Optimal vaccine allocation for COVID-19 in the Netherlands: A data-driven prioritization

For the control of COVID-19, vaccination programmes provide a long-term solution. The amount of available vaccines is often limited, and thus it is crucial to determine the allocation strategy. While mathematical modelling approaches have been used to find an optimal distribution of vaccines, there is an excessively large number of possible allocation schemes to be simulated. Here, we propose an algorithm to find a near-optimal allocation scheme given an intervention objective such as minimization of new infections, hospitalizations, or deaths, where multiple vaccines are available. The proposed principle for allocating vaccines is to target subgroups with the largest reduction in the outcome of interest. We use an approximation method to reconstruct the age-specific transmission intensity (the next generation matrix), and express the expected impact of vaccinating each subgroup in terms of the observed incidence of infection and force of infection. The proposed approach is firstly evaluated with a simulated epidemic and then applied to the epidemiological data on COVID-19 in the Netherlands. Our results reveal how the optimal allocation depends on the objective of infection control. In the case of COVID-19, if we wish to minimize deaths, the optimal allocation strategy is not efficient for minimizing other outcomes, such as infections. In simulated epidemics, an allocation strategy optimized for an outcome outperforms other strategies such as the allocation from young to old, from old to young, and at random. Our simulations clarify that the current policy in the Netherlands (i.e., allocation from old to young) was concordant with the allocation scheme that minimizes deaths. The proposed method provides an optimal allocation scheme, given routine surveillance data that reflect ongoing transmissions. This approach to allocation is useful for providing plausible simulation scenarios for complex models, which give a more robust basis to determine intervention strategies.     

Investigating associations between COVID-19 mortality and population-level health and socioeconomic indicators in the United States: A modeling study

BackgroundWith the availability of multiple Coronavirus Disease 2019 (COVID-19) vaccines and the predicted shortages in supply for the near future, it is necessary to allocate vaccines in a manner that minimizes severe outcomes, particularly deaths. To date, vaccination strategies in the United States have focused on individual characteristics such as age and occupation. Here, we assess the utility of population-level health and socioeconomic indicators as additional criteria for geographical allocation of vaccines.Methods and findingsCounty-level estimates of 14 indicators associated with COVID-19 mortality were extracted from public data sources. Effect estimates of the individual indicators were calculated with univariate models. Presence of spatial autocorrelation was established using Moran’s I statistic. Spatial simultaneous autoregressive (SAR) models that account for spatial autocorrelation in response and predictors were used to assess (i) the proportion of variance in county-level COVID-19 mortality that can explained by identified health/socioeconomic indicators (R²); and (ii) effect estimates of each predictor.Adjusting for case rates, the selected indicators individually explain 24%–29% of the variability in mortality. Prevalence of chronic kidney disease and proportion of population residing in nursing homes have the highest R². Mortality is estimated to increase by 43 per thousand residents (95% CI: 37–49; p < 0.001) with a 1% increase in the prevalence of chronic kidney disease and by 39 deaths per thousand (95% CI: 34–44; p < 0.001) with 1% increase in population living in nursing homes. SAR models using multiple health/socioeconomic indicators explain 43% of the variability in COVID-19 mortality in US counties, adjusting for case rates. R² was found to be not sensitive to the choice of SAR model form. Study limitations include the use of mortality rates that are not age standardized, a spatial adjacency matrix that does not capture human flows among counties, and insufficient accounting for interaction among predictors.ConclusionsSignificant spatial autocorrelation exists in COVID-19 mortality in the US, and population health/socioeconomic indicators account for a considerable variability in county-level mortality. In the context of vaccine rollout in the US and globally, national and subnational estimates of burden of disease could inform optimal geographical allocation of vaccines.

Sasikiran Kandula and Jeffrey Shaman study population health and COVID-19 mortality in the United States.     

Developing inexpensive malaria vaccines from plants and algae

Malaria is a parasitic, mosquito-borne, infectious disease that threatens nearly half of the global population. The last decade has seen a dramatic drop in the number of malaria-related deaths because of vector control methods and anti-malarial drugs. Unfortunately, this strategy is not sustainable because of the emergence of insecticide-resistant mosquitoes and drug-resistant Plasmodium parasites. Eradication of malaria will ultimately require low-cost easily administered vaccines that work in concert with current control methods. Low cost and ease of administration will be essential components of any vaccine, because malaria endemic regions are poor and often lack an adequate healthcare infrastructure. Recently, several groups have begun addressing these issues using inexpensive photosynthetic organisms for producing vaccine antigens and exploring oral delivery strategies. Immune responses from plant-based injectable malaria vaccines are promising, but attempts to adapt these for oral delivery suggest we are far from a feasible strategy. Here, we review examples of these technologies and discuss the progress and potential of this research, as well as the obstacles ahead.     

SARS: understanding the virus and development of rational therapy

In late 2002 a new disease, severe atypical respiratory syndrome (SARS), emerged in China. A hitherto unknown animal coronavirus (CoV) that had crossed the species barrier through close contact of humans with infected animals was identified as the etiological agent. It rapidly adapted to the new host and not only became readily transmissible between humans but also more pathogenic. Air travel spread it rapidly around the world and ultimately the virus infected 8096 people and caused 774 deaths in 26 countries on 5 continents. Aggressive quarantine measures successfully terminated the disease. Currently, there are no SARS cases recorded and most likely the virus no longer circulates in the human population. In this review we present an overview over SARS-Co virus biology, the disease and discuss strategies to develop antiviral drugs and vaccines.     

Sampling strategy optimization to increase statistical power in landscape genomics: A simulation‐based approach

An increasing number of studies are using landscape genomics to investigate local adaptation in wild and domestic populations. Implementation of this approach requires the sampling phase to consider the complexity of environmental settings and the burden of logistical constraints. These important aspects are often underestimated in the literature dedicated to sampling strategies. In this study, we computed simulated genomic data sets to run against actual environmental data in order to trial landscape genomics experiments under distinct sampling strategies. These strategies differed by design approach (to enhance environmental and/or geographical representativeness at study sites), number of sampling locations and sample sizes. We then evaluated how these elements affected statistical performances (power and false discoveries) under two antithetical demographic scenarios. Our results highlight the importance of selecting an appropriate sample size, which should be modified based on the demographic characteristics of the studied population. For species with limited dispersal, sample sizes above 200 units are generally sufficient to detect most adaptive signals, while in random mating populations this threshold should be increased to 400 units. Furthermore, we describe a design approach that maximizes both environmental and geographical representativeness of sampling sites and show how it systematically outperforms random or regular sampling schemes. Finally, we show that although having more sampling locations (between 40 and 50 sites) increase statistical power and reduce false discovery rate, similar results can be achieved with a moderate number of sites (20 sites). Overall, this study provides valuable guidelines for optimizing sampling strategies for landscape genomics experiments.     

Patterns of molecular evolution and diversification in a biodiversity hotspot: the California Floristic Province

The California Floristic Province harbours more endemic plant and animal taxa and more identifiable subspecies than any other area of comparable size in North America. We present evidence that physical historical processes have resulted in congruent patterns of genetic diversity over the past 2-10 million years. Using a molecular clock approach we show that diversification and establishment of spatial genetic structure across six taxonomic groups coincide with the putative age of California's mountain ranges and aridification in the region. Our results demonstrate the importance of geographical barriers and climatological events to species diversification and the overall geographical structure of biodiversity. These results should facilitate conservation efforts in this biodiversity hotspot for taxa whose population genetic structure is still unknown and may suggest the potential utility of this approach in regional conservation planning efforts.     

北京市2010年手足口病死亡病例流行病学及临床特征分析

目的：通过描述2010年北京市手足口病死亡病例的流行病学特点和临床特点,分析手足口病重症病例发生死亡的危险因素,为防制手足口病,减少手足口病的死亡提供科学依据。方法：采用现况描述及病例对照研究的方法,对2010年北京市手足口病死亡病例的特点进行分析。结果：2010年北京市共报告18例手足口病死亡病例,88．9％分布在5．8月份;死亡病例的男女比例为3．5：1;年龄均小于4岁;流动人口聚集区的死亡病例较多;散居儿童及外地户籍儿童比例高,分别为83．3％和94．4％。死亡病例均出现出疹、发热及神经精神系统症状,整体精神状况,呼吸功能指标以及循环系统受累明显。与重症痊愈病例相比,感染EV71型病毒是发生死亡的危险因素（x2=4．774,P=0．029）。结论：手足口病死亡病例分布与重症病例分布基本一致。应对流动人口聚集地区的4岁以下婴幼儿重症病例进行重点防控。     

Efficient Mitigation Strategies for Epidemics in Rural Regions

Containing an epidemic at its origin is the most desirable mitigation. Epidemics have often originated in rural areas, with rural communities among the first affected. Disease dynamics in rural regions have received limited attention, and results of general studies cannot be directly applied since population densities and human mobility factors are very different in rural regions from those in cities. We create a network model of a rural community in Kansas, USA, by collecting data on the contact patterns and computing rates of contact among a sampled population. We model the impact of different mitigation strategies detecting closely connected groups of people and frequently visited locations. Within those groups and locations, we compare the effectiveness of random and targeted vaccinations using a Susceptible-Exposed-Infected-Recovered compartmental model on the contact network. Our simulations show that the targeted vaccinations of only 10% of the sampled population reduced the size of the epidemic by 34.5%. Additionally, if 10% of the population visiting one of the most popular locations is randomly vaccinated, the epidemic size is reduced by 19%. Our results suggest a new implementation of a highly effective strategy for targeted vaccinations through the use of popular locations in rural communities.     

Chronic Wasting Disease: Transmission Mechanisms and the Possibility of Harvest Management

We develop a model of CWD management by nonselective deer harvest, currently the most feasible approach available for managing CWD in wild populations. We use the model to explore the effects of 6 common harvest strategies on disease prevalence and to identify potential optimal harvest policies for reducing disease prevalence without population collapse. The model includes 4 deer categories (juveniles, adult females, younger adult males, older adult males) that may be harvested at different rates, a food-based carrying capacity, which influences juvenile survival but not adult reproduction or survival, and seasonal force of infection terms for each deer category under differing frequency-dependent transmission dynamics resulting from environmental and direct contact mechanisms. Numerical experiments show that the interval of transmission coefficients β where the disease can be controlled is generally narrow and efficiency of a harvest policy to reduce disease prevalence depends crucially on the details of the disease transmission mechanism, in particular on the intensity of disease transmission to juveniles and the potential differences in the behavior of older and younger males that influence contact rates. Optimal harvest policy to minimize disease prevalence for each of the assumed transmission mechanisms is shown to depend on harvest intensity. Across mechanisms, a harvest that focuses on antlered deer, without distinguishing between age classes reduces disease prevalence most consistently, whereas distinguishing between young and older antlered deer produces higher uncertainty in the harvest effects on disease prevalence. Our results show that, despite uncertainties, a modelling approach can determine classes of harvest strategy that are most likely to be effective in combatting CWD.     

Optimizing one-dose and two-dose cholera vaccine allocation in outbreak settings: A modeling study

BackgroundA global stockpile of oral cholera vaccine (OCV) was established in 2013 for use in outbreak response and are licensed as two-dose regimens. Vaccine availability, however, remains limited. Previous studies have found that a single dose of OCV may provide substantial protection against cholera.MethodsUsing a mathematical model with two age groups paired with optimization algorithms, we determine the optimal vaccination strategy with one and two doses of vaccine to minimize cumulative overall infections, symptomatic infections, and deaths. We explore counterfactual vaccination scenarios in three distinct settings: Maela, the largest refugee camp in Thailand, with high in- and out-migration; N’Djamena, Chad, a densely populated region; and Haiti, where departments are connected by rivers and roads.ResultsOver the short term under limited vaccine supply, the optimal strategies for all objectives prioritize one dose to the older age group (over five years old), irrespective of setting and level of vaccination coverage. As more vaccine becomes available, it is optimal to administer a second dose for long-term protection. With enough vaccine to cover the whole population with one dose, the optimal strategies can avert up to 30% to 90% of deaths and 36% to 92% of symptomatic infections across the three settings over one year. The one-dose optimal strategies can avert 1.2 to 1.8 times as many cases and deaths compared to the standard two-dose strategy.ConclusionsIn an outbreak setting, speedy vaccination campaigns with a single dose of OCV is likely to avert more cases and deaths than a two-dose pro-rata campaign under a limited vaccine supply.     

Assessing the best time interval between doses in a two-dose vaccination regimen to reduce the number of deaths in an ongoing epidemic of SARS-CoV-2

The SARS-CoV-2 pandemic is a major concern all over the world and, as vaccines became available at the end of 2020, optimal vaccination strategies were subjected to intense investigation. Considering their critical role in reducing disease burden, the increasing demand outpacing production, and that most currently approved vaccines follow a two-dose regimen, the cost-effectiveness of delaying the second dose to increment the coverage of the population receiving the first dose is often debated. Finding the best solution is complex due to the trade-off between vaccinating more people with lower level of protection and guaranteeing higher protection to a fewer number of individuals. Here we present a novel extended age-structured SEIR mathematical model that includes a two-dose vaccination schedule with a between-doses delay modelled through delay differential equations and linear optimization of vaccination rates. By maintaining the minimum stock of vaccines under a given production rate, we evaluate the dose interval that minimizes the number of deaths. We found that the best strategy depends on an interplay between the vaccine production rate and the relative efficacy of the first dose. In the scenario of low first-dose efficacy, it is always better to vaccinate the second dose as soon as possible, while for high first-dose efficacy, the best strategy of time window depends on the production rate and also on second-dose efficacy provided by each type of vaccine. We also found that the rate of spread of the infection does not affect significantly the thresholds of the best window, but is an important factor in the absolute number of total deaths. These conclusions point to the need to carefully take into account both vaccine characteristics and roll-out speed to optimize the outcome of vaccination strategies.     

Spatial event cluster detection using a compound Poisson distribution

Geographic disease surveillance methods identify regions that have higher disease rates than expected. These approaches are generally applied to incident or prevalent cases of disease. In some contexts, disease-related events rather than individuals are the appropriate units of analysis for geographic surveillance. We propose a compound Poisson approach that detects event clusters by testing individual areas that may be combined with their nearest neighbors. The method is applicable to situations where the population sizes are diverse and the population distribution by important strata may differ by area. For example, a geographical region might have sparse population in the northern areas, and other areas which are predominantly retirement communities. The approach requires a coarse geographical relationship and administrative data for the numbers of population, cases, and events in each area. Pediatric self-inflicted injuries requiring presentation to Alberta emergency departments provide an illustration.     

Early real-time estimation of the basic reproduction number of emerging or reemerging infectious diseases in a community with heterogeneous contact pattern: Using data from Hong Kong 2009 H1N1 Pandemic Influenza as an illustrative example

Emerging and re-emerging infections such as SARS (2003) and pandemic H1N1 (2009) have caused concern for public health researchers and policy makers due to the increased burden of these diseases on health care systems. This concern has prompted the use of mathematical models to evaluate strategies to control disease spread, making these models invaluable tools to identify optimal intervention strategies. A particularly important quantity in infectious disease epidemiology is the basic reproduction number, R_0. Estimation of this quantity is crucial for effective control responses in the early phase of an epidemic. In our previous study, an approach for estimating the basic reproduction number in real time was developed. This approach uses case notification data and the structure of potential transmission contacts to accurately estimate R₀ from the limited amount of information available at the early stage of an outbreak. Based on this approach, we extend the existing methodology; the most recent method features intra- and inter-age groups contact heterogeneity. Given the number of newly reported cases at the early stage of the outbreak, with parsimony assumptions on removal distribution and infectivity profile of the diseases, experiments to estimate real time R₀ under different levels of intra- and inter-group contact heterogeneity using two age groups are presented. We show that the new method converges more quickly to the actual value of R₀ than the previous one, in particular when there is high-level intra-group and inter-group contact heterogeneity. With the age specific contact patterns, number of newly reported cases, removal distribution, and information about the natural history of the 2009 pandemic influenza in Hong Kong, we also use the extended model to estimate R₀ and age-specific R₀.     

Epidemiological and numerical simulation of rabies spreading from canines to various human populations in mainland China

BackgroundThe mortality of humans due to rabies in China has been declining in recent years, but it is still a significant public health problem. According to the global framework, China strives to achieve the goal of eliminating human rabies before 2030.MethodsWe reviewed the epidemiology of human deaths from rabies in mainland China from 2004 to 2018. We identified high risk regions, age and occupational groups, and used a continuous deterministic susceptibility-exposure-infection-recovery (SEIR) model with periodic transmission rate to explore seasonal rabies prevalence in different human populations. The SEIR model was used to simulate the data of human deaths from rabies reported by the Chinese Center for Disease Control and Prevention (China CDC). We calculated the relative transmission intensity of rabies from canines to different human groups, and they provided a reliable epidemiological basis for further control and prevention of human rabies.ResultsResults showed that human deaths from rabies exhibited regional differences and seasonal characteristics in mainland China. The annual human death from rabies in different regions, age groups and occupational groups decreased steadily across time. Nevertheless, the decreasing rates and the calculated R₀s of canines of various human groups were different. The transmission intensity of rabies from canines to human populations was the highest in the central regions of China, in people over 45 years old, and in farmers.ConclusionsAlthough the annual cases of human deaths from rabies have decreased steadily since 2007, the proportion of human deaths from rabies varies with region, age, gender, and occupation. Further enhancement of public awareness and immunization status in high-risk population groups and blocking the transmission routes of rabies from canines to humans are necessary. The concept of One Health should be abided and human, animal, and environmental health should be considered simultaneously to achieve the goal of eradicating human rabies before 2030.     

Projecting contact matrices in 177 geographical regions: An update and comparison with empirical data for the COVID-19 era

Mathematical models have played a key role in understanding the spread of directly-transmissible infectious diseases such as Coronavirus Disease 2019 (COVID-19), as well as the effectiveness of public health responses. As the risk of contracting directly-transmitted infections depends on who interacts with whom, mathematical models often use contact matrices to characterise the spread of infectious pathogens. These contact matrices are usually generated from diary-based contact surveys. However, the majority of places in the world do not have representative empirical contact studies, so synthetic contact matrices have been constructed using more widely available setting-specific survey data on household, school, classroom, and workplace composition combined with empirical data on contact patterns in Europe. In 2017, the largest set of synthetic contact matrices to date were published for 152 geographical locations. In this study, we update these matrices with the most recent data and extend our analysis to 177 geographical locations. Due to the observed geographic differences within countries, we also quantify contact patterns in rural and urban settings where data is available. Further, we compare both the 2017 and 2020 synthetic matrices to out-of-sample empirically-constructed contact matrices, and explore the effects of using both the empirical and synthetic contact matrices when modelling physical distancing interventions for the COVID-19 pandemic. We found that the synthetic contact matrices show qualitative similarities to the contact patterns in the empirically-constructed contact matrices. Models parameterised with the empirical and synthetic matrices generated similar findings with few differences observed in age groups where the empirical matrices have missing or aggregated age groups. This finding means that synthetic contact matrices may be used in modelling outbreaks in settings for which empirical studies have yet to be conducted.     

The Burden of Cholera in Uganda

Introduction

In 2010, the World Health Organization released a new cholera vaccine position paper, which recommended the use of cholera vaccines in high-risk endemic areas. However, there is a paucity of data on the burden of cholera in endemic countries. This article reviewed available cholera surveillance data from Uganda and assessed the sufficiency of these data to inform country-specific strategies for cholera vaccination.

Methods

The Uganda Ministry of Health conducts cholera surveillance to guide cholera outbreak control activities. This includes reporting the number of cases based on a standardized clinical definition plus systematic laboratory testing of stool samples from suspected cases at the outset and conclusion of outbreaks. This retrospective study analyzes available data by district and by age to estimate incidence rates. Since surveillance activities focus on more severe hospitalized cases and deaths, a sensitivity analysis was conducted to estimate the number of non-severe cases and unrecognized deaths that may not have been captured.

Results

Cholera affected all ages, but the geographic distribution of the disease was very heterogeneous in Uganda. We estimated that an average of about 11,000 cholera cases occurred in Uganda each year, which led to approximately 61–182 deaths. The majority of these cases (81%) occurred in a relatively small number of districts comprising just 24% of Uganda''s total population. These districts included rural areas bordering the Democratic Republic of Congo, South Sudan, and Kenya as well as the slums of Kampala city. When outbreaks occurred, the average duration was about 15 weeks with a range of 4–44 weeks.

Discussion

There is a clear subdivision between high-risk and low-risk districts in Uganda. Vaccination efforts should be focused on the high-risk population. However, enhanced or sentinel surveillance activities should be undertaken to better quantify the endemic disease burden and high-risk populations prior to introducing the vaccine.     

Pneumococci in the African Meningitis Belt: Meningitis Incidence and Carriage Prevalence in Children and Adults

Background

The development of optimal vaccination strategies for pneumococcal conjugate vaccines requires serotype-specific data on disease incidence and carriage prevalence. This information is lacking for the African meningitis belt.

Methods

We conducted hospital-based surveillance of acute bacterial meningitis in an urban and rural population of Burkina Faso during 2007–09. Cerebrospinal fluid was evaluated by polymerase chain reaction for species and serotype. In 2008, nasopharyngeal swabs were obtained from a representative population sample (1 month to 39 years; N = 519) and additional oropharyngeal swabs from 145 participants. Swabs were evaluated by culture.

Results

Annual pneumococcal meningitis incidence rates were highest among <6-month-old (58/100,000) and 15- to 19-year-old persons (15/100,000). Annual serotype 1 incidence was around 5/100,000 in all age groups. Pneumococcal carriage prevalence in nasopharyngeal swabs was 63% among <5-year-old children and 22% among ≥5-year-old persons, but adding oropharyngeal to nasopharyngeal swabs increased the estimated carriage prevalence by 60%. Serotype 1 showed high propensity for invasive disease, particularly among persons aged ≥5 years.

Conclusions

Serotype 1 causes the majority of cases with a relatively constant age-specific incidence. Pneumococcal carriage is common in all age groups including adults. Vaccination programs in this region may need to include older target age groups for optimal impact on disease burden.     

Age-structured non-pharmaceutical interventions for optimal control of COVID-19 epidemic

In an epidemic, individuals can widely differ in the way they spread the infection depending on their age or on the number of days they have been infected for. In the absence of pharmaceutical interventions such as a vaccine or treatment, non-pharmaceutical interventions (e.g. physical or social distancing) are essential to mitigate the pandemic. We develop an original approach to identify the optimal age-stratified control strategy to implement as a function of the time since the onset of the epidemic. This is based on a model with a double continuous structure in terms of host age and time since infection. By applying optimal control theory to this model, we identify a solution that minimizes deaths and costs associated with the implementation of the control strategy itself. We also implement this strategy for three countries with contrasted age distributions (Burkina-Faso, France, and Vietnam). Overall, the optimal strategy varies throughout the epidemic, with a more intense control early on, and depending on host age, with a stronger control for the older population, except in the scenario where the cost associated with the control is low. In the latter scenario, we find strong differences across countries because the control extends to the younger population for France and Vietnam 2 to 3 months after the onset of the epidemic, but not for Burkina Faso. Finally, we show that the optimal control strategy strongly outperforms a constant uniform control exerted over the whole population or over its younger fraction. This improved understanding of the effect of age-based control interventions opens new perspectives for the field, especially for age-based contact tracing.     

Optimal treatment of an SIR epidemic model with time delay

In this paper the optimal control strategies of an SIR (susceptible–infected–recovered) epidemic model with time delay are introduced. In order to do this, we consider an optimally controlled SIR epidemic model with time delay where a control means treatment for infectious hosts. We use optimal control approach to minimize the probability that the infected individuals spread and to maximize the total number of susceptible and recovered individuals. We first derive the basic reproduction number and investigate the dynamical behavior of the controlled SIR epidemic model. We also show the existence of an optimal control for the control system and present numerical simulations on real data regarding the course of Ebola virus in Congo. Our results indicate that a small contact rate(probability of infection) is suitable for eradication of the disease (Ebola virus) and this is one way of optimal treatment strategies for infectious hosts.     

A simple model of linkage disequilibrium and genetic drift in human genomic SNPs: importance of demography and SNP age

We propose a simple model of evolution at a pair of SNP loci, under mutation, genetic drift and recombination. The developed model allows to consider evolution of SNPs under different demographic scenarios. We applied it to SNP data containing polymorphisms spanning 19 gene regions. We initially matched the linkage disequilibrium (LD) data only, and then we reconciled both LD and heterozygosity data. The imbalance between LD and heterozygosity data, observed for some of the analyzed genomic regions, may be a signature of selection acting in these regions. However, assuming neutrality, we obtain estimates of the age of population expansion of modern humans, which are consistent with the consensus estimates. In addition, we are able to estimate the ages of the polymorphisms observed in different genomic regions and we find that they vary widely with respect to their age. Polymorphisms at loci implicated in human disease, seem to be younger than average. Our results supplement the conclusions originally obtained by Reich and co-workers for the same set of data.