数学中国 Ww. madio net Better Lioing through Math 135 Better Living through Math: An Analysis of Healthcare Systems Denis Aleshin cc Lampe Parousia rockstroh Harvey Mudd College Claremont, CA Advisor: Darryl Yong Summary Compelled by the great disparities among healthcare sys stems across the ment in stunted healthcare systems. We first establish a framework for discussing and comparing healthcare systems: using data taken from the World Health Organization, we use this framework to rank the systems of the U.S., Sweden, and Nigeria. Our rankings agree with previous studies. Using a probabilistic model incorporating economic factors, we inves- egy to improve its rank. Our results indicate that the U.S. shouldppat. tigate the effects of various changes to the U.S. system and develop a st more emphasis on the prevention of illness, and it should shift toward a more-centralized system so as to make care more accessible to lower- and middle-class individuals Introduction While the U.S. has historically spent more per capita on healthcare than most other countries, the U.S. has seen little improvement in healthcare, and even the U.S. Congress admits that the system is far from the best [1993]. Aithough healthcare is a significant voting issue, Americans re- main confused as to what the remedy for their healthcare should be[hitti 2008]. Additionally, recent problems such as medical tourism-traveling The UMAPJournal29(2)(2008)135-154 @Copyright 2008 by COMAP Inc. Allrightsreserved Permission to make digital or hand copies of part or all of this work for personal or classroom use is granted without fee povided that copies are not made or distributd for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for componerts of this work owned by others than CoMAP must be honored. To copy otherwise, o republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
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数学中国 Ww. madio. net 136 The UMAP Journal 29.2(2008) to foreign countries for healthcare-have reinforced the apparent need for reform[Kher 2006 but uncertainty remains as to what reforms should be implemented We provide a guideline for improving U. S healthcare. We offera frame- york for comparing and predicting various aspects of healthcare systems We define important terms and identify metrics for measuring quality. We use the combined metrics to rank the healthcare systems of the U.S. Nige- ria, and Sweden; these rankings agree with previous literature and support he effectiveness of our metrics We present a predictive model for a healthcare system that can ac- count for different economic classes. Tests run with this model suggest that putting more emphasis on prevention of illness and shifting toward more-centralized healthcare would greatly benefit the U.S Defining Healthcare What is Healthcare? Healthcare is the utilization of medical knowledge with the intent of maintaining or restoring an individuals health of body or mind. A health care system is a network of facilities and workers with the purpose of ad ministering healthcare to a country s population Quality of Healthcare The quality of a healthcare system should reflect how proficient it is t keeping individuals healthy. However, what is considered healthy can change over time, so we define our terms to accommodate changes in med ical opinions ower time The Organization for Economic Co-operation and Development(OECD), a large organization concerned with improving international living stan- dards, defines quality of a national healthcare system as: The degree to which health services for individuals and popula- tions increase the likelihood of desired health outcomes and are con- sistent with current professional knowledge. [2004 A health outcome is a measurable statistic associated with some feature of the overall health of a nation. We take desirable health outcomes to be universal, and we classify a health outcome as desirable or undesirable de- pending on the current consensus of the medical community. For example, an increase in a population'saverage lifespanshould always bedesiredover a decrease, and fewer smokers in a population should always be desired over more smokers [Peto and Lopez. 2000
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数学中国 Ww. madio. net Better Liming fhrough Math 137 Metrics for Assessing Quality We define a metric for the quality of a country s healthcare system as a measurement of something that is capable of impacting a health outcome a desirable metric is associated with a desirable health outcome (e.g,av- erage access to medical care, frequency of contraceptive use, frequency of immunizations), and vice versa for an undesirable metric(e g occurrence of diseases, waiting times for doctors, unaffordable costs). Due to the large differences in how healthcare is provided through out the world, some metrics- especially those impacted by culture or geo graphic conditions-might be inappropriate for comparisons between na- tions. That is, for a metric to be an effective measure of quality, it should measure something thatisimpacted directly by healthsystemsand it should be influenced by as few outside factors as possible Quality Criteri The OECD has identified three primary components of success of any healthcare systen promotion of good health, prevention of illness, and treatment and diagnosis of illness [Kelley and Hurst 2006] Additionally, the OECd has compiled a list of metrics that best measure the quality of each of these components [2004]. We use a slightly modified version of the OECD's description for a healthcare system; we consider a system to consist of the following components Prevention. Since promotion of good health and prevention of illness pri- marilyapply only to healthy populations, we treat these two components for their prevention and promotion o metrics suggested by the OECD mponents [2004] Accessibility. People are kept away from treatment or diagnosis by the ack of proximity of healthcare facilities, unavailability of staff, and the price of care[Feldstein 2006). A healthcare system cannot be effective if it cannot be reached by its population. Metrics for this parameter should measure the systems ability to accommodate people's needs in these Treatment. This component is unchanged from the OECd definition; the quality of this component should be measured by metrics suggested bi the OECD for their treatment and diagnosis component[2004]
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数学中国 Ww. madio. net 138 The UMAP Journal 29.2(2008) Which metrics to Use Two common metrics for healthcare quality are life expectancy and in fant mortality rate, but both are influenced by factors beyond the control of a reasonable healthcare system IO Neill and ONeill 20071. Life expectancy it does not distinguish between treatable causes of death(e. g, disease)and other causes(e.g, war). Similarly, infant mortality rates are strongly influ enced by cultural, social, and educational factors. Because of the outside forces, comparisons made with only these metrics are not reliable IO/Neill and ONeill 2007] We follow guidelines of the OECD, which has concluded that an effective metric is best characterized by three things: First, it [must] capture an important performance aspect [of the healthcare system]. Second, it [must] be scientifically sound. And third, it Imust be] potentially feasible 2004] Data for Metrics The World Health Organization(WHO)offers an abundance of statis- tics relating to healthcare, which are widely believed to be accurate and unbiased. We rely on the wHO as the primary source for health outcomes ssociated with We choose metrics based on the recommendations of OECD [2004]and the availability of data in the wHO database [2008]. we group them by omponent of health, as set out earlier. Prevention Obesity. This metric reflects the emphasis that a healthcare system places on healthy dietary habits as well as the publics desire to adopt those habits. Data for this metric are readily available from the wHo as Adults aged>15 years who are obese. Prevalence of contraceptives. Contraceptives prevent both unwanted preg nancies and the spread of sexually-transmitted diseases. The majority of abortions are performed due to unwanted pregnancies; abortions have substantial long-term consequences in women, both psychologically and medically OECD [2004]. This metric responds to measures taken by a healthcare system to reduce risks of unprotected sex. Data are available from WHO as"contraceptive prevalence rate
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数学中国 Ww. madio. net Better Living through Math 139 Smoking. Reducing smoking has traditionally been the responsibility of healthcare systems. This metric is a measure of how susceptible the pub lic is to beneficial influence from the healthcare system [OECD 2004 Data are available from WHO as"prevalence of current tobacco use among adults aged >15 years. Immunizations. These metrics quantify how proficient healthcare system s at preventing and controlling communicable diseases JOECD 200 WHO offers data for diphtheria, measles, tetanus, hepatitis B, toxoid and pertussis immunizations in one-year-olds WHO 2008]. We take an additional data set for polio immunizations from Earth Trends [n.d. ] Low birth weight. This metric is an indicator of the prenatal care that at- risk mothers receive. It reflects a healthcare systems ability to identify risk factors in patients as well as its capacity for preventing those factors from causing serious harm [OECD 2004]. Data are available from WHO as"low birth weight, newboms." Accessibility Abundance of medical personnel. This indicates the availability of pro- fessionals capable of administering care to the population. The WHo provides several data sets for this metric, induding the proportions of physicians, nurses, midwives, dentists, and pharmacists in the popula tion Abundance of medical facilities. This metric measures the proximity to healthcare systems. Data for this metric is limited; the WHO provides data only for"medical beds per 100,000 population. Affordability forindividuals. This metric measures how much money in- dividuals pay for care. Data for this metric are not directly available from WHO but instead we derive them from its"private spending"and"out of pocket spending Treatment Success of treatments. This metricshould reflect a healthcare systemslevel of care. The OECD suggests using the readmission rates for patient who have suffered congestive heart failure[2004 but these data are not widely available. Hence, we resort to using the"tuberculosis detection rate"and" tuberculosis treatment success"data provided by the WHO as an alternative Meta-Metrics It would be convenient tu tuinbine all the metrius in a meaningful way; we propose an algorithm for computing what we call meta-metrics. Begin
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数学中国 Ww. madio net 140 The UMAP Journal 29.2(2008) by selecting a healthcare component; for each of the metrics corresponding Determine the maximum and minimum values of the metric for a large sampleof countries; if a large samplenot available, then the metriccannot be used reliabl Scale each countrys datum linearly into the interval [0, 1] where the minimum value is mapped to o and the maximum value to I If the metric is undesirable (e. g, prevalence of obesity), subtract the scaled values from 1 to transform the metric into a desirable metric(e.g, lack of obesity). Then calculate the average value of all metrics associated with a country and define this number to be the country smeta-metric value for the chosen healthcare component. A meta-metric represent how well a country performs, on average, rel- ative to the rest of the world for a given healthcare component. A value close to 1 signifies that the component delivers care of the highest quality currently available; a value near O signifies that the country delivers some f the poorest quality care. Because of their compactness, meta-metrics are easy to use for comparisons between existing and potential healthcare systems Comparing Healthcare Systems United States The U.S. is the only developed country that does not employ universal coverage [Torrens 1978]. Instead, healthcare is different for every person, and consists of a loose association of coverage plans provided by private sources,the government and employers. The average middle-class person is usually covered by some sort of insurance and employs a private physi cian in sole charge of managing the individuals healthcare. Physicians exercise substantial influence on the U.S. system, because of their position in healthcare administration, as well as general tendencies of policy to favor private medical practice. This influence leads to the question of whether or not physicians or the federal government should control healthcare. More pressing issues are also troubling the U.S., as the increasing health budget is yielding little advance in the overall quality of care Torrens 1978]. To test the effectiveness of the meta-metrics, we compare several coun- tries for which there is a clear ranking of healthcare already established Based on "financial faimess, " the WHO ranked the healthcare systems of Sweden, the U.S., and Nigeria as 12th, 54th, and 180th in the world [2000bl Meta-metric values, calculated from the metrics and processes described earlier, are given in Table 1
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数学中国 Ww. madio net Better Living through Mat 141 Table 1. Meta-metric US. Sweden Nigeria Accessible Treatment 38 Sweden Sweden operates a nationalized healthcare system that every citizen contributes to based on a proportion of income. As a result, the OECD asserts that citizens enjoy roughly equal benefits, regardless of economic status ITengstam 1975]. The system is heavily regulated and is run by the National Board of Health and Welfare, which is responsible for sup medical care in both the public and private sectors. In addition, this Board is in charge of certifying ph sIclans, nurses, an nd midwives, and also su- pervises and reviews the decisions of the County Councils, where most of the responsibility for funding and maintaining healthcare falls [Tengstam 1975]. Anderson [1972] suggests that in many ways the Swedish system is superior to that of the United States because of Sweden s longstanding commitment to, and enforcement of, universal healthcare eas but treatment. However, the treatment meta-metric is calculated w me Nigeria Nigeria operates a three-tiered health system comprised of a national healthcare system financed by all citizens; government health insurance that is provided for government employees; and firms that contract with private healthcare providers. However, a significant number of Nigerians do not enjoy all the benefits of this system. Like many other African coun- tries, the roots of the Nigerian healthcare system can be traced back to a British colonial era. During this period, the health system was equipped to provide care only asmal portion of the population; the system wasnever adequately adapted to handle the region's growing population[ World Bank 1994. An additional hindrance in the system is an incredible disparity of wealth between upper- and lower-class citizens [World Bank 1994].Exam ples of failures in the health system abound. In one case, a 1985 outbreak of yellow fever devastated a small town(killing more than 1,000 people) despite the fact that a vaccine has been available since 1930[ Vogel 19931 Compared to the U.S. and Sweden, Nigeria'smeta-metrics place it at the ottom
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数学中国 Ww. madio. net 142 The UMAP Journal 29.2 Strengths and Limitations of Meta-Metrics Our meta-metrics demonstrate the following advantages: Flexibility. Additional metrics can be easily incorporated into the meta- metrics Relevance. Meta-metrics convey the average performance of a country's healthcare system relative to the rest of the worl Accuracy. The WHO and our meta-metrics both rank the US, Sweden, and Nigeria in the same relative order. These meta-metrics also demonstrate the following disadvantages Data is not concurrent. Data sets reported by the WHOcan often be several older than other data Demanding Data are required from a large number of countries in order to determine the worldwide maximum and minimum values for metrics Simplicity. It may be wiser to weight the metrics in the calculation of meta metrics instead of taking just their mean A Model for a Healthcare System Assumptions We assume that for a given nation: Wealth is not distributed equally. This is especially true for the U. S. Wolff 2004, which is the focus of most of our attention wHO data for that nation is recent and reliable. This assumption is not entirely valid, since some statistics from the wHO that we use date back to 2000. However, this should be less of a problem as data become more widely and frequently reported. The healthcare system operates in a consistent way. This is not at all true, but for the sake of simplicity we must assume that the system is pre- dictable Meta-metrics accurately reflect the performance of the health system Our results for the U.S, Sweden, and Nigeria support this assumption for all but the treatment meta-metric Certain meta-metrics scale with income. Measures taken by a healthcare system to prevent illness affect all people equally [Torrens 1978]. To ac- count for economic factors, we assume that accessibility and treatment scale with wealth. That is, an individual with more money has an easier time finding care and paying for treatment. This is a gross oversimplifi cation, but it allows the model to convey more information
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数学中国 Ww. madio. net Better Living through Math 143 Definition of the model Let A, T, and P be the countrys accessibility, treatment, and prevention meta-metrics, We treat them as probabilities of certain events occurring within the healthcare system P: the probability that an individual will be in good health; A: the probability that an individual win have access to affordable health care, should they need it; and T: the probability that a sick individual will be correctly diagnosed and properly treated. We model a healthcare system as the stochastic process pictured in Fig- ure 1, and we repeatedly apply this process to track the flow of healthy individuals through the system. Healthy Success 99 Figure L Model of the healtheare process, with four states and probabilities of transitions among If at some time n we have a population of H, healthy individuals, then weexpect Hn (1- P)of those people to fall into poor health in the next time interval. Of those who fall ill, a proportion AT of them will find access to treatment and become healthy. Hence, we predict the number of healthy individuals after n+ l units of time to be Hx+I= H,-H(1-P)+H,(1-P)AT. For an initial healthy population Ho, this simplifies to Hn=H0(P+AT-APT)
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数学中国 Ww. madio. net 144 The UMAP Journal 29.2(2008) Retention of the model To quantify the efficiency ofa healthcare system, we consider how many iterations n of the healthcare process are required before Hn falls below some threshold Hinin. Hence, we substitute H,= Hnin into(1)and solve for n to find the retention R In Hmin In Ho+In(P+ AT-APT) The retention R measures how long the modeled system can operate, start- ing from a healthy population, before an overwhelming majority of the population is no longer healthy. A larger retention value indicates a more ifective system. For all calculations of R, we take Ho and Hmin to be 100 and I Economic Weighting One of the primary discriminatory factors of healthcare in the U.Sis economic status; we would Hke to take this into consideration. To do so we consider three economic dasses Group 1: Those who control the lowest quartile of wealth. Group 2: Those in the middle quartiles for wealth Group 3: Those in the upper quartile for wealth, We adjust the parameters A and T based on the wealth of a group Since our meta-metrics describe the average performance of the system our model-without the economic weightings presented in this section- describes the effect of the system on the average person, "a person of median wealth(hence in Group 2). Analogously, we treat the median per son in the lower quartile as a representative of Group I and the median person in the upper quartile as representative of Group 3 We adjust the probabilities A and T for Group 1 by a factor of C, the ratio of the median wealth of an individual in the lowest quartile to that of the average person. Since wealth in the U.S. is so unevenly distributed, comparing the me- dian individual in the upper quartile to the average person would be mis leading. Instead, we adjust the probabilities A and T for Group 3 by factor of C, which now represents the wealth of the median individual in the upper quartile with respect to the richest person in Group 3. This gives us a weight based on how the wealth is distributed in the upper quartile Simply put, these factors give us a sense of the economic disparity be tween the groups; quality of accessibly and treatment scales with wealth, and C and C, and appropriate scaling factors. We calculate their values in the Appendix
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