Estimating Nuisance Parameters in Inverse Problems

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.Comment: 16 pages, 5 figure

A parametric level-set method for partially discrete tomography

This paper introduces a parametric level-set method for tomographic reconstruction of partially discrete images. Such images consist of a continuously varying background and an anomaly with a constant (known) grey-value. We represent the geometry of the anomaly using a level-set function, which we represent using radial basis functions. We pose the reconstruction problem as a bi-level optimization problem in terms of the background and coefficients for the level-set function. To constrain the background reconstruction we impose smoothness through Tikhonov regularization. The bi-level optimization problem is solved in an alternating fashion; in each iteration we first reconstruct the background and consequently update the level-set function. We test our method on numerical phantoms and show that we can successfully reconstruct the geometry of the anomaly, even from limited data. On these phantoms, our method outperforms Total Variation reconstruction, DART and P-DART.Comment: Paper submitted to 20th International Conference on Discrete Geometry for Computer Imager

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Variation in the COVID-19 infection-fatality ratio by age, time, and geography during the pre-vaccine era: a systematic analysis

Background The infection-fatality ratio (IFR) is a metric that quantifies the likelihood of an individual dying once infected with a pathogen. Understanding the determinants of IFR variation for COVID-19, the disease caused by the SARS-CoV-2 virus, has direct implications for mitigation efforts with respect to clinical practice, non-pharmaceutical interventions, and the prioritisation of risk groups for targeted vaccine delivery. The IFR is also a crucial parameter in COVID-19 dynamic transmission models, providing a way to convert a population's mortality rate into an estimate of infections.Methods We estimated age-specific and all-age IFR by matching seroprevalence surveys to total COVID-19 mortality rates in a population. The term total COVID-19 mortality refers to an estimate of the total number of deaths directly attributable to COVID-19. After applying exclusion criteria to 5131 seroprevalence surveys, the IFR analyses were informed by 2073 all-age surveys and 718 age-specific surveys (3012 age-specific observations). When seroprevalence was reported by age group, we split total COVID-19 mortality into corresponding age groups using a Bayesian hierarchical model to characterise the non-linear age pattern of reported deaths for a given location. To remove the impact of vaccines on the estimated IFR age pattern, we excluded age-specific observations of seroprevalence and deaths that occurred after vaccines were introduced in a location. We estimated age-specific IFR with a non-linear meta-regression and used the resulting age pattern to standardise all-age IFR observations to the global age distribution. All IFR observations were adjusted for baseline and waning antibody-test sensitivity. We then modelled age-standardised IFR as a function of time, geography, and an ensemble of 100 of the top-performing covariate sets. The covariates included seven clinical predictors (eg, age-standardised obesity prevalence) and two measures of health system performance. Final estimates for 190 countries and territories, as well as subnational locations in 11 countries and territories, were obtained by predicting age-standardised IFR conditional on covariates and reversing the age standardisation.Findings We report IFR estimates for April 15, 2020, to January 1, 2021, the period before the introduction of vaccines and widespread evolution of variants. We found substantial heterogeneity in the IFR by age, location, and time. Age-specific IFR estimates form a J shape, with the lowest IFR occurring at age 7 years (0-0023%, 95% uncertainty interval [UI] 0-0015-0-0039) and increasing exponentially through ages 30 years (0-0573%, 0-0418-0-0870), 60 years (1-0035%, 0-7002-1-5727), and 90 years (20-3292%, 14-6888-28-9754). The countries with the highest IFR on July 15, 2020, were Portugal (2-085%, 0-946-4-395), Monaco (1-778%, 1-265-2-915), Japan (1-750%, 1-302-2-690), Spain (1-710%, 0-991-2-718), and Greece (1-637%, 1-155-2-678). All-age IFR varied by a factor of more than 30 among 190 countries and territories.After age standardisation, the countries with the highest IFR on July 15, 2020, were Peru (0-911%, 0-636-1-538), Portugal (0-850%, 0-386-1-793), Oman (0-762%, 0-381-1-399), Spain (0-751%, 0-435-1-193), and Mexico (0-717%, 0-426-1-404). Subnational locations with high IFRs also included hotspots in the UK and southern and eastern states of the USA. Sub-Saharan African countries and Asian countries generally had the lowest all-age and age-standardised IFRs. Population age structure accounted for 74% of logit-scale variation in IFRs estimated for 39 in-sample countries on July 15, 2020. A post-hoc analysis showed that high rates of transmission in the care home population might account for higher IFRs in some locations. Among all countries and territories, we found that the median IFR decreased from 0-466% (interquartile range 0-223-0-840) to 0-314% (0-143-0-551) between April 15, 2020, and Jan 1, 2021.Interpretation Estimating the IFR for global populations helps to identify relative vulnerabilities to COVID-19. Information about how IFR varies by age, time, and location informs clinical practice and non-pharmaceutical interventions like physical distancing measures, and underpins vaccine risk stratification. IFR and mortality risk form a J shape with respect to age, which previous research, such as that by Glynn and Moss in 2020, has identified to be a common pattern among infectious diseases. Understanding the experience of a population with COVID-19 mortality requires consideration for local factors; IFRs varied by a factor of more than 30 among 190 countries and territories in this analysis. In particular, the presence of elevated age-standardised IFRs in countries with well resourced health-care systems indicates that factors beyond health-care capacity are important. Potential extenuating circumstances include outbreaks among care home residents, variable burdens of severe cases, and the population prevalence of comorbid conditions that increase the severity of COVID-19 disease. During the pre-vaccine period, the estimated 33% decrease in median IFR over 8 months suggests that treatment for COVID-19 has improved over time. Estimating IFR for the pre-vaccine era provides an important baseline for describing the progression of COVID-19 mortality patterns.Funding Bill & Melinda Gates Foundation, J Stanton, T Gillespie, and J and E Nordstrom Copyright (c) 2022 The Author(s). Published by Elsevier Ltd. This is an Open Access article under the CC BY-NC-ND 4.0 license

Global, regional, and national burden of meningitis and its aetiologies, 1990–2019: a systematic analysis for the Global Burden of Disease Study 2019

Background: Although meningitis is largely preventable, it still causes hundreds of thousands of deaths globally each year. WHO set ambitious goals to reduce meningitis cases by 2030, and assessing trends in the global meningitis burden can help track progress and identify gaps in achieving these goals. Using data from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019, we aimed to assess incident cases and deaths due to acute infectious meningitis by aetiology and age from 1990 to 2019, for 204 countries and territories. Methods: We modelled meningitis mortality using vital registration, verbal autopsy, sample-based vital registration, and mortality surveillance data. Meningitis morbidity was modelled with a Bayesian compartmental model, using data from the published literature identified by a systematic review, as well as surveillance data, inpatient hospital admissions, health insurance claims, and cause-specific meningitis mortality estimates. For aetiology estimation, data from multiple causes of death, vital registration, hospital discharge, microbial laboratory, and literature studies were analysed by use of a network analysis model to estimate the proportion of meningitis deaths and cases attributable to the following aetiologies: Neisseria meningitidis, Streptococcus pneumoniae, Haemophilus influenzae, group B Streptococcus, Escherichia coli, Klebsiella pneumoniae, Listeria monocytogenes, Staphylococcus aureus, viruses, and a residual other pathogen category. Findings: In 2019, there were an estimated 236 000 deaths (95% uncertainty interval [UI] 204 000–277 000) and 2·51 million (2·11–2·99) incident cases due to meningitis globally. The burden was greatest in children younger than 5 years, with 112 000 deaths (87 400–145 000) and 1·28 million incident cases (0·947–1·71) in 2019. Age-standardised mortality rates decreased from 7·5 (6·6–8·4) per 100 000 population in 1990 to 3·3 (2·8–3·9) per 100 000 population in 2019. The highest proportion of total all-age meningitis deaths in 2019 was attributable to S pneumoniae (18·1% [17·1–19·2]), followed by N meningitidis (13·6% [12·7–14·4]) and K pneumoniae (12·2% [10·2–14·3]). Between 1990 and 2019, H influenzae showed the largest reduction in the number of deaths among children younger than 5 years (76·5% [69·5–81·8]), followed by N meningitidis (72·3% [64·4–78·5]) and viruses (58·2% [47·1–67·3]). Interpretation: Substantial progress has been made in reducing meningitis mortality over the past three decades. However, more meningitis-related deaths might be prevented by quickly scaling up immunisation and expanding access to health services. Further reduction in the global meningitis burden should be possible through low-cost multivalent vaccines, increased access to accurate and rapid diagnostic assays, enhanced surveillance, and early treatment. Funding: Bill & Melinda Gates Foundation

The burden of antimicrobial resistance in the Americas in 2019: a cross-country systematic analysis

Background Antimicrobial resistance (AMR) is an urgent global health challenge and a critical threat to modern health care. Quantifying its burden in the WHO Region of the Americas has been elusive—despite the region’s long history of resistance surveillance. This study provides comprehensive estimates of AMR burden in the Americas to assess this growing health threat. Methods We estimated deaths and disability-adjusted life-years (DALYs) attributable to and associated with AMR for 23 bacterial pathogens and 88 pathogen–drug combinations for countries in the WHO Region of the Americas in 2019. We obtained data from mortality registries, surveillance systems, hospital systems, systematic literature reviews, and other sources, and applied predictive statistical modelling to produce estimates of AMR burden for all countries in the Americas. Five broad components were the backbone of our approach: the number of deaths where infection had a role, the proportion of infectious deaths attributable to a given infectious syndrome, the proportion of infectious syndrome deaths attributable to a given pathogen, the percentage of pathogens resistant to an antibiotic class, and the excess risk of mortality (or duration of an infection) associated with this resistance. We then used these components to estimate the disease burden by applying two counterfactual scenarios: deaths attributable to AMR (compared to an alternative scenario where resistant infections are replaced with susceptible ones), and deaths associated with AMR (compared to an alternative scenario where resistant infections would not occur at all). We generated 95% uncertainty intervals (UIs) for final estimates as the 25th and 975th ordered values across 1000 posterior draws, and models were cross-validated for out-of-sample predictive validity. Findings We estimated 569,000 deaths (95% UI 406,000–771,000) associated with bacterial AMR and 141,000 deaths (99,900–196,000) attributable to bacterial AMR among the 35 countries in the WHO Region of the Americas in 2019. Lower respiratory and thorax infections, as a syndrome, were responsible for the largest fatal burden of AMR in the region, with 189,000 deaths (149,000–241,000) associated with resistance, followed by bloodstream infections (169,000 deaths [94,200–278,000]) and peritoneal/intra-abdominal infections (118,000 deaths [78,600–168,000]). The six leading pathogens (by order of number of deaths associated with resistance) were Staphylococcus aureus, Escherichia coli, Klebsiella pneumoniae, Streptococcus pneumoniae, Pseudomonas aeruginosa, and Acinetobacter baumannii. Together, these pathogens were responsible for 452,000 deaths (326,000–608,000) associated with AMR. Methicillin-resistant S. aureus predominated as the leading pathogen–drug combination in 34 countries for deaths attributable to AMR, while aminopenicillin-resistant E. coli was the leading pathogen–drug combination in 15 countries for deaths associated with AMR. Interpretation Given the burden across different countries, infectious syndromes, and pathogen–drug combinations, AMR represents a substantial health threat in the Americas. Countries with low access to antibiotics and basic health-care services often face the largest age-standardised mortality rates associated with and attributable to AMR in the region, implicating specific policy interventions. Evidence from this study can guide mitigation efforts that are tailored to the needs of each country in the region while informing decisions regarding funding and resource allocation. Multisectoral and joint cooperative efforts among countries will be a key to success in tackling AMR in the Americas.publishedVersio

Global mortality associated with 33 bacterial pathogens in 2019: a systematic analysis for the Global Burden of Disease Study 2019

Background: Reducing the burden of death due to infection is an urgent global public health priority. Previous studies have estimated the number of deaths associated with drug-resistant infections and sepsis and found that infections remain a leading cause of death globally. Understanding the global burden of common bacterial pathogens (both susceptible and resistant to antimicrobials) is essential to identify the greatest threats to public health. To our knowledge, this is the first study to present global comprehensive estimates of deaths associated with 33 bacterial pathogens across 11 major infectious syndromes. Methods: We estimated deaths associated with 33 bacterial genera or species across 11 infectious syndromes in 2019 using methods from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019, in addition to a subset of the input data described in the Global Burden of Antimicrobial Resistance 2019 study. This study included 343 million individual records or isolates covering 11 361 study-location-years. We used three modelling steps to estimate the number of deaths associated with each pathogen: deaths in which infection had a role, the fraction of deaths due to infection that are attributable to a given infectious syndrome, and the fraction of deaths due to an infectious syndrome that are attributable to a given pathogen. Estimates were produced for all ages and for males and females across 204 countries and territories in 2019. 95% uncertainty intervals (UIs) were calculated for final estimates of deaths and infections associated with the 33 bacterial pathogens following standard GBD methods by taking the 2·5th and 97·5th percentiles across 1000 posterior draws for each quantity of interest. Findings: From an estimated 13·7 million (95% UI 10·9–17·1) infection-related deaths in 2019, there were 7·7 million deaths (5·7–10·2) associated with the 33 bacterial pathogens (both resistant and susceptible to antimicrobials) across the 11 infectious syndromes estimated in this study. We estimated deaths associated with the 33 bacterial pathogens to comprise 13·6% (10·2–18·1) of all global deaths and 56·2% (52·1–60·1) of all sepsis-related deaths in 2019. Five leading pathogens—Staphylococcus aureus, Escherichia coli, Streptococcus pneumoniae, Klebsiella pneumoniae, and Pseudomonas aeruginosa—were responsible for 54·9% (52·9–56·9) of deaths among the investigated bacteria. The deadliest infectious syndromes and pathogens varied by location and age. The age-standardised mortality rate associated with these bacterial pathogens was highest in the sub-Saharan Africa super-region, with 230 deaths (185–285) per 100 000 population, and lowest in the high-income super-region, with 52·2 deaths (37·4–71·5) per 100 000 population. S aureus was the leading bacterial cause of death in 135 countries and was also associated with the most deaths in individuals older than 15 years, globally. Among children younger than 5 years, S pneumoniae was the pathogen associated with the most deaths. In 2019, more than 6 million deaths occurred as a result of three bacterial infectious syndromes, with lower respiratory infections and bloodstream infections each causing more than 2 million deaths and peritoneal and intra-abdominal infections causing more than 1 million deaths. Interpretation: The 33 bacterial pathogens that we investigated in this study are a substantial source of health loss globally, with considerable variation in their distribution across infectious syndromes and locations. Compared with GBD Level 3 underlying causes of death, deaths associated with these bacteria would rank as the second leading cause of death globally in 2019; hence, they should be considered an urgent priority for intervention within the global health community. Strategies to address the burden of bacterial infections include infection prevention, optimised use of antibiotics, improved capacity for microbiological analysis, vaccine development, and improved and more pervasive use of available vaccines. These estimates can be used to help set priorities for vaccine need, demand, and development. Funding: Bill & Melinda Gates Foundation, Wellcome Trust, and Department of Health and Social Care, using UK aid funding managed by the Fleming Fund

Global, regional, and national burden of diabetes from 1990 to 2021, with projections of prevalence to 2050: a systematic analysis for the Global Burden of Disease Study 2021

Background: Diabetes is one of the leading causes of death and disability worldwide, and affects people regardless of country, age group, or sex. Using the most recent evidentiary and analytical framework from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD), we produced location-specific, age-specific, and sex-specific estimates of diabetes prevalence and burden from 1990 to 2021, the proportion of type 1 and type 2 diabetes in 2021, the proportion of the type 2 diabetes burden attributable to selected risk factors, and projections of diabetes prevalence through 2050. Methods: Estimates of diabetes prevalence and burden were computed in 204 countries and territories, across 25 age groups, for males and females separately and combined; these estimates comprised lost years of healthy life, measured in disability-adjusted life-years (DALYs; defined as the sum of years of life lost [YLLs] and years lived with disability [YLDs]). We used the Cause of Death Ensemble model (CODEm) approach to estimate deaths due to diabetes, incorporating 25 666 location-years of data from vital registration and verbal autopsy reports in separate total (including both type 1 and type 2 diabetes) and type-specific models. Other forms of diabetes, including gestational and monogenic diabetes, were not explicitly modelled. Total and type 1 diabetes prevalence was estimated by use of a Bayesian meta-regression modelling tool, DisMod-MR 2.1, to analyse 1527 location-years of data from the scientific literature, survey microdata, and insurance claims; type 2 diabetes estimates were computed by subtracting type 1 diabetes from total estimates. Mortality and prevalence estimates, along with standard life expectancy and disability weights, were used to calculate YLLs, YLDs, and DALYs. When appropriate, we extrapolated estimates to a hypothetical population with a standardised age structure to allow comparison in populations with different age structures. We used the comparative risk assessment framework to estimate the risk-attributable type 2 diabetes burden for 16 risk factors falling under risk categories including environmental and occupational factors, tobacco use, high alcohol use, high body-mass index (BMI), dietary factors, and low physical activity. Using a regression framework, we forecast type 1 and type 2 diabetes prevalence through 2050 with Socio-demographic Index (SDI) and high BMI as predictors, respectively. Findings: In 2021, there were 529 million (95% uncertainty interval [UI] 500–564) people living with diabetes worldwide, and the global age-standardised total diabetes prevalence was 6·1% (5·8–6·5). At the super-region level, the highest age-standardised rates were observed in north Africa and the Middle East (9·3% [8·7–9·9]) and, at the regional level, in Oceania (12·3% [11·5–13·0]). Nationally, Qatar had the world's highest age-specific prevalence of diabetes, at 76·1% (73·1–79·5) in individuals aged 75–79 years. Total diabetes prevalence—especially among older adults—primarily reflects type 2 diabetes, which in 2021 accounted for 96·0% (95·1–96·8) of diabetes cases and 95·4% (94·9–95·9) of diabetes DALYs worldwide. In 2021, 52·2% (25·5–71·8) of global type 2 diabetes DALYs were attributable to high BMI. The contribution of high BMI to type 2 diabetes DALYs rose by 24·3% (18·5–30·4) worldwide between 1990 and 2021. By 2050, more than 1·31 billion (1·22–1·39) people are projected to have diabetes, with expected age-standardised total diabetes prevalence rates greater than 10% in two super-regions: 16·8% (16·1–17·6) in north Africa and the Middle East and 11·3% (10·8–11·9) in Latin America and Caribbean. By 2050, 89 (43·6%) of 204 countries and territories will have an age-standardised rate greater than 10%. Interpretation: Diabetes remains a substantial public health issue. Type 2 diabetes, which makes up the bulk of diabetes cases, is largely preventable and, in some cases, potentially reversible if identified and managed early in the disease course. However, all evidence indicates that diabetes prevalence is increasing worldwide, primarily due to a rise in obesity caused by multiple factors. Preventing and controlling type 2 diabetes remains an ongoing challenge. It is essential to better understand disparities in risk factor profiles and diabetes burden across populations, to inform strategies to successfully control diabetes risk factors within the context of multiple and complex drivers. Funding: Bill & Melinda Gates Foundation

Global, regional, and national prevalence and mortality burden of sickle cell disease, 2000–2021: a systematic analysis from the Global Burden of Disease Study 2021

Background Previous global analyses, with known underdiagnosis and single cause per death attribution systems, provide only a small insight into the suspected high population health effect of sickle cell disease. Completed as part of the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2021, this study delivers a comprehensive global assessment of prevalence of sickle cell disease and mortality burden by age and sex for 204 countries and territories from 2000 to 2021. Methods We estimated cause-specific sickle cell disease mortality using standardised GBD approaches, in which each death is assigned to a single underlying cause, to estimate mortality rates from the International Classification of Diseases (ICD)-coded vital registration, surveillance, and verbal autopsy data. In parallel, our goal was to estimate a more accurate account of sickle cell disease health burden using four types of epidemiological data on sickle cell disease: birth incidence, age-specific prevalence, with-condition mortality (total deaths), and excess mortality (excess deaths). Systematic reviews, supplemented with ICD-coded hospital discharge and insurance claims data, informed this modelling approach. We employed DisMod-MR 2.1 to triangulate between these measures—borrowing strength from predictive covariates and across age, time, and geography—and generated internally consistent estimates of incidence, prevalence, and mortality for three distinct genotypes of sickle cell disease: homozygous sickle cell disease and severe sickle cell β-thalassaemia, sickle-haemoglobin C disease, and mild sickle cell β-thalassaemia. Summing the three models yielded final estimates of incidence at birth, prevalence by age and sex, and total sickle cell disease mortality, the latter of which was compared directly against cause-specific mortality estimates to evaluate differences in mortality burden assessment and implications for the Sustainable Development Goals (SDGs). Findings Between 2000 and 2021, national incidence rates of sickle cell disease were relatively stable, but total births of babies with sickle cell disease increased globally by 13·7% (95% uncertainty interval 11·1–16·5), to 515 000 (425 000–614 000), primarily due to population growth in the Caribbean and western and central sub-Saharan Africa. The number of people living with sickle cell disease globally increased by 41·4% (38·3–44·9), from 5·46 million (4·62–6·45) in 2000 to 7·74 million (6·51–9·2) in 2021. We estimated 34 400 (25 000–45 200) cause-specific all-age deaths globally in 2021, but total sickle cell disease mortality burden was nearly 11-times higher at 376 000 (303 000–467 000). In children younger than 5 years, there were 81 100 (58 800–108 000) deaths, ranking total sickle cell disease mortality as 12th (compared to 40th for cause-specific sickle cell disease mortality) across all causes estimated by the GBD in 2021. Interpretation Our findings show a strikingly high contribution of sickle cell disease to all-cause mortality that is not apparent when each death is assigned to only a single cause. Sickle cell disease mortality burden is highest in children, especially in countries with the greatest under-5 mortality rates. Without comprehensive strategies to address morbidity and mortality associated with sickle cell disease, attainment of SDG 3.1, 3.2, and 3.4 is uncertain. Widespread data gaps and correspondingly high uncertainty in the estimates highlight the urgent need for routine and sustained surveillance efforts, further research to assess the contribution of conditions associated with sickle cell disease, and widespread deployment of evidence-based prevention and treatment for those with sickle cell disease.publishedVersio