Optimization by Adaptive Stochastic Descent

Estimating the optimal linear combination of predictors using spherically constrained optimization

BACKGROUND

In the context of a binary classification problem, the optimal linear combination of continuous predictors can be estimated by maximizing the area under the receiver operating characteristic curve. For ordinal responses, the optimal predictor combination can similarly be obtained by maximization of the hypervolume under the manifold (HUM). Since the empirical HUM is discontinuous, non-differentiable, and possibly multi-modal, solving this maximization problem requires a global optimization technique. Estimation of the optimal coefficient vector using existing global optimization techniques is computationally expensive, becoming prohibitive as the number of predictors and the number of outcome categories increases.

RESULTS

We propose an efficient derivative-free black-box optimization technique based on pattern search to solve this problem, which we refer to as Spherically Constrained Optimization Routine (SCOR). Through extensive simulation studies, we demonstrate that the proposed method achieves better performance than existing methods including the step-down algorithm. Finally, we illustrate the proposed method to predict the severity of swallowing difficulty after radiation therapy for oropharyngeal cancer based on radiation dose to various structures in the head and neck.

CONCLUSIONS

Our proposed method addresses an important challenge in combining multiple biomarkers to predict an ordinal outcome. This problem is particularly relevant to medical research, where it may be of interest to diagnose a disease with various stages of progression or a toxicity with multiple grades of severity. We provide the implementation of our proposed SCOR method as an R package, available online at https://CRAN.R-project.org/package=SCOR .

Cost optimisation analysis of the expanded programme for immunisation: balancing equity and coverage in Pakistan

Introduction

With limited resources, attaining maximal average health service coverage can be at odds with maximising equity which attempts to promote greater reach among underserved populations. In this study, we examined the trade-offs in immunisation coverage levels and equity for children under 5 years of age in Pakistan across various subpopulations who can be targeted with different combinations of immunisation service modalities.

Methods

We conducted a detailed costing exercise across 16 geographically and demographically diverse districts in Pakistan. These data were the basis for (a) technical efficiency benchmarking via Data Envelopment Analysis to identify potential efficiency gains by location, delivery model and cost ingredient; (b) allocative efficiency optimisation modelling to understand how resource allocations could be optimised and to devise recommended budget allocations and operational metrics. Finally, the hypothetical overall efficiency gains attainable were estimated if available resources were allocated with the optimal emphases, and if service delivery models operated at productivity levels at the benchmarked frontier of efficiency.

Results

Benchmarking suggests that ~44% of delivery models are running efficiently and 37% are highly inefficient. While coverage and equity are usually at odds, surprisingly, the optimisation modelling revealed that substantial improvements in equity between subpopulations does not necessarily cost very much in overall immunisation coverage: theoretically, equity can be achieved while still attaining close to maximal immunisation coverage. Overall, analyses suggest greater emphases should be placed on outreach delivery models which particularly target rural areas and slum populations.

Conclusion

The unit cost differentials within districts are not sufficiently large for there to be a large reduction in potential Fully Immunised Children coverage if one focuses on maximising equity. However, reallocations of programme budgets can have a significant impact on equity outcomes, particularly at current low spending amounts. Therefore, it is recommended to address equity as the key objective in national immunisation programming.

Optima TB: A tool to help optimally allocate tuberculosis spending

Approximately 85% of tuberculosis (TB) related deaths occur in low- and middle-income countries where health resources are scarce. Effective priority setting is required to maximise the impact of limited budgets. The Optima TB tool has been developed to support analytical capacity and inform evidence-based priority setting processes for TB health benefits package design. This paper outlines the Optima TB framework and how it was applied in Belarus, an upper-middle income country in Eastern Europe with a relatively high burden of TB. Optima TB is a population-based disease transmission model, with programmatic cost functions and an optimisation algorithm. Modelled populations include age-differentiated general populations and higher-risk populations such as people living with HIV. Populations and prospective interventions are defined in consultation with local stakeholders. In partnership with the latter, demographic, epidemiological, programmatic, as well as cost and spending data for these populations and interventions are then collated. An optimisation analysis of TB spending was conducted in Belarus, using program objectives and constraints defined in collaboration with local stakeholders, which included experts, decision makers, funders and organisations involved in service delivery, support and technical assistance. These analyses show that it is possible to improve health impact by redistributing current TB spending in Belarus. Specifically, shifting funding from inpatient- to outpatient-focused care models, and from mass screening to active case finding strategies, could reduce TB prevalence and mortality by up to 45% and 50%, respectively, by 2035. In addition, an optimised allocation of TB spending could lead to a reduction in drug-resistant TB infections by 40% over this period. This would support progress towards national TB targets without additional financial resources. The case study in Belarus demonstrates how reallocations of spending across existing and new interventions could have a substantial impact on TB outcomes. This highlights the potential for Optima TB and similar modelling tools to support evidence-based priority setting.

Tuberculosis (TB) remains a leading global cause of death and morbidity, and 85% of deaths occur in countries where resources for TB care and control are limited. Many countries cannot finance all TB interventions or technologies, which means difficult decisions on what to prioritise and publically finance. Modelling tools can help decision-makers set priorities based on evidence, in a systematic and transparent way. This study presents Optima TB, a tool that estimates which allocations of spending across interventions will most likely maximise specified objectives—such as minimising TB deaths, prevalence and incidence. In partnership with local decision-makers and stakeholders, Optima TB was applied in Belarus. Recommendations from the model findings include focussing investment on outpatient rather than inpatient care and actively finding people with TB (e.g. through contact tracing) rather than mass testing of the population. The recommended reallocations of spending could reduce TB prevalence and deaths by up to 45% and 50%, respectively, by 2035 for the same amount of spending. Key stakeholders were engaged throughout the analysis and findings and uncertainty around the results were clearly communicated with decision-makers. The timeliness of the results helped inform national dialogue on TB care reform, among other key policy discussions.

Covasim: An agent-based model of COVID-19 dynamics and interventions

The COVID-19 pandemic has created an urgent need for models that can project epidemic trends, explore intervention scenarios, and estimate resource needs. Here we describe the methodology of Covasim (COVID-19 Agent-based Simulator), an open-source model developed to help address these questions. Covasim includes country-specific demographic information on age structure and population size; realistic transmission networks in different social layers, including households, schools, workplaces, long-term care facilities, and communities; age-specific disease outcomes; and intrahost viral dynamics, including viral-load-based transmissibility. Covasim also supports an extensive set of interventions, including non-pharmaceutical interventions, such as physical distancing and protective equipment; pharmaceutical interventions, including vaccination; and testing interventions, such as symptomatic and asymptomatic testing, isolation, contact tracing, and quarantine. These interventions can incorporate the effects of delays, loss-to-follow-up, micro-targeting, and other factors. Implemented in pure Python, Covasim has been designed with equal emphasis on performance, ease of use, and flexibility: realistic and highly customized scenarios can be run on a standard laptop in under a minute. In collaboration with local health agencies and policymakers, Covasim has already been applied to examine epidemic dynamics and inform policy decisions in more than a dozen countries in Africa, Asia-Pacific, Europe, and North America.

Covasim: An agent-based model of COVID-19 dynamics and interventions

The COVID-19 pandemic has created an urgent need for models that can project epidemic trends, explore intervention scenarios, and estimate resource needs. Here we describe the methodology of Covasim (COVID-19 Agent-based Simulator), an open-source model developed to help address these questions. Covasim includes country-specific demographic information on age structure and population size; realistic transmission networks in different social layers, including households, schools, workplaces, long-term care facilities, and communities; age-specific disease outcomes; and intrahost viral dynamics, including viral-load-based transmissibility. Covasim also supports an extensive set of interventions, including non-pharmaceutical interventions, such as physical distancing and protective equipment; pharmaceutical interventions, including vaccination; and testing interventions, such as symptomatic and asymptomatic testing, isolation, contact tracing, and quarantine. These interventions can incorporate the effects of delays, loss-to-follow-up, micro-targeting, and other factors. Implemented in pure Python, Covasim has been designed with equal emphasis on performance, ease of use, and flexibility: realistic and highly customized scenarios can be run on a standard laptop in under a minute. In collaboration with local health agencies and policymakers, Covasim has already been applied to examine epidemic dynamics and inform policy decisions in more than a dozen countries in Africa, Asia-Pacific, Europe, and North America.

Optimization Models for HIV/AIDS Resource Allocation: A Systematic Review

OBJECTIVE

This study reviews optimization models for human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS) resource allocation.

METHODS

We searched 2 databases for peer-reviewed articles published from January 1985 through August 2019 that describe optimization models for resource allocation in HIV/AIDS. We included models that consider 2 or more competing HIV/AIDS interventions. We extracted data on selected characteristics and identified similarities and differences across models. We also assessed the quality of mathematical disease transmission models based on the best practices identified by a 2010 task force.

RESULTS

The final qualitative synthesis included 23 articles that used 14 unique optimization models. The articles shared several characteristics, including the use of dynamic transmission modeling to estimate health benefits and the inclusion of specific high-risk groups in the study population. The models explored similar HIV/AIDS interventions that span primary and secondary prevention and antiretroviral treatment. Most articles were focused on sub-Saharan African countries (57%) and the United States (39%). There was notable variation in the types of optimization objectives across the articles; the most common was minimizing HIV incidence or maximizing infections averted (87%). Articles that utilized mathematical modeling of HIV disease and transmission displayed variable quality.

CONCLUSIONS

This systematic review of the literature identified examples of optimization models that have been applied in different settings, many of which displayed similar features. There were similarities in objective functions across optimization models, but they did not align with global HIV/AIDS goals or targets. Future work should be applied in countries facing the largest declines in HIV/AIDS funding.

Opportunities for improved HIV prevention and treatment through budget optimization in Eswatini

INTRODUCTION

Eswatini achieved a 44% decrease in new HIV infections from 2014 to 2019 through substantial scale-up of testing and treatment. However, it still has one of the highest rates of HIV incidence in the world, with 14 infections per 1,000 adults 15-49 years estimated for 2017. The Government of Eswatini has called for an 85% reduction in new infections by 2023 over 2017 levels. To make further progress towards this target and to achieve maximum health gains, this study aims to model optimized investments of available HIV resources.

METHODS

The Optima HIV model was applied to estimate the impact of efficiency strategies to accelerate prevention of HIV infections and HIV-related deaths. We estimated the number of infections and deaths that could be prevented by optimizing HIV investments. We optimize across HIV programs, then across service delivery modalities for voluntary medical male circumcision (VMMC), HIV testing, and antiretroviral refill, as well as switching to a lower cost antiretroviral regimen.

FINDINGS

Under an optimized budget, prioritising HIV testing for the general population followed by key preventative interventions may result in approximately 1,000 more new infections (2% more) being averted by 2023. More infections could be averted with further optimization between service delivery modalities across the HIV cascade. Scaling-up index and self-testing could lead to 100,000 more people getting tested for HIV (25% more tests) with the same budget. By prioritizing Fast-Track, community-based, and facility-based antiretroviral refill options, an estimated 30,000 more people could receive treatment, 17% more than baseline or US$5.5 million could be saved, 4% of the total budget. Finally, switching non-pregnant HIV-positive adults to a Dolutegravir-based antiretroviral therapy regimen and concentrating delivery of VMMC to existing fixed facilities over mobile clinics, US$4.5 million (7% of total budget) and US$6.6 million (10% of total budget) could be saved, respectively.

SIGNIFICANCE

With a relatively short five-year timeframe, even under a substantially increased and optimized budget, Eswatini is unlikely to reach their ambitious national prevention target by 2023. However, by optimizing investment of the same budget towards highly cost-effective VMMC, testing, and treatment modalities, further reductions in HIV incidence and cost savings could be realized.

The Cascade Analysis Tool: software to analyze and optimize care cascades

Introduction: Cascades, which track the progressive stages of engagement on the path towards a successful outcome, are increasingly being employed to quantitatively assess progress towards targets associated with health and development responses. Maximizing the proportion of people with successful outcomes within a budget-constrained context requires identifying and implementing interventions that are not only effective, but also cost-effective.

Methods: We developed a software application called the Cascade Analysis Tool that implements advanced analysis and optimization methods for understanding cascades, combined with the flexibility to enable application across a wide range of areas in health and development. The tool allows users to design the cascade, collate and enter data, and then use the built-in analysis methods in order to answer key policy questions, such as: understanding where the biggest drop-offs along the cascade are; visualizing how the cascade varies by population; investigating the impact of introducing a new intervention or scaling up/down existing interventions; and estimating how available funding should be optimally allocated among available interventions in order to achieve a variety of different objectives selectable by the user (such as optimizing cascade outcomes in target years). The Cascade Analysis Tool is available via a user-friendly web-based application, and comes with a user guide, a library of pre-made examples, and training materials.

Discussion: Whilst the Cascade Analysis Tool is still in the early stages of existence, it has already shown promise in preliminary applications, and we believe there is potential for it to help make sense of the increasing quantities of data on cascades.

Optimal allocation of HIV resources among geographical regions

BACKGROUND

Health resources are limited, which means spending should be focused on the people, places and programs that matter most. Choosing the mix of programs to maximize a health outcome is termed allocative efficiency. Here, we extend the methodology of allocative efficiency to answer the question of how resources should be distributed among different geographic regions.

METHODS

We describe a novel geographical optimization algorithm, which has been implemented as an extension to the Optima HIV model. This algorithm identifies an optimal funding of services and programs across regions, such as multiple countries or multiple districts within a country. The algorithm consists of three steps: (1) calibrating the model to each region, (2) determining the optimal allocation for each region across a range of different budget levels, and (3) finding the budget level in each region that minimizes the outcome (such as reducing new HIV infections and/or HIV-related deaths), subject to the constraint of fixed total budget across all regions. As a case study, we applied this method to determine an illustrative allocation of HIV program funding across three representative oblasts (regions) in Ukraine (Mykolayiv, Poltava, and Zhytomyr) to minimize the number of new HIV infections.

RESULTS

Geographical optimization was found to identify solutions with better outcomes than would be possible by considering region-specific allocations alone. In the case of Ukraine, prior to optimization (i.e. with status quo spending), a total of 244,000 HIV-related disability-adjusted life years (DALYs) were estimated to occur from 2016 to 2030 across the three oblasts. With optimization within (but not between) oblasts, this was estimated to be reduced to 181,000. With geographical optimization (i.e., allowing reallocation of funds between oblasts), this was estimated to be further reduced to 173,000.

CONCLUSIONS

With the increasing availability of region- and even facility-level data, geographical optimization is likely to play an increasingly important role in health economic decision making. Although the largest gains are typically due to reallocating resources to the most effective interventions, especially treatment, further gains can be achieved by optimally reallocating resources between regions. Finally, the methods described here are not restricted to geographical optimization, and can be applied to other problems where competing resources need to be allocated with constraints, such as between diseases.

The Cascade Analysis Tool: software to analyze and optimize care cascades

Introduction: Cascades, which track the progressive stages of engagement on the path towards a successful outcome, are increasingly being employed to quantitatively assess progress towards targets associated with health and development responses. Maximizing the proportion of people with successful outcomes within a budget-constrained context requires identifying and implementing interventions that are not only effective, but also cost-effective. Methods: We developed a software application called the Cascade Analysis Tool that implements advanced analysis and optimization methods for understanding cascades, combined with the flexibility to enable application across a wide range of areas in health and development. The tool allows users to design the cascade, collate and enter data, and then use the built-in analysis methods in order to answer key policy questions, such as: understanding where the biggest drop-offs along the cascade are; visualizing how the cascade varies by population; investigating the impact of introducing a new intervention or scaling up/down existing interventions; and estimating how available funding should be optimally allocated among available interventions in order to achieve a variety of different objectives selectable by the user (such as optimizing cascade outcomes in target years). The Cascade Analysis Tool is available via a user-friendly web-based application, and comes with a user guide, a library of pre-made examples, and training materials. Discussion: Whilst the Cascade Analysis Tool is still in the early stages of existence, it has already shown promise in preliminary applications, and we believe there is potential for it to help make sense of the increasing quantities of data on cascades.

The global Optima HIV allocative efficiency model: targeting resources in efforts to end AIDS

BACKGROUND

To move towards ending AIDS by 2030, HIV resources should be allocated cost-effectively. We used the Optima HIV model to estimate how global HIV resources could be retargeted for greatest epidemiological effect and how many additional new infections could be averted by 2030.

METHODS

We collated standard data used in country modelling exercises (including demographic, epidemiological, behavioural, programmatic, and expenditure data) from Jan 1, 2000, to Dec 31, 2015 for 44 countries, capturing 80% of people living with HIV worldwide. These data were used to parameterise separate subnational and national models within the Optima HIV framework. To estimate optimal resource allocation at subnational, national, regional, and global levels, we used an adaptive stochastic descent optimisation algorithm in combination with the epidemic models and cost functions for each programme in each country. Optimal allocation analyses were done with international HIV funds remaining the same to each country and by redistributing these funds between countries.

FINDINGS

Without additional funding, if countries were to optimally allocate their HIV resources from 2016 to 2030, we estimate that an additional 7·4 million (uncertainty range 3·9 million-14·0 million) new infections could be averted, representing a 26% (uncertainty range 13-50%) incidence reduction. Redistribution of international funds between countries could avert a further 1·9 million infections, which represents a 33% (uncertainty range 20-58%) incidence reduction overall. To reduce HIV incidence by 90% relative to 2010, we estimate that more than a three-fold increase of current annual funds will be necessary until 2030. The most common priorities for optimal resource reallocation are to scale up treatment and prevention programmes targeting key populations at greatest risk in each setting. Prioritisation of other HIV programmes depends on the epidemiology and cost-effectiveness of service delivery in each setting as well as resource availability.

INTERPRETATION

Further reductions in global HIV incidence are possible through improved targeting of international and national HIV resources.

FUNDING

World Bank and Australian NHMRC.

Getting it right when budgets are tight: Using optimal expansion pathways to prioritize responses to concentrated and mixed HIV epidemics

Background

Prioritizing investments across health interventions is complicated by the nonlinear relationship between intervention coverage and epidemiological outcomes. It can be difficult for countries to know which interventions to prioritize for greatest epidemiological impact, particularly when budgets are uncertain.

Methods

We examined four case studies of HIV epidemics in diverse settings, each with different characteristics. These case studies were based on public data available for Belarus, Peru, Togo, and Myanmar. The Optima HIV model and software package was used to estimate the optimal distribution of resources across interventions associated with a range of budget envelopes. We constructed “investment staircases”, a useful tool for understanding investment priorities. These were used to estimate the best attainable cost-effectiveness of the response at each investment level.

Findings

We find that when budgets are very limited, the optimal HIV response consists of a smaller number of ‘core’ interventions. As budgets increase, those core interventions should first be scaled up, and then new interventions introduced. We estimate that the cost-effectiveness of HIV programming decreases as investment levels increase, but that the overall cost-effectiveness remains below GDP per capita.

Significance

It is important for HIV programming to respond effectively to the overall level of funding availability. The analytic tools presented here can help to guide program planners understand the most cost-effective HIV responses and plan for an uncertain future.