📚 Mathematical Models for Environment, Health and Well-being | 环境、健康与福祉的数学模型
Environmental and public health data are rarely deterministic; they vary from place to place, person to person, and year to year. A-level Mathematics, especially the statistics and modelling strands, gives you the tools to describe this variability, test health claims, and predict the impact of environmental change on human well-being.
环境与公共卫生数据很少是确定性的；它们因地点、人群和年份而变化。A-level 数学，尤其是统计与建模部分，为你提供了描述这种变异性、检验健康声明以及预测环境变化对人类福祉影响的工具。
1. Why Mathematics Matters in Environmental Health | 为什么数学对环境健康至关重要
Environmental health issues such as air pollution, water contamination, and disease outbreaks are studied using quantitative evidence. Mathematical models turn raw measurements into comparable rates, risk estimates, and predictions that can guide policy.
环境健康问题，如空气污染、水污染和疾病暴发，都是通过定量证据来研究的。数学模型将原始测量值转化为可比较的比率、风险估计和预测，从而指导政策。
In the Edexcel specification, skills such as sampling, probability, hypothesis testing, and exponential modelling are directly applicable. A question might give pollutant concentrations and ask you to estimate the mean, test whether a limit is exceeded, or model the decay of a chemical.
在 Edexcel 大纲中，抽样、概率、假设检验和指数建模等技能可直接应用。题目可能给出污染物浓度，要求你估计均值、检验是否超过限值，或对化学物质的衰减建模。
2. Sampling and Data Collection | 抽样与数据收集
Before any model is built, data must be collected in a representative way. Random sampling removes selection bias, while stratified sampling ensures that subgroups such as age bands or districts are included proportionally.
在建立任何模型之前，必须以具有代表性的方式收集数据。随机抽样可消除选择偏差，而分层抽样可确保年龄组或地区等子群体按比例纳入。
For example, a study of lung function in a city might stratify by postcode and age, then randomly select individuals within each stratum. This gives a more reliable estimate than a convenience sample taken from one clinic.
例如，一项城市肺功能研究可能按邮编和年龄分层，然后在每一层内随机选择个体。这比从一家诊所获得的便利样本更可靠。
The sampling frame and sample size also matter. A larger random sample reduces the standard error of the mean, making confidence intervals narrower.
抽样框和样本量也很重要。较大的随机样本会降低均值的标准误差，从而使置信区间更窄。
3. Measures of Location and Spread | 集中趋势与离散程度
The mean, median, and mode summarise a typical value in a dataset, but they are not enough. Public health data often contain outliers, so the median may be more robust than the mean for reporting pollutant exposures.
均值、中位数和众数概括了数据集中的典型值，但这还不够。公共卫生数据通常包含异常值，因此在报告污染物暴露时，中位数可能比均值更稳健。
Measures of spread such as range, interquartile range (IQR), and standard deviation tell us how consistent the measurements are. A small standard deviation in blood lead levels suggests that most children have similar exposure, while a large one indicates inequality.
极差、四分位距（IQR）和标准差等离散程度指标告诉我们测量值的一致性程度。血铅水平的标准差较小，表明大多数儿童的暴露水平相似；标准差较大则表明存在不平等。
Sample standard deviation: s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
The sample standard deviation is given by s = √[ Σ(xᵢ − x̄)² / (n − 1) ], where x̄ is the sample mean and n is the sample size.
样本标准差由 s = √[ Σ(xᵢ − x̄)² / (n − 1) ] 给出，其中 x̄ 是样本均值，n 是样本量。
4. Probability and Risk Communication | 概率与风险沟通
Probability is the language of risk. If a water supply test finds E. coli in 2 out of 50 samples, the observed proportion is 0.04, but the true contamination rate may be different due to sampling variability.
概率是风险的语言。如果供水检测在 50 个样本中发现 2 个大肠杆菌，观测比例为 0.04，但由于抽样变异性，真实污染率可能不同。
Rules such as P(A or B) = P(A) + P(B) − P(A and B) help calculate the chance that at least one of several health risks occurs. Conditional probability is essential when interpreting screening test results.
P(A 或 B) = P(A) + P(B) − P(A 且 B) 等规则有助于计算若干健康风险中至少一个发生的概率。在解释筛查检测结果时，条件概率必不可少。
For example, a positive test for a rare disease may still imply a low probability of actually having the disease if the test has imperfect specificity. This is a common source of public misunderstanding.
例如，如果一种罕见疾病的检测特异性不完善，即使检测结果为阳性，实际患病的概率仍可能很低。这是公众常见的误解来源。
5. Exponential Growth and Decay Models | 指数增长与衰减模型
Many environmental processes are modelled using exponential functions. Pollutant decay in water, drug elimination from the body, and the early growth of an epidemic can all be described by N(t) = N₀e^(−λt) or N(t) = N₀e^(λt).
许多环境过程使用指数函数建模。水中污染物的衰减、药物在体内的消除以及流行病的早期增长都可以用 N(t) = N₀e^(−λt) 或 N(t) = N₀e^(λt) 来描述。
In the decay case, λ is the decay constant, and the half-life is t₁/₂ = ln 2 / λ. If a pesticide has a half-life of 15 days, you can predict the concentration after 45 days by repeated halving or by using logarithms.
在衰减情形下，λ 是衰减常数，半衰期为 t₁/₂ = ln 2 / λ。如果一种农药的半衰期为 15 天，你可以通过反复减半或使用对数来预测 45 天后的浓度。
Exponential models are common in Edexcel exam questions. You may be asked to find λ from data, interpret ln N against t, or decide whether a linear model is more appropriate.
指数模型在 Edexcel 考试题中很常见。你可能需要根据数据求 λ，解释 ln N 对 t 的图形，或判断线性模型是否更合适。
6. Modelling Air Pollution Levels | 空气污染水平建模
Air quality is often reported by an Air Quality Index (AQI) based on concentrations of PM2.5, PM10, NO₂, and ozone. These continuous variables can be summarised by frequency tables, histograms, and cumulative frequency curves.
空气质量通常由基于 PM2.5、PM10、NO₂ 和臭氧浓度的空气质量指数 (AQI) 报告。这些连续变量可以通过频数表、直方图和累积频率曲线进行汇总。
A cumulative frequency graph allows you to estimate medians, quartiles, and percentiles. A percentile such as the 95th percentile of daily PM2.5 concentrations is often used to judge compliance with air quality standards.
累积频率图可以用于估计中位数、四分位数和百分位数。每日 PM2.5 浓度的第 95 百分位数通常用于判断是否符合空气质量标准。
| AQI category | PM2.5 (μg/m³, 24h) | Health advice |
|---|---|---|
| Good | 0-12 | No restrictions |
| Moderate | 12.1-35.4 | Sensitive groups caution |
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