Synthesizing epidemiological and economic optima for control of immunizing infections

Forward-backward and period doubling bifurcations in a discrete epidemic model with vaccination and limited medical resources

A discrete epidemic model with vaccination and limited medical resources is proposed to understand its underlying dynamics. The model induces a nonsmooth two dimensional map that exhibits a surprising array of dynamical behavior including the phenomena of the forward-backward bifurcation and period doubling route to chaos with feasible parameters in an invariant region. We demonstrate, among other things, that the model generates the above described phenomena as the transmission rate or the basic reproduction number of the disease gradually increases provided that the immunization rate is low, the vaccine failure rate is high and the medical resources are limited. Finally, the numerical simulations are provided to illustrate our main results.

Comparative performance of between-population vaccine allocation strategies with applications for emerging pandemics

Vaccine allocation decisions during emerging pandemics have proven to be challenging due to competing ethical, practical, and political considerations. Complicating decision making, policy makers need to consider vaccine allocation strategies that balance needs both within and between populations. When vaccine stockpiles are limited, doses should be allocated in locations to maximize their impact. Using a susceptible-exposed-infectious-recovered (SEIR) model we examine optimal vaccine allocation decisions across two populations considering the impact of characteristics of the population (e.g., size, underlying immunity, heterogeneous risk structure, interaction), vaccine (e.g., vaccine efficacy), pathogen (e.g., transmissibility), and delivery (e.g., varying speed and timing of rollout). Across a wide range of characteristics considered, we find that vaccine allocation proportional to population size (i.e., pro-rata allocation) performs either better or comparably to nonproportional allocation strategies in minimizing the cumulative number of infections. These results may argue in favor of sharing of vaccines between locations in the context of an epidemic caused by an emerging pathogen, where many epidemiologic characteristics may not be known.

Considering equity in priority setting using transmission models: Recommendations and data needs

OBJECTIVES

Disease transmission models are used in impact assessment and economic evaluations of infectious disease prevention and treatment strategies, prominently so in the COVID-19 response. These models rarely consider dimensions of equity relating to the differential health burden between individuals and groups. We describe concepts and approaches which are useful when considering equity in the priority setting process, and outline the technical choices concerning model structure, outputs, and data requirements needed to use transmission models in analyses of health equity.

METHODS

We reviewed the literature on equity concepts and approaches to their application in economic evaluation and undertook a technical consultation on how equity can be incorporated in priority setting for infectious disease control. The technical consultation brought together health economists with an interest in equity-informative economic evaluation, ethicists specialising in public health, mathematical modellers from various disease backgrounds, and representatives of global health funding and technical assistance organisations, to formulate key areas of consensus and recommendations.

RESULTS

We provide a series of recommendations for applying the Reference Case for Economic Evaluation in Global Health to infectious disease interventions, comprising guidance on 1) the specification of equity concepts; 2) choice of evaluation framework; 3) model structure; and 4) data needs. We present available conceptual and analytical choices, for example how correlation between different equity- and disease-relevant strata should be considered dependent on available data, and outline how assumptions and data limitations can be reported transparently by noting key factors for consideration.

CONCLUSIONS

Current developments in economic evaluations in global health provide a wide range of methodologies to incorporate equity into economic evaluations. Those employing infectious disease models need to use these frameworks more in priority setting to accurately represent health inequities. We provide guidance on the technical approaches to support this goal and ultimately, to achieve more equitable health policies.

Comparative performance of between-population vaccine allocation strategies with applications for emerging pandemics

Vaccine allocation decisions during emerging pandemics have proven to be challenging due to competing ethical, practical, and political considerations. Complicating decision making, policy makers need to consider vaccine allocation strategies that balance needs both within and between populations. Due to limited vaccine stockpiles, vaccine doses should be allocated in locations where their impact will be maximized. Using a susceptible-exposed-infectious-recovered (SEIR) model we examine optimal vaccine allocation decisions across two populations considering the impact of population size, underlying immunity, continuous vaccine roll-out, heterogeneous population risk structure, and differences in disease transmissibility. We find that in the context of an emerging pathogen where many epidemiologic characteristics might not be known, equal vaccine allocation between populations performs optimally in most scenarios. In the specific case considering heterogeneous population risk structure, first targeting individuals at higher risk of transmission or death due to infection leads to equal resource allocation across populations.

Optimal control of the spatial allocation of COVID-19 vaccines: Italy as a case study

While campaigns of vaccination against SARS-CoV-2 are underway across the world, communities face the challenge of a fair and effective distribution of a limited supply of doses. Current vaccine allocation strategies are based on criteria such as age or risk. In the light of strong spatial heterogeneities in disease history and transmission, we explore spatial allocation strategies as a complement to existing approaches. Given the practical constraints and complex epidemiological dynamics, designing effective vaccination strategies at a country scale is an intricate task. We propose a novel optimal control framework to derive the best possible vaccine allocation for given disease transmission projections and constraints on vaccine supply and distribution logistics. As a proof-of-concept, we couple our framework with an existing spatially explicit compartmental COVID-19 model tailored to the Italian geographic and epidemiological context. We optimize the vaccine allocation on scenarios of unfolding disease transmission across the 107 provinces of Italy, from January to April 2021. For each scenario, the optimal solution significantly outperforms alternative strategies that prioritize provinces based on incidence, population distribution, or prevalence of susceptibles. Our results suggest that the complex interplay between the mobility network and the spatial heterogeneities implies highly non-trivial prioritization strategies for effective vaccination campaigns. Our work demonstrates the potential of optimal control for complex and heterogeneous epidemiological landscapes at country, and possibly global, scales.

Incorporating economic constraints for optimal control of immunizing infections

It is well-known that the interruption of transmission of a disease can be achieved, provided the vaccinated population reaches a threshold depending on, among others, the efficacy of vaccines. The purpose of this paper is to address the optimal vaccination strategy by imposing the economic constraints. In particular, an S--(I,V)--S model used to describe the spreading of the disease in a well-mixed population and a cost function consisting of vaccination and infection costs are proposed. The well-definedness of the above-described modeling is provided. We were then able to provide an optimal strategy to minimize the cost for all parameters. In particular, the optimal vaccination level to minimize the cost can be completely characterized for all parameters. For instance, the optimal vaccination level can be classified by the magnitude of the failure rate of the vaccine with other parameters being given. Under these circumstances, the optimal strategy to minimize the cost is roughly to eliminate the disease locally (respectively, choose an economic optimum resulting in not to wipe out the disease completely or take no vaccination for anyone) provided the vaccine failure rate is relatively small (respectively, intermediate or large). Numerical simulations to illustrate our main results are also provided. Moreover, the data collected at the height of the Covid-19 pandemic in Taiwan are also numerically simulated to provide the corresponding optimal vaccination strategy.

Equitable access to COVID-19 vaccines makes a life-saving difference to all countries

Despite broad agreement on the negative consequences of vaccine inequity, the distribution of COVID-19 vaccines is imbalanced. Access to vaccines in high-income countries (HICs) is far greater than in low- and middle-income countries (LMICs). As a result, there continue to be high rates of COVID-19 infections and deaths in LMICs. In addition, recent mutant COVID-19 outbreaks may counteract advances in epidemic control and economic recovery in HICs. To explore the consequences of vaccine (in)equity in the face of evolving COVID-19 strains, we examine vaccine allocation strategies using a multistrain metapopulation model. Our results show that vaccine inequity provides only limited and short-term benefits to HICs. Sharper disparities in vaccine allocation between HICs and LMICs lead to earlier and larger outbreaks of new waves. Equitable vaccine allocation strategies, in contrast, substantially curb the spread of new strains. For HICs, making immediate and generous vaccine donations to LMICs is a practical pathway to protect everyone.

Economic evaluation of disease elimination: An extension to the net-benefit framework and application to human African trypanosomiasis

The global health community has earmarked a number of diseases for elimination or eradication, and these goals have often been praised on the premise of long-run cost savings. However, decision makers must contend with a multitude of demands on health budgets in the short or medium term, and costs per case often rise as the burden of a disease falls, rendering such efforts beyond the cost-effective use of scarce resources. In addition, these decisions must be made in the presence of substantial uncertainty regarding the feasibility and costs of elimination or eradication efforts. Therefore, analytical frameworks are necessary to consider the additional effort for reaching global goals, like elimination or eradication, that are beyond the cost-effective use of country resources. We propose a modification to the net-benefit framework to consider the implications of switching from an optimal strategy, in terms of cost-per-burden averted, to a strategy with a higher likelihood of meeting the global target of elimination or eradication. We illustrate the properties of our framework by considering the economic case of efforts to eliminate the transmission of gambiense human African trypanosomiasis (gHAT), a vector-borne, parasitic disease in West and Central Africa, by 2030.

Challenges in evaluating risks and policy options around endemic establishment or elimination of novel pathogens

When a novel pathogen emerges there may be opportunities to eliminate transmission - locally or globally - whilst case numbers are low. However, the effort required to push a disease to elimination may come at a vast cost at a time when uncertainty is high. Models currently inform policy discussions on this question, but there are a number of open challenges, particularly given unknown aspects of the pathogen biology, the effectiveness and feasibility of interventions, and the intersecting political, economic, sociological and behavioural complexities for a novel pathogen. In this overview, we detail how models might identify directions for better leveraging or expanding the scope of data available on the pathogen trajectory, for bounding the theoretical context of emergence relative to prospects for elimination, and for framing the larger economic, behavioural and social context that will influence policy decisions and the pathogen’s outcome.

Vaccine nationalism and the dynamics and control of SARS-CoV-2

Vaccines provide powerful tools to mitigate the enormous public health and economic costs that the ongoing severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic continues to exert globally, yet vaccine distribution remains unequal among countries. To examine the potential epidemiological and evolutionary impacts of “vaccine nationalism,” we extend previous models to include simple scenarios of stockpiling between two regions. In general, when vaccines are widely available and the immunity they confer is robust, sharing doses minimizes total cases across regions. A number of subtleties arise when the populations and transmission rates in each region differ, depending on evolutionary assumptions and vaccine availability. When the waning of natural immunity contributes most to evolutionary potential, sustained transmission in low-access regions results in an increased potential for antigenic evolution, which may result in the emergence of novel variants that affect epidemiological characteristics globally. Overall, our results stress the importance of rapid, equitable vaccine distribution for global control of the pandemic.

Reducing dengue fever cases at the lowest budget: a constrained optimization approach applied to Thailand

Background

With the challenges that dengue fever (DF) presents to healthcare systems and societies, public health officials must determine where best to allocate scarce resources and restricted budgets. Constrained optimization (CO) helps to address some of the acknowledged limitations of conventional health economic analyses and has typically been used to identify the optimal allocation of resources across interventions subject to a variety of constraints.

Methods

A dynamic transmission model was developed to predict the number of dengue cases in Thailand at steady state. A CO was then applied to identify the optimal combination of interventions (release of Wolbachia-infected mosquitoes and paediatric vaccination) within the constraints of a fixed budget, set no higher than cost estimates of the current vector control programme, to minimize the number of dengue cases and disability-adjusted life years (DALYs) lost. Epidemiological, cost, and effectiveness data were informed by national data and the research literature. The time horizon was 10 years. Scenario analyses examined different disease management and intervention costs, budget constraints, vaccine efficacy, and optimization time horizon.

Results

Under base-case budget constraints, the optimal coverage of the two interventions to minimize dengue incidence was predicted to be nearly equal (Wolbachia 50%; paediatric vaccination 49%) with corresponding coverages under lower bound (Wolbachia 54%; paediatric vaccination 10%) and upper bound (Wolbachia 67%; paediatric vaccination 100%) budget ceilings. Scenario analyses indicated that the most impactful situations related to the costs of Wolbachia and paediatric vaccination with decreases/ increases in costs of interventions demonstrating a direct correlation with coverage (increases/ decreases) of the respective control strategies under examination.

Conclusions

Determining the best investment strategy for dengue control requires the identification of the optimal mix of interventions to implement in order to maximize public health outcomes, often under fixed budget constraints. A CO model was developed with the objective of minimizing dengue cases (and DALYs lost) over a 10-year time horizon, within the constraints of the estimated budgets for vector control in the absence of vaccination and Wolbachia. The model provides a tool for developing estimates of optimal coverage of combined dengue control strategies that minimize dengue burden at the lowest budget.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12889-021-10747-3.

Pandemic Equation for Describing and Predicting COVID19 Evolution

The purpose of this work is to describe the dynamics of the COVID-19 pandemics accounting for the mitigation measures, for the introduction or removal of the quarantine, and for the effect of vaccination when and if introduced. The methods used include the derivation of the Pandemic Equation describing the mitigation measures via the evolution of the growth time constant in the Pandemic Equation resulting in an asymmetric pandemic curve with a steeper rise than a decrease and mitigation measures. The Pandemic Equation predicts how the quarantine removal and business opening lead to a spike in the pandemic curve. The effective vaccination reduces the new daily infections predicted by the Pandemic Equation. The pandemic curves in many localities have similar time dependencies but shifted in time. The Pandemic Equation parameters extracted from the well advanced pandemic curves can be used for predicting the pandemic evolution in the localities, where the pandemics is still in the initial stages. Using the multiple pandemic locations for the parameter extraction allows for the uncertainty quantification in predicting the pandemic evolution using the introduced Pandemic Equation. Compared with other pandemic models our approach allows for easier parameter extraction amenable to using Artificial Intelligence models.

Epidemiologically and Socio-economically Optimal Policies via Bayesian Optimization

Mass public quarantining, colloquially known as a lock-down, is a non-pharmaceutical intervention to check spread of disease. This paper presents ESOP (Epidemiologically and Socio-economically Optimal Policies), a novel application of active machine learning techniques using Bayesian optimization, that interacts with an epidemiological model to arrive at lock-down schedules that optimally balance public health benefits and socio-economic downsides of reduced economic activity during lock-down periods. The utility of ESOP is demonstrated using case studies with VIPER (Virus-Individual-Policy-EnviRonment), a stochastic agent-based simulator that this paper also proposes. However, ESOP is flexible enough to interact with arbitrary epidemiological simulators in a black-box manner, and produce schedules that involve multiple phases of lock-downs.

Applying the 'no-one worse off' criterion to design Pareto efficient HIV responses in Sudan and Togo

INTRODUCTION

Globally, there is increased focus on getting the greatest impact from available health funding. However, the pursuit of overall welfare maximization may mean some are left worse off than before. Pareto efficiency takes welfare shifts into account by ruling out funding reallocations that worsen outcomes for any person or group.

METHODS

Using the Optima HIV model, studies of HIV response efficiency were conducted in Sudan in 2014 and Togo in 2015. In this article, we estimate the welfare maximizing and Pareto efficient allocations for these two national HIV budgets, using data from the original studies.

RESULTS

We estimate that, if the 2013 HIV budget for Sudan was annually available to 2020 but with funds reallocated according to the welfare maximizing allocation, a 36% reduction in cumulative new infections could be achieved between 2014 and 2020. We also find that this is Pareto efficient. In Togo, however, we find that it is possible to reduce overall new infections but applying the Pareto efficiency criterion means that shifts in emphases cannot occur in the HIV response without additional resources.

DISCUSSION

Protecting service coverage for key population groups is not necessarily equivalent to protecting health outcomes. In some cases, requiring Pareto efficiency may reduce the potential for population-wide welfare gains, but this is not always the case.

CONCLUSION

Pareto efficiency may be an appropriate addition to the quantitative toolset for evaluating HIV responses.

Incentivizing hospital infection control

Healthcare-associated infections (HAIs) pose a significant burden to patient safety. Institutions can implement hospital infection control (HIC) measures to reduce the impact of HAIs. Since patients can carry pathogens between institutions, there is an economic incentive for hospitals to free ride on the HIC investments of other facilities. Subsidies for infection control by public health authorities could encourage regional spending on HIC. We develop coupled mathematical models of epidemiology and hospital behavior in a game-theoretic framework to investigate how hospitals may change spending behavior in response to subsidies. We demonstrate that under a limited budget, a dollar-for-dollar matching grant outperforms both a fixed-amount subsidy and a subsidy on uninfected patients in reducing the number of HAIs in a single institution. Additionally, when multiple hospitals serve a community, funding priority should go to the hospital with a lower transmission rate. Overall, subsidies incentivize HIC spending and reduce the overall prevalence of HAIs.

Dose-Optimal Vaccine Allocation over Multiple Populations

Vaccination is an effective way to prevent an epidemic. It results in immunity for the vaccinated individuals, but it also reduces the infection pressure for unvaccinated people. Thus people may actually escape infection without being vaccinated: the so-called "herd effect." We analytically study the relation between the herd effect and the vaccination fraction for the seminal SIR compartmental model, which consists of a set of differential equations describing the time course of an epidemic. We prove that the herd effect is in general convex-concave in the vaccination fraction and give precise conditions on the epidemic for the convex part to arise. We derive the significant consequences of these structural insights for allocating a limited vaccine stockpile to multiple non-interacting populations. We identify for each population a unique vaccination fraction that is most efficient per dose of vaccine: our dose-optimal coverage. We characterize the solution of the vaccine allocation problem and we show the crucial importance of the dose-optimal coverage. A single dose of vaccine may be a drop in the ocean, but multiple doses together can save a population. To benefit from this, policy makers should select a subset of populations to which the vaccines are allocated. Focusing on a limited number of populations can make a significant difference, whereas allocating equally to all populations would be substantially less effective.

Essential information: Uncertainty and optimal control of Ebola outbreaks

Early resolution of uncertainty during an epidemic outbreak can lead to rapid and efficient decision making, provided that the uncertainty affects prioritization of actions. The wide range in caseload projections for the 2014 Ebola outbreak caused great concern and debate about the utility of models. By coding and running 37 published Ebola models with five candidate interventions, we found that, despite this large variation in caseload projection, the ranking of management options was relatively consistent. Reducing funeral transmission and reducing community transmission were generally ranked as the two best options. Value of information (VoI) analyses show that caseloads could be reduced by 11% by resolving all model-specific uncertainties, with information about model structure accounting for 82% of this reduction and uncertainty about caseload only accounting for 12%. Our study shows that the uncertainty that is of most interest epidemiologically may not be the same as the uncertainty that is most relevant for management. If the goal is to improve management outcomes, then the focus of study should be to identify and resolve those uncertainties that most hinder the choice of an optimal intervention. Our study further shows that simplifying multiple alternative models into a smaller number of relevant groups (here, with shared structure) could streamline the decision-making process and may allow for a better integration of epidemiological modeling and decision making for policy.

Essential information: Uncertainty and optimal control of Ebola outbreaks

Early resolution of uncertainty during an epidemic outbreak can lead to rapid and efficient decision making, provided that the uncertainty affects prioritization of actions. The wide range in caseload projections for the 2014 Ebola outbreak caused great concern and debate about the utility of models. By coding and running 37 published Ebola models with five candidate interventions, we found that, despite this large variation in caseload projection, the ranking of management options was relatively consistent. Reducing funeral transmission and reducing community transmission were generally ranked as the two best options. Value of information (VoI) analyses show that caseloads could be reduced by 11% by resolving all model-specific uncertainties, with information about model structure accounting for 82% of this reduction and uncertainty about caseload only accounting for 12%. Our study shows that the uncertainty that is of most interest epidemiologically may not be the same as the uncertainty that is most relevant for management. If the goal is to improve management outcomes, then the focus of study should be to identify and resolve those uncertainties that most hinder the choice of an optimal intervention. Our study further shows that simplifying multiple alternative models into a smaller number of relevant groups (here, with shared structure) could streamline the decision-making process and may allow for a better integration of epidemiological modeling and decision making for policy.

Self-enforcing regional vaccination agreements

In a highly interconnected world, immunizing infections are a transboundary problem, and their control and elimination require international cooperation and coordination. In the absence of a global or regional body that can impose a universal vaccination strategy, each individual country sets its own strategy. Mobility of populations across borders can promote free-riding, because a country can benefit from the vaccination efforts of its neighbours, which can result in vaccination coverage lower than the global optimum. Here we explore whether voluntary coalitions that reward countries that join by cooperatively increasing vaccination coverage can solve this problem. We use dynamic epidemiological models embedded in a game-theoretic framework in order to identify conditions in which coalitions are self-enforcing and therefore stable, and thus successful at promoting a cooperative vaccination strategy. We find that countries can achieve significantly greater vaccination coverage at a lower cost by forming coalitions than when acting independently, provided a coalition has the tools to deter free-riding. Furthermore, when economically or epidemiologically asymmetric countries form coalitions, realized coverage is regionally more consistent than in the absence of coalitions.

Systematic Review of Health Economic Analyses of Measles and Rubella Immunization Interventions

Economic analyses for vaccine-preventable diseases provide important insights about the value of prevention. We reviewed the literature to identify all of the peer-reviewed, published economic analyses of interventions related to measles and rubella immunization options to assess the different types of analyses performed and characterize key insights. We searched PubMed, the Science Citation Index, and references from relevant articles for studies in English and found 67 analyses that reported primary data and quantitative estimates of benefit-cost or cost-effectiveness analyses for measles and/or rubella immunization interventions. We removed studies that we characterized as cost-minimization analyses from this sample because they generally provide insights that focused on more optimal strategies to achieve the same health outcome. The 67 analyses we included demonstrate the large economic benefits associated with preventing measles and rubella infections using vaccines and the benefit of combining measles and rubella antigens into a formulation that saves the costs associated with injecting the vaccines separately. Despite the importance of population immunity and dynamic viral transmission, most of the analyses used static models to estimate cases prevented and characterize benefits, although the use of dynamic models continues to increase. Many of the analyses focused on characterizing the most significant adverse outcomes (e.g., mortality for measles, congenital rubella syndrome for rubella) and/or only direct costs, and the most complete analyses present data from high-income countries.

Optimizing Real-Time Vaccine Allocation in a Stochastic SIR Model

Real-time vaccination following an outbreak can effectively mitigate the damage caused by an infectious disease. However, in many cases, available resources are insufficient to vaccinate the entire at-risk population, logistics result in delayed vaccine deployment, and the interaction between members of different cities facilitates a wide spatial spread of infection. Limited vaccine, time delays, and interaction (or coupling) of cities lead to tradeoffs that impact the overall magnitude of the epidemic. These tradeoffs mandate investigation of optimal strategies that minimize the severity of the epidemic by prioritizing allocation of vaccine to specific subpopulations. We use an SIR model to describe the disease dynamics of an epidemic which breaks out in one city and spreads to another. We solve a master equation to determine the resulting probability distribution of the final epidemic size. We then identify tradeoffs between vaccine, time delay, and coupling, and we determine the optimal vaccination protocols resulting from these tradeoffs.

Limiting the spread of disease through altered migration patterns

We consider a model for an epidemic in a population that occupies geographically distinct locations. The disease is spread within subpopulations by contacts between infective and susceptible individuals, and is spread between subpopulations by the migration of infected individuals. We show how susceptible individuals can act collectively to limit the spread of disease during the initial phase of an epidemic by specifying the distribution that minimises the growth rate of the epidemic when the infectives are migrating so as to maximise the growth rate. We also give an explicit strategy that minimises the basic reproduction number, which is also shown be optimal in terms of the probability of extinction and total size of the epidemic.

Vaccine Induced Herd Immunity for Control of Respiratory Syncytial Virus Disease in a Low-Income Country Setting

Background

Respiratory syncytial virus (RSV) is globally ubiquitous, and infection during the first six months of life is a major risk for severe disease and hospital admission; consequently RSV is the most important viral cause of respiratory morbidity and mortality in young children. Development of vaccines for young infants is complicated by the presence of maternal antibodies and immunological immaturity, but vaccines targeted at older children avoid these problems. Vaccine development for young infants has been unsuccessful, but this is not the case for older children (> 6m). Would vaccinating older children have a significant public health impact? We developed a mathematical model to explore the benefits of a vaccine against RSV.

Methods and Findings

We have used a deterministic age structured model capturing the key epidemiological characteristics of RSV and performed a statistical maximum-likelihood fit to age-specific hospitalization data from a developing country setting. To explore the effects of vaccination under different mixing assumptions, we included two versions of contact matrices: one from a social contact diary study, and the second a synthesised construction based on demographic data. Vaccination is assumed to elicit an immune response equivalent to primary infection. Our results show that immunisation of young children (5–10m) is likely to be a highly effective method of protection of infants (<6m) against hospitalisation. The majority benefit is derived from indirect protection (herd immunity). A full sensitivity and uncertainty analysis using Latin Hypercube Sampling of the parameter space shows that our results are robust to model structure and model parameters.

Conclusions

This result suggests that vaccinating older infants and children against RSV can have a major public health benefit.

The Economics of Epidemic Diseases

Epidemic, infectious, diseases affect a large number of individuals across developing as well as developed countries. With reference to some very simple diffusion models, in this paper we consider how available economic resources could be optimally allocated by health authorities to mitigate, possibly eradicate, the disease. Optimality was defined as the minimization of the long run number of infected people. The main goal of the work has been to introduce a methodology for deciding if it would be best to concentrate resources to prevent contact between individuals and with an external source, or to develop a new treatment for curing the disease, or both. The analysis suggests that this depends on the cost functions, that is the available technology, for controlling the relevant parameters underlying the epidemics as well as on the available financial resources. In the case of the recent Ebola outbreak, the suggestions of the model have been consistent with the policies adopted.

Reactive vaccination in the presence of disease hotspots

Reactive vaccination has recently been adopted as an outbreak response tool for cholera and other infectious diseases. Owing to the global shortage of oral cholera vaccine, health officials must quickly decide who and where to distribute limited vaccine. Targeted vaccination in transmission hotspots (i.e. areas with high transmission efficiency) may be a potential approach to efficiently allocate vaccine, however its effectiveness will likely be context-dependent. We compared strategies for allocating vaccine across multiple areas with heterogeneous transmission efficiency. We constructed metapopulation models of a cholera-like disease and compared simulated epidemics where: vaccine is targeted at areas of high or low transmission efficiency, where vaccine is distributed across the population, and where no vaccine is used. We find that connectivity between populations, transmission efficiency, vaccination timing and the amount of vaccine available all shape the performance of different allocation strategies. In highly connected settings (e.g. cities) when vaccinating early in the epidemic, targeting limited vaccine at transmission hotspots is often optimal. Once vaccination is delayed, targeting the hotspot is rarely optimal, and strategies that either spread vaccine between areas or those targeted at non-hotspots will avert more cases. Although hotspots may be an intuitive outbreak control target, we show that, in many situations, the hotspot-epidemic proceeds so fast that hotspot-targeted reactive vaccination will prevent relatively few cases, and vaccination shared across areas where transmission can be sustained is often best.

Modeling infectious disease dynamics in the complex landscape of global health

Despite some notable successes in the control of infectious diseases, transmissible pathogens still pose an enormous threat to human and animal health. The ecological and evolutionary dynamics of infections play out on a wide range of interconnected temporal, organizational, and spatial scales, which span hours to months, cells to ecosystems, and local to global spread. Moreover, some pathogens are directly transmitted between individuals of a single species, whereas others circulate among multiple hosts, need arthropod vectors, or can survive in environmental reservoirs. Many factors, including increasing antimicrobial resistance, increased human connectivity and changeable human behavior, elevate prevention and control from matters of national policy to international challenge. In the face of this complexity, mathematical models offer valuable tools for synthesizing information to understand epidemiological patterns, and for developing quantitative evidence for decision-making in global health.

Seven challenges in modeling vaccine preventable diseases

Vaccination has been one of the most successful public health measures since the introduction of basic sanitation. Substantial mortality and morbidity reductions have been achieved via vaccination against many infections, and the list of diseases that are potentially controllable by vaccines is growing steadily. We introduce key challenges for modeling in shaping our understanding and guiding policy decisions related to vaccine preventable diseases.

Optimal vaccination in a stochastic epidemic model of two non-interacting populations

Developing robust, quantitative methods to optimize resource allocations in response to epidemics has the potential to save lives and minimize health care costs. In this paper, we develop and apply a computationally efficient algorithm that enables us to calculate the complete probability distribution for the final epidemic size in a stochastic Susceptible-Infected-Recovered (SIR) model. Based on these results, we determine the optimal allocations of a limited quantity of vaccine between two non-interacting populations. We compare the stochastic solution to results obtained for the traditional, deterministic SIR model. For intermediate quantities of vaccine, the deterministic model is a poor estimate of the optimal strategy for the more realistic, stochastic case.

Estimating drivers of autochthonous transmission of chikungunya virus in its invasion of the americas

Background Chikungunya is an emerging arbovirus that has caused explosive outbreaks in Africa and Asia for decades and invaded the Americas just over a year ago. During this ongoing invasion, it has spread to 45 countries where it has been transmitted autochthonously, infecting nearly 1.3 million people in total. Methods Here, we made use of weekly, country-level case reports to infer relationships between transmission and two putative climatic drivers: temperature and precipitation averaged across each country on a monthly basis. To do so, we used a TSIR model that enabled us to infer a parametric relationship between climatic drivers and transmission potential, and we applied a new method for incorporating a probabilistic description of the serial interval distribution into the TSIR framework. Results We found significant relationships between transmission and linear and quadratic terms for temperature and precipitation and a linear term for log incidence during the previous pathogen generation. The lattermost suggests that case numbers three to four weeks ago are largely predictive of current case numbers. This effect is quite nonlinear at the country level, however, due to an estimated mixing parameter of 0.74. Relationships between transmission and the climatic variables that we estimated were biologically plausible and in line with expectations. Conclusions Our analysis suggests that autochthonous transmission of Chikungunya in the Americas can be correlated successfully with putative climatic drivers, even at the coarse scale of countries and using long-term average climate data. Overall, this provides a preliminary suggestion that successfully forecasting the future trajectory of a Chikungunya outbreak and the receptivity of virgin areas may be possible. Our results also provide tentative estimates of timeframes and areas of greatest risk, and our extension of the TSIR model provides a novel tool for modeling vector-borne disease transmission.

Six challenges in the eradication of infectious diseases

Eradication and elimination are increasingly a part of the global health agenda. Once control measures have driven infection to low levels, the ecology of disease may change posing challenges for eradication efforts. These challenges vary from identifying pockets of susceptibles, improving monitoring during and after the endgame, to quantifying the economics of disease eradication versus sustained control, all of which are shaped and influenced by processes of loss of immunity, susceptible build-up, emergence of resistance, population heterogeneities and non-compliance with control measures. Here we discuss how modelling can be used to address these challenges.

Reconstructing propagation networks with natural diversity and identifying hidden sources

Our ability to uncover complex network structure and dynamics from data is fundamental to understanding and controlling collective dynamics in complex systems. Despite recent progress in this area, reconstructing networks with stochastic dynamical processes from limited time series remains to be an outstanding problem. Here we develop a framework based on compressed sensing to reconstruct complex networks on which stochastic spreading dynamics take place. We apply the methodology to a large number of model and real networks, finding that a full reconstruction of inhomogeneous interactions can be achieved from small amounts of polarized (binary) data, a virtue of compressed sensing. Further, we demonstrate that a hidden source that triggers the spreading process but is externally inaccessible can be ascertained and located with high confidence in the absence of direct routes of propagation from it. Our approach thus establishes a paradigm for tracing and controlling epidemic invasion and information diffusion in complex networked systems.

The structure of many complex systems is usually difficult to determine. Zhesi Shen et al. adapt a signal-processing technique known as compressed sensing to reconstruct the dynamics and structure of a complex propagation network from a small amount of time series data.

Enabling implementation of the Global Vaccine Action Plan: developing investment cases to achieve targets for measles and rubella prevention

Global prevention and control of infectious diseases requires significant investment of financial and human resources and well-functioning leadership and management structures. The reality of competing demands for limited resources leads to trade-offs and questions about the relative value of specific investments. Developing investment cases can help to provide stakeholders with information about the benefits, costs, and risks associated with available options, including examination of social, political, governance, and ethical issues. We describe the process of developing investment cases for globally coordinated management of action plans for measles and rubella as tools for enabling the implementation of the Global Vaccine Action Plan (GVAP). We focus on considerations related to the timing of efforts to achieve measles and rubella goals independently and within the context of ongoing polio eradication efforts, other immunization priorities, and other efforts to control communicable diseases or child survival initiatives. Our analysis suggests that the interactions between the availability and sustainability of financial support, sufficient supplies of vaccines, capacity of vaccine delivery systems, and commitments at all levels will impact the feasibility and timing of achieving national, regional, and global goals. The timing of investments and achievements will determine the net financial and health benefits obtained. The methodology, framing, and assumptions used to characterize net benefits and uncertainties in the investment cases will impact estimates and perceptions about the value of prevention achieved overall by the GVAP. We suggest that appropriately valuing the benefits of investments of measles and rubella prevention will require the use of integrated dynamic disease, economic, risk, and decision analytic models in combination with consideration of qualitative factors, and that synthesizing information in the form of investment cases may help stakeholders manage expectations as they chart the course ahead and navigate the decade of vaccines.

Economic epidemiology of avian influenza on smallholder poultry farms

Highly pathogenic avian influenza (HPAI) is often controlled through culling of poultry. Compensating farmers for culled chickens or ducks facilitates effective culling and control of HPAI. However, ensuing price shifts can create incentives that alter the disease dynamics of HPAI. Farmers control certain aspects of the dynamics by setting a farm size, implementing infection control measures, and determining the age at which poultry are sent to market. Their decisions can be influenced by the market price of poultry which can, in turn, be set by policy makers during an HPAI outbreak. Here, we integrate these economic considerations into an epidemiological model in which epidemiological parameters are determined by an outside agent (the farmer) to maximize profit from poultry sales. Our model exhibits a diversity of behaviors which are sensitive to (i) the ability to identify infected poultry, (ii) the average price of infected poultry, (iii) the basic reproductive number of avian influenza, (iv) the effect of culling on the market price of poultry, (v) the effect of market price on farm size, and (vi) the effect of poultry density on disease transmission. We find that under certain market and epidemiological conditions, culling can increase farm size and the total number of HPAI infections. Our model helps to inform the optimization of public health outcomes that best weigh the balance between public health risk and beneficial economic outcomes for farmers.

Country- and age-specific optimal allocation of dengue vaccines

Several dengue vaccines are under development, and some are expected to become available imminently. Concomitant with the anticipated release of these vaccines, vaccine allocation strategies for dengue-endemic countries in Southeast Asia and Latin America are currently under development. We developed a model of dengue transmission that incorporates the age-specific distributions of dengue burden corresponding to those in Thailand and Brazil, respectively, to determine vaccine allocations that minimize the incidence of dengue hemorrhagic fever, taking into account limited availability of vaccine doses in the initial phase of production. We showed that optimal vaccine allocation strategies vary significantly with the demographic burden of dengue hemorrhagic fever. Consequently, the strategy that is optimal for one country may be sub-optimal for another country. More specifically, we showed that, during the first years following introduction of a dengue vaccine, it is optimal to target children for dengue mass vaccination in Thailand, whereas young adults should be targeted in Brazil.

Evaluating roles of nodes in optimal allocation of vaccines with economic considerations

Since the allocation of vaccines is often constrained by limited resources, designing an economical vaccination strategy is a fundamental goal of the epidemiological modelling. In this study, with the objective of reducing costs, we determine the optimal allocation of vaccines for a general class of infectious diseases that spread mainly via contact. We use an optimization routine to identify the roles of nodes with distinct degrees as depending on the cost of treatment to that of vaccination (relative cost of treatment). The optimal allocation drives vaccination priority to medium-degree nodes at a low relative cost of treatment or to high-degree nodes at a high relative cost of treatment. According to the presented results, we may adjust the vaccination priority in the face of an endemic situation.

Vacated niches, competitive release and the community ecology of pathogen eradication

A recurring theme in the epidemiological literature on disease eradication is that each pathogen occupies an ecological niche, and eradication of one pathogen leaves a vacant niche that favours the emergence of new pathogens to replace it. However, eminent figures have rejected this view unequivocally, stating that there is no basis to fear pathogen replacement and even that pathogen niches do not exist. After exploring the roots of this controversy, I propose resolutions to disputed issues by drawing on broader ecological theory, and advance a new consensus based on robust mechanistic principles. I argue that pathogen eradication (and cessation of vaccination) leads to a 'vacated niche', which could be re-invaded by the original pathogen if introduced. Consequences for other pathogens will vary, with the crucial mechanisms being competitive release, whereby the decline of one species allows its competitors to perform better, and evolutionary adaptation. Hence, eradication can cause a quantitative rise in the incidence of another infection, but whether this leads to emergence as an endemic pathogen depends on additional factors. I focus on the case study of human monkeypox and its rise following smallpox eradication, but also survey how these ideas apply to other pathogens and discuss implications for eradication policy.

Towards the endgame and beyond: complexities and challenges for the elimination of infectious diseases

Successful control measures have interrupted the local transmission of human infectious diseases such as measles, malaria and polio, and saved and improved billions of lives. Similarly, control efforts have massively reduced the incidence of many infectious diseases of animals, such as rabies and rinderpest, with positive benefits for human health and livelihoods across the globe. However, disease elimination has proven an elusive goal, with only one human and one animal pathogen globally eradicated. As elimination targets expand to regional and even global levels, hurdles may emerge within the endgame when infections are circulating at very low levels, turning the last mile of these public health marathons into the longest mile. In this theme issue, we bring together recurring challenges that emerge as we move towards elimination, highlighting the unanticipated consequences of particular ecologies and pathologies of infection, and approaches to their management.

Economic considerations for the eradication endgame

An infectious disease will be eradicated only if it is eliminated everywhere, including in the hardest-to-reach, most vaccine-wary communities. If eradication is successful, it promises a dividend in the form of avoided infections and vaccinations. However, success is never certain unless and until eradication is achieved, and claiming the dividend means bearing the possibly great risk of re-emergence. Economic analysis of eradication evaluates these risks and rewards relative to the alternative of 'optimal control', and also exposes the incentives for achieving and capitalizing on eradication. Eradication is a 'game', because some countries may be willing to eliminate the disease within their borders only if assured that all others will eliminate the disease within their borders. International financing is also a game, because each country would rather free ride than contribute. Finally, for diseases such as polio, capitalizing on eradication is a game, for should any country continue to vaccinate in the post-eradication era using the live-attenuated polio vaccine, the countries that stop vaccinating will be exposed to the risk of vaccine-derived polioviruses. In the framework developed in this paper, eradication is a seductive goal, its attainment fraught with peril.

Efficient control of epidemics spreading on networks: balance between treatment and recovery

We analyse two models describing disease transmission and control on regular and small-world networks. We use simulations to find a control strategy that minimizes the total cost of an outbreak, thus balancing the costs of disease against that of the preventive treatment. The models are similar in their epidemiological part, but differ in how the removed/recovered individuals are treated. The differences in models affect choice of the strategy only for very cheap treatment and slow spreading disease. However for the combinations of parameters that are important from the epidemiological perspective (high infectiousness and expensive treatment) the models give similar results. Moreover, even where the choice of the strategy is different, the total cost spent on controlling the epidemic is very similar for both models.

Optimal vaccine allocation for the early mitigation of pandemic influenza

With new cases of avian influenza H5N1 (H5N1AV) arising frequently, the threat of a new influenza pandemic remains a challenge for public health. Several vaccines have been developed specifically targeting H5N1AV, but their production is limited and only a few million doses are readily available. Because there is an important time lag between the emergence of new pandemic strain and the development and distribution of a vaccine, shortage of vaccine is very likely at the beginning of a pandemic. We coupled a mathematical model with a genetic algorithm to optimally and dynamically distribute vaccine in a network of cities, connected by the airline transportation network. By minimizing the illness attack rate (i.e., the percentage of people in the population who become infected and ill), we focus on optimizing vaccine allocation in a network of 16 cities in Southeast Asia when only a few million doses are available. In our base case, we assume the vaccine is well-matched and vaccination occurs 5 to 10 days after the beginning of the epidemic. The effectiveness of all the vaccination strategies drops off as the timing is delayed or the vaccine is less well-matched. Under the best assumptions, optimal vaccination strategies substantially reduced the illness attack rate, with a maximal reduction in the attack rate of 85%. Furthermore, our results suggest that cooperative strategies where the resources are optimally distributed among the cities perform much better than the strategies where the vaccine is equally distributed among the network, yielding an illness attack rate 17% lower. We show that it is possible to significantly mitigate a more global epidemic with limited quantities of vaccine, provided that the vaccination campaign is extremely fast and it occurs within the first weeks of transmission.

Self-boosting vaccines and their implications for herd immunity

Advances in vaccine technology over the past two centuries have facilitated far-reaching impact in the control of many infections, and today's emerging vaccines could likewise open new opportunities in the control of several diseases. Here we consider the potential, population-level effects of a particular class of emerging vaccines that use specific viral vectors to establish long-term, intermittent antigen presentation within a vaccinated host: in essence, "self-boosting" vaccines. In particular, we use mathematical models to explore the potential role of such vaccines in situations where current immunization raises only relatively short-lived protection. Vaccination programs in such cases are generally limited in their ability to raise lasting herd immunity. Moreover, in certain cases mass vaccination can have the counterproductive effect of allowing an increase in severe disease, through reducing opportunities for immunity to be boosted through natural exposure to infection. Such dynamics have been proposed, for example, in relation to pertussis and varicella-zoster virus. In this context we show how self-boosting vaccines could open qualitatively new opportunities, for example by broadening the effective duration of herd immunity that can be achieved with currently used immunogens. At intermediate rates of self-boosting, these vaccines also alleviate the potential counterproductive effects of mass vaccination, through compensating for losses in natural boosting. Importantly, however, we also show how sufficiently high boosting rates may introduce a new regime of unintended consequences, wherein the unvaccinated bear an increased disease burden. Finally, we discuss important caveats and data needs arising from this work.

Optimizing reactive responses to outbreaks of immunizing infections: balancing case management and vaccination

For vaccine-preventable infections, immunization generally needs to be supplemented by palliative care of individuals missed by the vaccination. Costs and availability of vaccine doses and palliative care vary by disease and by region. In many situations, resources for delivery of palliative care are independent of resources required for vaccination; however we also need to consider the conservative scenario where there is some trade-off between efforts, which is of potential relevance for resource-poor settings. We formulate an SEIR model that includes those two control strategies – vaccination and palliative care. We consider their relative merit and optimal allocation in the context of a highly efficacious vaccine, and under the assumption that palliative care may reduce transmission. We investigate the utility of a range of mixed or pure strategies that can be implemented after an epidemic has started, and look for rule-of-thumb principles of how best to reduce the burden of disease during an acute outbreak over a spectrum of vaccine-preventable infections. Intuitively, we expect the best strategy to initially focus on vaccination, and enhanced palliative care after the infection has peaked, but a number of plausible realistic constraints for control result in important qualifications on the intervention strategy. The time in the epidemic when one should switch strategy depends sensitively on the relative cost of vaccine to palliative care, the available budget, and . Crucially, outbreak response vaccination may be more effective in managing low- diseases, while high scenarios enhance the importance of routine vaccination and case management.

Optimal but unequitable prophylactic distribution of vaccine

The final epidemic size (R(∞)) remains one of the fundamental outcomes of an epidemic, and measures the total number of individuals infected during a "free-fall" epidemic when no additional control action is taken. As such, it provides an idealised measure for optimising control policies before an epidemic arises. Although the generality of formulae for calculating the final epidemic size have been discussed previously, we offer an alternative probabilistic argument and then use this formula to consider the optimal deployment of vaccine in spatially segregated populations that minimises the total number of cases. We show that for a limited stockpile of vaccine, the optimal policy is often to immunise one population to the exclusion of others. However, as greater realism is included, this extreme and arguably unethical policy, is replaced by an optimal strategy where vaccine supply is more evenly spatially distributed.