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

Unifying incidence and prevalence under a time-varying general branching process

Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.

Contemporary prevalence estimates of undiagnosed and diagnosed atrial fibrillation in the United States

BACKGROUND

Atrial fibrillation (AF) prevalence estimates vary and have been based on cohorts with clinically established or diagnosed disease. Undiagnosed AF prevalence estimates are less certain as they are based on nongeneralizable convenience samples.

HYPOTHESIS

Because AF is often asymptomatic, it my remain undiagnosed until the development of complications such as stroke or heart failure. Consequently, the observed prevalence of diagnosed AF from the literature may underestimate total disease burden. We therefore sought to estimate the total prevalence of both diagnosed and undiagnosed AF.

METHODS

We performed a retrospective cohort study from 2012 to 2017 using data from five US medical claims data sets. Undiagnosed AF prevalence was estimated based on the observed incidence of ischemic stroke, systemic embolism (SE), and AF incidence after a stroke/SE. The diagnosed AF cohort included AF patients between Q1 2014 and Q3 2015. The undiagnosed AF cohort were patients with assumed undiagnosed AF in the year before a stroke/SE and who were newly diagnosed with AF in the 3-month poststroke/SE. Stroke/SE incidence was calculated among all AF patients and the ratio of number of undiagnosed AF patients to stroke rate was created. Age- and sex-adjusted estimates were stratified by period of assumed undiagnosed AF before poststroke/SE AF diagnosis (1 or 2 years).

RESULTS

The estimated US prevalence of AF (diagnosed and undiagnosed) in Q3 2015 was 5 628 000 cases, of which 591 000 cases (11%) were undiagnosed. The assumed 2-year undiagnosed AF prevalence was 23% (1 531 000) of the total prevalent patients with AF (6 568 000). Undiagnosed (vs. diagnosed) AF patients were older and had higher CHA2DS2-VASc scores. Of undiagnosed AF, 93% had CHA2DS2-VASc ≥2 and met OAC criteria.

CONCLUSIONS

These contemporary estimates demonstrate the high prevalence of undiagnosed AF in the United States. Undiagnosed AF patients are composed of primarily elderly individuals who if diagnosed, would meet criteria for stroke prevention therapy.

Statistical Deconvolution for Inference of Infection Time Series

Accurate measurement of daily infection incidence is crucial to epidemic response. However, delays in symptom onset, testing, and reporting obscure the dynamics of transmission, necessitating methods to remove the effects of stochastic delays from observed data. Existing estimators can be sensitive to model misspecification and censored observations; many analysts have instead used methods that exhibit strong bias. We develop an estimator with a regularization scheme to cope with stochastic delays, which we term the robust incidence deconvolution estimator. We compare the method to existing estimators in a simulation study, measuring accuracy in a variety of experimental conditions. We then use the method to study COVID-19 records in the United States, highlighting its stability in the face of misspecification and right censoring. To implement the robust incidence deconvolution estimator, we release incidental, a ready-to-use R implementation of our estimator that can aid ongoing efforts to monitor the COVID-19 pandemic.

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

When responding to infectious disease outbreaks, rapid and accurate estimation of the epidemic trajectory is critical. However, two common data collection problems affect the reliability of the epidemiological data in real time: missing information on the time of first symptoms, and retrospective revision of historical information, including right censoring. Here, we propose an approach to construct epidemic curves in near real time that addresses these two challenges by 1) imputation of dates of symptom onset for reported cases using a dynamically-estimated "backward" reporting delay conditional distribution, and 2) adjustment for right censoring using the NobBS software package to nowcast cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number (Rt) in real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We evaluate how these real-time estimates compare with more complete epidemiological data that became available later. We explore the impact of the different assumptions on the estimates, and compare our estimates with those obtained from commonly used surveillance approaches. Our framework can help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health systems in other locations.

The second wave: estimating the hidden asymptomatic prevalence of COVID-19 in Ireland as we plan for imminent immunisation

Since the first case of COVID-19 in Ireland was recorded policy makers have introduced mitigation measures to control the spread of infection. Infection is spread by both known cases and hidden, undetected asymptomatic cases.  Asymptomatic individuals are people who transmit the virus but display no clinical symptoms. Current evidence reveals that this population is a major contributing factor to the spread of the disease. There is little or no knowledge of the scale of the hidden prevalence of all infections both asymptomatic and symptomatic in Ireland. Furthermore, as governments plan for the roll out of imminent immunisation programmes, the need to know the scale of the hidden prevalence and hence knowledge of the level of immunisation required is essential.

We describe and analyse the numbers of reported cases of COVID-19 in Ireland from the first case in February 2020 to mid-December 2020. Using the method of back-calculation we provide estimates of the asymptomatic prevalence of cases from June to December 2020.

The descriptive analysis highlighted two epidemic waves of known cases in the time period. Wave two from June to December included twice as many cases as wave one and cases were significantly younger. The back-calculation estimates of asymptomatic prevalence during this time period revealed that for every case known there was an additional unknown case and total prevalence in wave two was estimated to be approximately 95,000 as opposed to the reported 48,390 cases.

As prevalence in wave two is known to be spreading within and from younger age groups the role of mixing patterns on spread needs to be disseminated to the wider public to adequately inform them how personal modifications in behaviour can contribute to the control of the epidemic. While universally imposed lockdowns and mitigation measures may be essential, personal behavioural mixing choices are powerful protectors.

Missing link survival analysis with applications to available pandemic data

It is shown how to overcome a new missing data problem in survival analysis. Iterative nonparametric techniques are utilized and the missing data information is both estimated and used for further estimation in each iterative step. Theory is developed and a good finite sample performance is illustrated by simulations. The main motivation is an application to French data on the temporal development of the number of hospitalized Covid-19 patients.

Estimating a time-to-event distribution from right-truncated data in an epidemic: A review of methods

Time-to-event data are right-truncated if only individuals who have experienced the event by a certain time can be included in the sample. For example, we may be interested in estimating the distribution of time from onset of disease symptoms to death and only have data on individuals who have died. This may be the case, for example, at the beginning of an epidemic. Right truncation causes the distribution of times to event in the sample to be biased towards shorter times compared to the population distribution, and appropriate statistical methods should be used to account for this bias. This article is a review of such methods, particularly in the context of an infectious disease epidemic, like COVID-19. We consider methods for estimating the marginal time-to-event distribution, and compare their efficiencies. (Non-)identifiability of the distribution is an important issue with right-truncated data, particularly at the beginning of an epidemic, and this is discussed in detail. We also review methods for estimating the effects of covariates on the time to event. An illustration of the application of many of these methods is provided, using data on individuals who had died with coronavirus disease by 5 April 2020.

Forecasting Multi-Wave Epidemics Through Bayesian Inference

We present a simple, near-real-time Bayesian method to infer and forecast a multiwave outbreak, and demonstrate it on the COVID-19 pandemic. The approach uses timely epidemiological data that has been widely available for COVID-19. It provides short-term forecasts of the outbreak's evolution, which can then be used for medical resource planning. The method postulates one- and multiwave infection models, which are convolved with the incubation-period distribution to yield competing disease models. The disease models' parameters are estimated via Markov chain Monte Carlo sampling and information-theoretic criteria are used to select between them for use in forecasting. The method is demonstrated on two- and three-wave COVID-19 outbreaks in California, New Mexico and Florida, as observed during Summer-Winter 2020. We find that the method is robust to noise, provides useful forecasts (along with uncertainty bounds) and that it reliably detected when the initial single-wave COVID-19 outbreaks transformed into successive surges as containment efforts in these states failed by the end of Spring 2020.

Thoughts on "AIDS and COVID-19: A Tale of Two Pandemics and the Role of Statisticians" by Susan S. Ellenberg and Jeffrey S. Morris

Human immunodeficiency virus and Covid-19 (or SARS-CoV-2) differ in their incubation distributions and in their susceptibility to immunologic defense. These features affect our ability to predict the course of these epidemics and to control them.

The second wave: estimating the hidden asymptomatic prevalence of COVID-19 in Ireland as we plan for imminent immunisation

Since the first case of COVID-19 in Ireland was recorded policy makers have introduced mitigation measures to control the spread of infection. Infection is spread by both known cases and hidden, undetected asymptomatic cases.  Asymptomatic individuals are people who transmit the virus but display no clinical symptoms. Current evidence reveals that this population is a major contributing factor to the spread of the disease. There is little or no knowledge of the scale of the hidden prevalence of all infections both asymptomatic and symptomatic in Ireland. Furthermore, as governments plan for the roll out of imminent immunisation programmes, the need to know the scale of the hidden prevalence and hence knowledge of the level of immunisation required is essential. We describe and analyse the numbers of reported cases of COVID-19 in Ireland from the first case in February 2020 to mid-December 2020. Using the method of back-calculation we provide estimates of the asymptomatic prevalence of cases from June to December 2020. The descriptive analysis highlighted two epidemic waves of known cases in the time period. Wave two from June to December included twice as many cases as wave one and cases were significantly younger. The back-calculation estimates of asymptomatic prevalence during this time period revealed that for every case known there was an additional unknown case and total prevalence in wave two was estimated to be approximately 95,000 as opposed to the reported 48,390 cases. As prevalence in wave two is known to be spreading within and from younger age groups the role of mixing patterns on spread needs to be disseminated to the wider public to adequately inform them how personal modifications in behaviour can contribute to the control of the epidemic. While universally imposed lockdowns and mitigation measures may be essential, personal behavioural mixing choices are powerful protectors.

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

Designing public health responses to outbreaks requires close monitoring of population-level health indicators in real-time. Thus, an accurate estimation of the epidemic curve is critical. We propose an approach to reconstruct epidemic curves in near real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We address two data collection problems that affected the reliability of the available real-time epidemiological data, namely, the frequent missing information documenting when a patient first experienced symptoms, and the frequent retrospective revision of historical information (including right censoring). This is done by using a novel back-calculating procedure based on imputing patients' dates of symptom onset from reported cases, according to a dynamically-estimated "backward" reporting delay conditional distribution, and adjusting for right censoring using an existing package, NobBS , to estimate in real time (nowcast) cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number ( R t ) in real-time. At each step, we evaluate how different assumptions affect the recovered epidemiological events and compare the proposed approach to the alternative procedure of merely using curves of case counts, by report day, to characterize the time-evolution of the outbreak. Finally, we assess how these real-time estimates compare with subsequently documented epidemiological information that is considered more reliable and complete that became available later in time. Our approach may help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health surveillance systems in other locations.

Estimating the prevalence of problem drug use from drug-related mortality data

BACKGROUND AND AIMS

Indirect estimation methods are required for estimating the size of populations where only a proportion of individuals are observed directly, such as problem drug users (PDUs). Capture-recapture and multiplier methods are widely used, but have been criticized as subject to bias. We propose a new approach to estimating prevalence of PDU from numbers of fatal drug-related poisonings (fDRPs) using linked databases, addressing the key limitations of simplistic 'mortality multipliers'.

METHODS

Our approach requires linkage of data on a large cohort of known PDUs to mortality registers and summary information concerning additional fDRPs observed outside this cohort. We model fDRP rates among the cohort and assume that rates in unobserved PDUs are equal to rates in the cohort during periods out of treatment. Prevalence is estimated in a Bayesian statistical framework, in which we simultaneously fit regression models to fDRP rates and prevalence, allowing both to vary by demographic factors and the former also by treatment status.

RESULTS

We report a case study analysis, estimating the prevalence of opioid dependence in England in 2008/09, by gender, age group and geographical region. Overall prevalence was estimated as 0.82% (95% credible interval = 0.74-0.94%) of 15-64-year-olds, which is similar to a published estimate based on capture-recapture analysis.

CONCLUSIONS

Our modelling approach estimates prevalence from drug-related mortality data, while addressing the main limitations of simplistic multipliers. This offers an alternative approach for the common situation where available data sources do not meet the strong assumptions required for valid capture-recapture estimation. In a case study analysis, prevalence estimates based on our approach were surprisingly similar to existing capture-recapture estimates but, we argue, are based on a much more objective and justifiable modelling approach.

Estimation of infection density and epidemic size of COVID-19 using the back-calculation algorithm

The novel coronavirus (COVID-19) is continuing its spread across the world, claiming more than 160,000 lives and sickening more than 2,400,000 people as of April 21, 2020. Early research has reported a basic reproduction number (R0) between 2.2 to 3.6, implying that the majority of the population is at risk of infection if no intervention measures were undertaken. The true size of the COVID-19 epidemic remains unknown, as a significant proportion of infected individuals only exhibit mild symptoms or are even asymptomatic. A timely assessment of the evolving epidemic size is crucial for resource allocation and triage decisions. In this article, we modify the back-calculation algorithm to obtain a lower bound estimate of the number of COVID-19 infected persons in China in and outside the Hubei province. We estimate the infection density among infected and show that the drastic control measures enforced throughout China following the lockdown of Wuhan City effectively slowed down the spread of the disease in two weeks. We also investigate the COVID-19 epidemic size in South Korea and find a similar effect of its "test, trace, isolate, and treat" strategy. Our findings are expected to provide guidelines and enlightenment for surveillance and control activities of COVID-19 in other countries around the world.

Characterization of partially observed epidemics through Bayesian inference: application to COVID-19

We demonstrate a Bayesian method for the "real-time" characterization and forecasting of partially observed COVID-19 epidemic. Characterization is the estimation of infection spread parameters using daily counts of symptomatic patients. The method is designed to help guide medical resource allocation in the early epoch of the outbreak. The estimation problem is posed as one of Bayesian inference and solved using a Markov chain Monte Carlo technique. The data used in this study was sourced before the arrival of the second wave of infection in July 2020. The proposed modeling approach, when applied at the country level, generally provides accurate forecasts at the regional, state and country level. The epidemiological model detected the flattening of the curve in California, after public health measures were instituted. The method also detected different disease dynamics when applied to specific regions of New Mexico.

Modeling methods for estimating HIV incidence: a mathematical review

Estimating HIV incidence is crucial for monitoring the epidemiology of this infection, planning screening and intervention campaigns, and evaluating the effectiveness of control measures. However, owing to the long and variable period from HIV infection to the development of AIDS and the introduction of highly active antiretroviral therapy, accurate incidence estimation remains a major challenge. Numerous estimation methods have been proposed in epidemiological modeling studies, and here we review commonly-used methods for estimation of HIV incidence. We review the essential data required for estimation along with the advantages and disadvantages, mathematical structures and likelihood derivations of these methods. The methods include the classical back-calculation method, the method based on CD4+ T-cell depletion, the use of HIV case reporting data, the use of cohort study data, the use of serial or cross-sectional prevalence data, and biomarker approach. By outlining the mechanistic features of each method, we provide guidance for planning incidence estimation efforts, which may depend on national or regional factors as well as the availability of epidemiological or laboratory datasets.

Extending Bayesian back-calculation to estimate age and time specific HIV incidence

CD4-based multi-state back-calculation methods are key for monitoring the HIV epidemic, providing estimates of HIV incidence and diagnosis rates by disentangling their inter-related contribution to the observed surveillance data. This paper, extends existing approaches to age-specific settings, permitting the joint estimation of age- and time-specific incidence and diagnosis rates and the derivation of other epidemiological quantities of interest. This allows the identification of specific age-groups at higher risk of infection, which is crucial in directing public health interventions. We investigate, through simulation studies, the suitability of various bivariate splines for the non-parametric modelling of the latent age- and time-specific incidence and illustrate our method on routinely collected data from the HIV epidemic among gay and bisexual men in England and Wales.

A METHOD FOR ESTIMATING THE PROPORTION OF HIV INFECTED PERSONS THAT HAVE BEEN DIAGNOSED AND APPLICATION TO CHINA

Estimation of the proportion of living HIV infected persons that have been diagnosed is critical for tracking progress toward meeting the UNAIDS goal that all persons who need HIV treatment receive it. The objective of this article is to develop a method for estimating that proportion. The methodological problem is that persons with undiagnosed HIV infection are not directly observable and are a "hidden" population. Here we propose a methodology for estimating the proportion diagnosed that is relatively simple to implement. The key idea is that in many settings certain health conditions such as pregnancy or an upcoming surgery lead to mandatory HIV tests. The size of the undiagnosed infected population can be estimated from the numbers of infected persons diagnosed by mandatory tests and an estimate of the rate that persons in the undiagnosed infected population receive mandatory tests. We discuss approaches for estimating the rate of mandatory testing in the undiagnosed population, such as surgical or pregnancy rates. We develop estimators of the proportion diagnosed and confidence interval procedures. Sample size considerations and sensitivity analyses to underlying assumptions are considered. The proposed methods can be performed at a local level and within demographic strata. Implementation of the method is simple and requires neither historical HIV/AIDS surveillance data nor biomarkers such as CD4 cell counts. The methods are applied to data from Dehong Prefecture in Yunnan Province, China.

Estimating the number of people living with HIV and the undiagnosed fraction in Spain in 2013

OBJECTIVE

We aim to estimate the number of people living with HIV and the undiagnosed fraction in Spain, where coverage of the HIV surveillance system has only recently become complete.

METHODS

The reconstruction of all HIV diagnoses and infections was obtained by combining HIV and AIDS surveillance data. The imputation of the diagnoses and back-calculation of the infection incidence are integrated in a Bayesian framework to take into account the uncertainty associated with unavailable data.

RESULTS

An estimated 141 000 [95% credible interval (CI) 128 000-155 000] persons were living with HIV by the end of 2013, in Spain and 18% (95% CI 14.3-22.1%) were unaware of it. A similar fraction of undiagnosed infections was obtained in men who have sex with men and heterosexuals (18.8 and 20.1%, respectively), but for injection drug users, this fraction was 3.5%.

CONCLUSION

This study provides the first estimates of the number of people living with HIV and the undiagnosed fraction in Spain, using routine surveillance data. The proposed method could be useful for countries where the geographical coverage of the HIV surveillance system is partial or was completed only recently.

Estimated prevalence of undiagnosed atrial fibrillation in the United States

Introduction

As atrial fibrillation (AF) is often asymptomatic, it may remain undiagnosed until or even after development of complications, such as stroke. Consequently the observed prevalence of AF may underestimate total disease burden.

Methods

To estimate the prevalence of undiagnosed AF in the United States, we performed a retrospective cohort modeling study in working age (18–64) and elderly (≥65) people using commercial and Medicare administrative claims databases. We identified patients in years 2004–2010 with incident AF following an ischemic stroke. Using a back-calculation methodology, we estimated the prevalence of undiagnosed AF as the ratio of the number of post-stroke AF patients and the CHADS₂-specific stroke probability for each patient, adjusting for age and gender composition based on United States census data.

Results

The estimated prevalence of AF (diagnosed and undiagnosed) was 3,873,900 (95%CI: 3,675,200–4,702,600) elderly and 1,457,100 (95%CI: 1,218,500–1,695,800) working age adults, representing 10.0% and 0.92% of the respective populations. Of these, 698,900 were undiagnosed: 535,400 (95%CI: 331,900–804,400) elderly and 163,500 (95%CI: 17,700–400,000) working age adults, representing 1.3% and 0.09% of the respective populations. Among all undiagnosed cases, 77% had a CHADS₂ score ≥1, and 56% had CHADS₂ score ≥2.

Conclusions

Using a back-calculation approach, we estimate that the total AF prevalence in 2009 was 5.3 million of which 0.7 million (13.1% of AF cases) were undiagnosed. Over half of the modeled population with undiagnosed AF was at moderate to high risk of stroke.

Population metrics for suicide events: A causal inference approach

Large-scale public health prevention initiatives and interventions are a very important component to current public health strategies. But evaluating effects of such large-scale prevention/intervention faces a lot of challenges due to confounding effects and heterogeneity of study population. In this paper, we will develop metrics to assess the risk for suicide events based on causal inference framework when the study population is heterogeneous. The proposed metrics deal with the confounding effect by first estimating the risk of suicide events within each of the risk levels, number of prior attempts, and then taking a weighted sum of the conditional probabilities. The metrics provide unbiased estimates of the risk of suicide events. Simulation studies and a real data example will be used to demonstrate the proposed metrics.

Data Mining in HIV-AIDS Surveillance System : Application to Portuguese Data

The Human Immunodeficiency Virus (HIV) is an infectious agent that attacks the immune system cells. Without a strong immune system, the body becomes very susceptible to serious life threatening opportunistic diseases. In spite of the great progresses on medication and prevention over the last years, HIV infection continues to be a major global public health issue, having claimed more than 36 million lives over the last 35 years since the recognition of the disease. Monitoring, through registries, of HIV-AIDS cases is vital to assess general health care needs and to support long-term health-policy control planning. Surveillance systems are therefore established in almost all developed countries. Typically, this is a complex system depending on several stakeholders, such as health care providers, the general population and laboratories, which challenges an efficient and effective reporting of diagnosed cases. One issue that often arises is the administrative delay in reports of diagnosed cases. This paper aims to identify the main factors influencing reporting delays of HIV-AIDS cases within the portuguese surveillance system. The used methodologies included multilayer artificial neural networks (MLP), naive bayesian classifiers (NB), support vector machines (SVM) and the k-nearest neighbor algorithm (KNN). The highest classification accuracy, precision and recall were obtained for MLP and the results suggested homogeneous administrative and clinical practices within the reporting process. Guidelines for reductions of the delays should therefore be developed nationwise and transversally to all stakeholders.

A review of back-calculation techniques and their potential to inform mitigation strategies with application to non-transmissible acute infectious diseases

Back-calculation is a process whereby generally unobservable features of an event leading to a disease outbreak can be inferred either in real-time or shortly after the end of the outbreak. These features might include the time when persons were exposed and the source of the outbreak. Such inferences are important as they can help to guide the targeting of mitigation strategies and to evaluate the potential effectiveness of such strategies. This article reviews the process of back-calculation with a particular emphasis on more recent applications concerning deliberate and naturally occurring aerosolized releases. The techniques can be broadly split into two themes: the simpler temporal models and the more sophisticated spatio-temporal models. The former require input data in the form of cases' symptom onset times, whereas the latter require additional spatial information such as the cases' home and work locations. A key aspect in the back-calculation process is the incubation period distribution, which forms the initial topic for consideration. Links between atmospheric dispersion modelling, within-host dynamics and back-calculation are outlined in detail. An example of how back-calculation can inform mitigation strategies completes the review by providing improved estimates of the duration of antibiotic prophylaxis that would be required in the response to an inhalational anthrax outbreak.

A Bayesian back-calculation method to estimate the risk of bovine spongiform encephalopathy (BSE) in Canada during the period 1996-2011

Seventeen typical cases of bovine spongiform encephalopathy (BSE) were detected in Canada the period of 2003-2011. The clinical incidence of BSE was censored by early slaughter, death, or exportation of infected cattle due to the long incubation period of BSE disease. The aim of this study was to estimate the infection incidence of BSE in birth cohorts during 1996-2004 and project infection frequency through to 2007. An estimate of the number of asymptomatic infected cattle slaughtered for human consumption is also provided. The number of incident, asymptomatic cases was assumed to follow a Poisson process. A Bayesian back-calculation approach was used to project the risk of contracting BSE in those birth cohorts. Model parameters and inputs were taken from scientific literature and governmental data sources. The projected number of infected cattle in birth cohorts spanning the period 1996-2007 was 492, with median 95% credible interval 258-830. If the requirement to remove specified risk material (SRM) from cattle prior to entering the food chain was not in place, the predicted number of slaughtered infected in the human food chain from 1996-2010 was 298, with a 95% credible interval 156-500. The magnitude of the BSE epidemic in Canada for 1996-2007 birth cohorts was estimated to be approximately 28-fold higher than the number of clinical cases detected through to October 2011. Although some of those cattle were slaughtered for human consumption, the requirement of SRM removal may have prevented most of the infectious material from entering the food chain.

Back-calculating the incidence of infection of leprosy in a Bayesian framework

BACKGROUND

The number of new leprosy cases reported annually is falling worldwide, but remains relatively high in some populations. Because of the long and variable periods between infection, onset of disease, and diagnosis, the recently detected cases are a reflection of infection many years earlier. Estimation of the numbers of sub-clinical and clinical infections would be useful for management of elimination programmes. Back-calculation is a methodology that could provide estimates of prevalence of undiagnosed infections, future diagnoses and the effectiveness of control.

METHODS

A basic back-calculation model to investigate the infection dynamics of leprosy has been developed using Markov Chain Monte Carlo in a Bayesian context. The incidence of infection and the detection delay both vary with calendar time. Public data from Thailand are used to demonstrate the results that are obtained as the incidence of diagnosed cases falls.

RESULTS

The results show that the underlying burden of infection and short-term future predictions of cases can be estimated with a simple model. The downward trend in new leprosy cases in Thailand is expected to continue. In 2015 the predicted total number of undiagnosed sub-clinical and clinical infections is 1,168 (846-1,546) of which 466 (381-563) are expected to be clinical infections.

CONCLUSIONS

Bayesian back-calculation has great potential to provide estimates of numbers of individuals in health/infection states that are as yet unobserved. Predictions of future cases provides a quantitative measure of understanding for programme managers and evaluators. We will continue to develop the approach, and suggest that it might be useful for other NTD in which incidence of diagnosis is not an immediate measure of infection.

Inferring HIV incidence from case surveillance with CD4+ cell counts

BACKGROUND

In some countries, HIV surveillance is based on case-reporting of newly diagnosed infections. We present a new back-projection method for estimating HIV-incidence trends using individuals' CD4 cell counts at diagnosis.

METHODS

On the basis of a review of CD4 cell count distributions among HIV-uninfected people, CD4 cell count following primary infection, and rates of CD4 cell count decline over time among people with HIV, we simulate the expected distribution in time between infection and diagnosis. Applying this to all diagnosed individuals provides a distribution of likely infection times and estimates for population incidence, level of undiagnosed HIV, and the average time from infection to diagnosis each year. We applied this method to the national HIV case surveillance data of Australia for 1983-2013.

RESULTS

The estimated number of new HIV infections in Australia in 2013 was 912 (95% uncertainty bound 835-1002). We estimate that 2280 (95% uncertainty bound 1900-2830) people were living with undiagnosed HIV at the end of 2013, corresponding to approximately 9.4% (95% uncertainty bound 7.8-10.1%) of all people living with HIV. With increases in the average CD4 count at diagnosis, the inferred HIV testing rate has been increasing over time and the estimated mean and median times between infection and diagnosis have decreased substantially. However, the estimated mean time between infection and diagnosis is considerably greater than the median, indicating that some people remain undiagnosed for long periods. Differences were found between cases attributable to male homosexual exposure versus other cases.

CONCLUSION

This methodology provides a novel way of estimating population incidence by combining diagnosis dates and CD4 cell counts at diagnosis.

A New Method for Estimating the Number of Undiagnosed HIV Infected Based on HIV Testing History, with an Application to Men Who Have Sex with Men in Seattle/King County, WA

We develop a new approach for estimating the undiagnosed fraction of HIV cases, the first step in the HIV Care Cascade. The goal is to address a critical blindspot in HIV prevention and treatment planning, with an approach that simplifies data requirements and can be implemented with open-source software. The primary data required is HIV testing history information on newly diagnosed cases. Two methods are presented and compared. The first is a general methodology based on simplified back-calculation that can be used to assess changes in the undiagnosed fraction over time. The second makes an assumption of constant incidence, allowing the estimate to be expressed as a simple closed formula calculation. We demonstrate the methods with an application to HIV diagnoses among men who have sex with men (MSM) from Seattle/King County. The estimates suggest that 6% of HIV-infected MSM in King County are undiagnosed, about one-third of the comparable national estimate. A sensitivity analysis on the key distributional assumption gives an upper bound of 11%. The undiagnosed fraction varies by race/ethnicity, with estimates of 4.9% among white, 8.6% of African American, and 9.3% of Hispanic HIV-infected MSM being undiagnosed.

Use of surveillance data on HIV diagnoses with HIV-related symptoms to estimate the number of people living with undiagnosed HIV in need of antiretroviral therapy

Background

It is important to have methods available to estimate the number of people who have undiagnosed HIV and are in need of antiretroviral therapy (ART).

Methods

The method uses the concept that a predictable level of occurrence of AIDS or other HIV-related clinical symptoms which lead to presentation for care, and hence diagnosis of HIV, arises in undiagnosed people with a given CD4 count. The method requires surveillance data on numbers of new HIV diagnoses with HIV-related symptoms, and the CD4 count at diagnosis. The CD4 count-specific rate at which HIV-related symptoms develop are estimated from cohort data. 95% confidence intervals can be constructed using a simple simulation method.

Results

For example, if there were 13 HIV diagnoses with HIV-related symptoms made in one year with CD4 count at diagnosis between 150–199 cells/mm³, then since the CD4 count-specific rate of HIV-related symptoms is estimated as 0.216 per person-year, the estimated number of person years lived in people with undiagnosed HIV with CD4 count 150–199 cells/mm3 is 13/0.216 = 60 (95% confidence interval: 29–100), which is considered an estimate of the number of people living with undiagnosed HIV in this CD4 count stratum.

Conclusions

The method is straightforward to implement within a short period once a surveillance system of all new HIV diagnoses, collecting data on HIV-related symptoms at diagnosis, is in place and is most suitable for estimating the number of undiagnosed people with CD4 count <200 cells/mm3 due to the low rate of developing HIV-related symptoms at higher CD4 counts. A potential source of bias is under-diagnosis and under-reporting of diagnoses with HIV-related symptoms. Although this method has limitations as with all approaches, it is important for prompting increased efforts to identify undiagnosed people, particularly those with low CD4 count, and for informing levels of unmet need for ART.

The impact of the activities of Ghana AIDS Commission on new HIV infections in Ghana: An intervention time series analysis

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Intervention time series was used to model the impact of the activities of GAC.

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Northern sector HIV/AIDS follows integrated moving average model.

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Southern sector HIV/AIDS follows autoregressive integrated moving average model.

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No significant impact of GAC activities on HIV/AIDS observed in the northern sector.

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Significant impact of GAC activities on HIV/AIDS observed in the southern sector.

The study examines the impact of Ghana AIDS Commission (GAC) activities on new HIV infections. Secondary data from 1 January 1996 to 31 December 2007 were obtained from the Ghana National AIDS Control Programme and Biostatistics Department of the Health Ministry. Intervention time series analysis was applied separately to the data from the northern and southern sectors due to data collection mechanism and location of two tertiary teaching hospitals. The impact of GAC activities is measured by the statistical significance of the coefficients of the intervention variable. The intervention variable is coded as zero (0) for the period before and one (1), the period after the GAC activities commenced. It was shown that the HIV incidence in the northern follows an integrated moving average model, whilst in the southern sectors an autoregressive integrated moving average model was the appropriate model. No significant impact of GAC activities was observed in the northern sector. In contrast the GAC activities in the southern sector were associated with a reduction in new HIV infections (male = −0.20 ± 0.13, p < 0.05; female = 0.14 ± 0.10, p < 0.05), corresponding to a 15% and 18% reduction in male and female new HIV infections respectively in the sector. It can therefore be concluded that the GAC activities were a success. However, greater focus of GAC efforts in the northern sector is required to ensure that its activities have an impact on the incidence there.

Generalization of Carey's equality and a theorem on stationary population

Carey's Equality pertaining to stationary models is well known. In this paper, we have stated and proved a fundamental theorem related to the formation of this Equality. This theorem will provide an in-depth understanding of the role of each captive subject, and their corresponding follow-up duration in a stationary population. We have demonstrated a numerical example of a captive cohort and the survival pattern of medfly populations. These results can be adopted to understand age-structure and aging process in stationary and non-stationary population models.

A dose and time response Markov model for the in-host dynamics of infection with intracellular bacteria following inhalation: with application to Francisella tularensis

In a novel approach, the standard birth–death process is extended to incorporate a fundamental mechanism undergone by intracellular bacteria, phagocytosis. The model accounts for stochastic interaction between bacteria and cells of the immune system and heterogeneity in susceptibility to infection of individual hosts within a population. Model output is the dose–response relation and the dose-dependent distribution of time until response, where response is the onset of symptoms. The model is thereafter parametrized with respect to the highly virulent Schu S4 strain of Francisella tularensis, in the first such study to consider a biologically plausible mathematical model for early human infection with this bacterium. Results indicate a median infectious dose of about 23 organisms, which is higher than previously thought, and an average incubation period of between 3 and 7 days depending on dose. The distribution of incubation periods is right-skewed up to about 100 organisms and symmetric for larger doses. Moreover, there are some interesting parallels to the hypotheses of some of the classical dose–response models, such as independent action (single-hit model) and individual effective dose (probit model). The findings of this study support experimental evidence and postulations from other investigations that response is, in fact, influenced by both in-host and between-host variability.

Dynamics of infectious diseases

Modern infectious disease epidemiology has a strong history of using mathematics both for prediction and to gain a deeper understanding. However the study of infectious diseases is a highly interdisciplinary subject requiring insights from multiple disciplines, in particular a biological knowledge of the pathogen, a statistical description of the available data and a mathematical framework for prediction. Here we begin with the basic building blocks of infectious disease epidemiology--the SIS and SIR type models--before considering the progress that has been made over the recent decades and the challenges that lie ahead. Throughout we focus on the understanding that can be developed from relatively simple models, although accurate prediction will inevitably require far greater complexity beyond the scope of this review. In particular, we focus on three critical aspects of infectious disease models that we feel fundamentally shape their dynamics: heterogeneously structured populations, stochasticity and spatial structure. Throughout we relate the mathematical models and their results to a variety of real-world problems.

Associations of age and sex with the clinical outcome and incubation period of Shiga toxin-producing Escherichia coli O104:H4 infections, 2011

We pooled data on adults who reported diarrhea or developed life-threatening hemolytic uremic syndrome (HUS) in any of 6 closed cohorts from 4 countries (1 cohort each in Denmark, France, and Sweden and 3 in Germany) that were investigated during a large outbreak of Shiga toxin-producing Escherichia coli (STEC) O104:H4 infection in 2011. Logistic regression and Weibull regression for interval censored data were used to assess the relation of age and sex with clinical outcome and with incubation period. Information on the latter was used in a nonparametric back-projection context to estimate when adult cases reported in Germany were exposed to STEC O104:H4. Overall, data from 119 persons (median age, 49 years; 80 women) were analyzed. Bloody diarrhea and HUS were recorded as the most severe outcome for 44 and 26 individuals, respectively. Older age was significantly associated with bloody diarrhea but not with HUS. Woman had nonsignificantly higher odds for bloody diarrhea (odds ratio = 1.81) and developing HUS (odds ratio = 1.83) than did men. Older participants had a statistically significantly reduced incubation period. The shortest interval that included 75% of exposures in adults spanned only 12 days and preceded outbreak detection. In conclusion, the frequency of bloody diarrhea but not of HUS and the length of the incubation period depended on the age of individuals infected with STEC O104:H4. A large number of people were exposed to STEC O104:H4 for a short period of time.

Estimating incidence curves of several infections using symptom surveillance data

We introduce a method for estimating incidence curves of several co-circulating infectious pathogens, where each infection has its own probabilities of particular symptom profiles. Our deconvolution method utilizes weekly surveillance data on symptoms from a defined population as well as additional data on symptoms from a sample of virologically confirmed infectious episodes. We illustrate this method by numerical simulations and by using data from a survey conducted on the University of Michigan campus. Last, we describe the data needs to make such estimates accurate.

Estimating the distribution of the window period for recent HIV infections: a comparison of statistical methods

In the past few years a number of antibody biomarkers have been developed to distinguish between recent and established Human Immunodeficiency Virus (HIV) infection. Typically, a specific threshold/cut-off of the biomarker is chosen, values below which are indicative of recent infections. Such biomarkers have attracted considerable interest as the basis for incidence estimation using a cross-sectional sample. An estimate of HIV incidence can be obtained from the prevalence of recent infection, as measured in the sample, and knowledge of the time spent in the recent infection state, known as the window period. However, such calculations are based on a number of assumptions concerning the distribution of the window period. We compare two statistical methods for estimating the mean and distribution of a window period using data on repeated measurements of an antibody biomarker from a cohort of HIV seroconverters. The methods account for the interval-censored nature of both the date of seroconversion and the date of crossing a specific threshold. We illustrate the methods using repeated measurements of the Avidity Index (AI) and make recommendations about the choice of threshold for this biomarker so that the resulting window period satisfies the assumptions for incidence estimation.

HIV in hiding: methods and data requirements for the estimation of the number of people living with undiagnosed HIV

Many people who are HIV positive are unaware of their infection status. Estimation of the number of people with undiagnosed HIV within a country or region is vital for understanding future need for treatment and for motivating testing programs. We review the available estimation approaches which are in current use. They can be broadly classified into those based on prevalence surveys and those based on reported HIV and AIDS cases. Estimation based on prevalence data requires data from regular prevalence surveys in different population groups together with estimates of the size of these groups. The recommended minimal case reporting data needed to estimate the number of patients with undiagnosed HIV are HIV diagnoses, including CD4 count at diagnosis and whether there has been an AIDS diagnosis in the 3 months before or after HIV diagnosis, and data on deaths in people with HIV. We would encourage all countries to implement several methods that will help develop our understanding of strengths and weaknesses of the various methods.

Trend analysis and short-term forecast of incident HIV infection in Ghana

The study uses time-series modelling to determine and predict trends in incident HIV infection in Ghana among specific age groups. The HIV data for Ghana were grouped according to northern and southern spatial sectors as they exhibited slightly different data collection formats. The trend of the epidemic is modelled using moving-average smoothing techniques, and the Box-Jenkins ARIMA model is used to forecast cases of newly acquired (incident) HIV infection. Trend analysis of past growth patterns reveals an increase in new cases of HIV infection in the northern sector, with the greatest increase occurring among persons aged 30 years and over. The epidemic in the southern sector appears to have levelled off. However, incident HIV infection in the 20-39-year-old age group of females in the sector is estimated to increase in the next three years. Moreover, the estimates suggest a higher increase in incident cases than that predicted by the National AIDS Control Programme. Nevertheless, incident HIV infection among persons aged 19 and below is found to be relatively stable. Thus, if efforts are made to reduce or prevent an increase in the number of new infections in the northern sector, and for the 20-39 years age group in the southern sector, Ghana will have a brighter future with regard to its response to the HIV epidemic. These findings can assist with developing strategic-intervention policy planning for Ghana and other countries in sub-Saharan Africa.

Multi-scale modelling in computational biomedicine

The inherent complexity of biomedical systems is well recognized; they are multi-scale, multi-science systems, bridging a wide range of temporal and spatial scales. This article reviews the currently emerging field of multi-scale modelling in computational biomedicine. Many exciting multi-scale models exist or are under development. However, an underpinning multi-scale modelling methodology seems to be missing. We propose a direction that complements the classic dynamical systems approach and introduce two distinct case studies, transmission of resistance in human immunodeficiency virus spreading and in-stent restenosis in coronary artery disease.