before the start of registration. The combination of steps 1+2 is defined as the counting method, to estimate the limited duration prevalence; step 3 is the completeness index method, to estimate the complete prevalence. For early established registries, steps 1+2 are more important than step 3, because observation time is long enough to include all past diagnosed cases still alive in the prevalence data. For more recently established registries, step 3 is by far the most critical because a large part of prevalence might have been diagnosed before the period of registration (Corazziari I, Mariotto A, Capocaccia R. Correcting the completeness bias of observed prevalence. Tumori 1999; 85: 370-81). The work by Clegg LX, Gail MH, Feuer EJ. Estimating the variance of disease-prevalence estimates from population-based registries. Biometrics 2002; 55: 1137-44. considers the problem of the variability of the estimated prevalence up to step 2. To our knowledge, no other work has considered the variability induced by correcting for the unobserved cases diagnosed before the period of registration, crucial to estimate the prevalence in recent registries. An analytic approach is considered to calculate the variance of step 3. A unified expression for the variance of the prevalence allowing for steps 1 through 3 is obtained. Some applications to cancer data are presented.

Wallinga and Teunis Respond to "Real-Time Tracking of Infection Control Measures"

We thank Drs. Lipsitch and Bergstrom (1) for their thoughtful commentary on our article (2), which appears in this issue of the Journal. They discuss the method we developed in the context of real-time analysis of case notifications. We would like to add a few more thoughts along these lines. The contribution of the method toward real-time analysis of incoming notifications is, as Lipsitch and Bergstrom aptly summarized, to provide “a means of transforming the time series of cases... into a time series of estimated values for the instantaneous reproductive number on each day” (1, p. 517). The transformation is from calendar time to generation time and from number to relative increase in number. The key variable involved in this transformation is the generation interval, which is the time interval between a primary case and a secondary case. Literature on this key variable has been scant and has been blurred by the use of confusing terminology. We cannot think of a better introduction to the concept of generation intervals than was provided by Professor Paul Fine in a recent article in this journal (3). Fine illustrates the relatedness of the generation interval, epidemic curves, transmission chains, and reproduction numbers. Our contribution is simply to specify these relations further. In our article (2), we emphasized that under certain technical conditions, only a few simple computational steps are needed to transform the time series of severe acute respiratory syndrome cases into a time series of estimated reproduction numbers. Here we would like to add that such a transformation could be carried out for any infectious disease that is transmitted from person to person. Lipsitch and Bergstrom (1) correctly note that a short generation interval facilitates real-time interpretation of incoming case reports; for infections with a long generation interval (such as human immunodeficiency virus, herpesvirus, or mycobacterial infection), the transformation might be less useful. At the moment, we are studying the possibility of extending the method to account for reporting delay, unobserved cases, and heterogeneous transmission. However, the finding that transformation is in any case feasible, and in many cases computationally trivial, opens up perspectives for estimating reproduction numbers from time series of case notifications. The reason for emphasizing the relevance of real-time estimation of reproduction numbers of emerging infectious diseases is that after syndromic surveillance systems have signaled the emergence of a new disease, we know little more about the novel infectious agent than that each case has been infected by another case and that successive generations of cases are separated by a typical generation interval. This is not enough information with which to build a mathematical transmission model, but it is sufficient for real-time estimation of reproduction numbers, and hence for real-time estimation of the additional control effort required. Lipsitch and Bergstrom assert that, when the next important outbreak arrives, such almost parameter-free estimation tools for reproduction numbers can be a valuable addition to syndromic surveillance systems and mathematical transmission models. We could not agree more.

A capture-recapture model to estimate prevalence of children born in Scotland with developmental eye defects.

Microphthalmia, anophthalmia and coloboma (MAC) are closely related structural developmental eye defects that have caused public concern in the UK following media reports of apparent clusters of cases. An extensive search was carried out to identify potential cases, and diagnoses were confirmed by ophthalmological review. Despite this effort, it was recognised that some cases would remain unidentified. Capture-recapture methods could improve the accuracy of the study's birth prevalence estimate. A register of all children born with the developmental eye defects of MAC in Scotland (1981-96) was compiled. This identified 188 cases from eight national ascertainment sources. The methods described by Hook and Regal were used to determine possible estimates for each source. A log-linear method of capture-recapture was used to fit a model to the number of children identified by the different combinations of the eight national sources, and the individual sources were included in the model as main effects. Maximum-likelihood estimates were calculated for tables of captured/not captured by one source versus any other source. The estimated number of unobserved cases for the most complete source, the register of congenital anomalies for Scotland, and the seven others combined is 78 (95% CI 53-115). Estimates calculated from two different log-linear models of the numbers of cases unobserved by all sources were 61 (95% CI 43-86) and 175 (95% CI 106-287). An external check with an independent congenital-anomalies register in one region, which is known to have high completeness, gave an estimate of 120. The study found a birth prevalence of MAC in Scotland of 1.9/10000, based on total live births during the study period. The capture-recapture analysis suggests that the true birth prevalence lies between 2.4/10 000 and 3.5/10000. Our experience shows that, even with multiple sources of ascertainment and considerable resources directed at case identification, a considerable level of under-ascertainment of congenital eye-anomalies still results, and suggests that the use of capture-recapture analysis would be a useful addition to congenital anomalies registers for needs assessment, disease-burden estimates and health-service planning.

Estimating the prevalence of multiple sclerosis in the United Kingdom by using capture-recapture methodology.

The geographic distribution of multiple sclerosis is nonrandom, as the disease is more prevalent in temperate than in tropical regions. Surveys conducted between 1970 and 1996 suggest that multiple sclerosis is more prevalent in the northern part of the United Kingdom than in the southern part. This north-south gradient ("the latitudinal gradient") might be a methodological artifact, because high prevalence figures from serial surveys of the northern part of the United Kingdom might have been the result of better ascertainment. By using capture-recapture methods, the authors found that case ascertainment was similar in the northern and southern parts of the United Kingdom. When prevalence figures for multiple sclerosis in the southern United Kingdom were increased to account for the number of unobserved cases, the difference persisted: The prevalence of multiple sclerosis in the northern part of the United Kingdom appeared to be at least 180 cases per 100,000 persons, whereas the maximum prevalence in the southern part of the United Kingdom was less than 160 cases per 100,000 persons. The distribution of multiple sclerosis in the United Kingdom is not uniform and is consistent with the hypothesis that populations with a high prevalence of multiple sclerosis may be genetically predisposed to the disease.

Structure of superdeformed bands in the A ≈ 150 mass region

The structure of superdeformed rotational bands recently discovered around 152Dy is discussed within the deformed shell model based on an average Woods-Saxon potential with a monopole pairing force. A comparison with available experimental data is provided and detailed predictions for yet unobserved cases are given. Pronounced variations in the observed rotational pattern are attributed to the angular momentum alignment of the high- N intruder (quasi)particles.

Optimal control-limit strategies for a partially observed replacement problem†

In this paper, we consider the problem of determining an optimal replacement strategy for a system subject to Markov deterioration, where replacement decisions can only be made on the basis of noise-corrupted observations of the system state. Three conditions are presented which guarantee that for the general case search for an optimal replacement strategy can be limited to control-limit strategies. Two of these conditions, both straightforward generalizations of assumptions due to Derman, are sufficient for the existence of control-limit strategies for the completely observed (Derman's result) and the completely unobserved (a new result) cases. The third condition is new and is concerned with the quality of data that the decision maker receives regarding the system state. Simplified assumptions which imply that this latter condition holds are also presented.