📚 Mathematics for Global Challenges: Modelling, Optimisation and Decision-Making | 数学应对全球挑战：建模、优化与决策
Contemporary global issues such as climate change, pandemics, resource scarcity and disaster relief require rigorous quantitative reasoning. A-level Mathematics provides the core tools: calculus, statistics, probability and decision mathematics. This article explores how these techniques are applied to model, analyse and resolve real-world problems, with Edexcel-style examples and interpretations.
气候变化、流行病、资源短缺和灾害救援等当代全球问题需要严谨的定量推理。A-level 数学提供了微积分、统计、概率和决策数学等核心工具。本文探讨如何运用这些技术对现实问题进行建模、分析和解决，并结合 Edexcel 风格的例题与解释。
1. Why Mathematics Matters in Global Problem-Solving | 数学为何对解决全球问题至关重要
Mathematical models convert vague policy questions into testable statements. For example, predicting the spread of a virus requires defining variables, parameters and assumptions before any data is collected. Without this structure, decisions rely on intuition and may fail under complexity.
数学模型将模糊的政策问题转化为可检验的陈述。例如，预测病毒传播需要在收集数据之前定义变量、参数和假设。没有这种结构，决策只能依赖直觉，在复杂情况下可能失败。
In Edexcel Mathematics, you build this skill through modelling cycles: formulate a model, analyse it mathematically, interpret results and refine assumptions. The same cycle is used by the Intergovernmental Panel on Climate Change (IPCC) and the World Health Organization (WHO).
在 Edexcel 数学中，你通过建模循环培养这一技能：建立模型、进行数学分析、解释结果并修正假设。政府间气候变化专门委员会（IPCC）和世界卫生组织（WHO）也使用同样的循环。
2. Modelling Climate Change with Exponential Functions | 用指数函数模拟气候变化
Atmospheric CO₂ concentration has been approximated by exponential growth over the industrial era. If the concentration C in parts per million satisfies the equation below, where k is the annual growth rate, then a small change in k causes a large long-term difference.
工业化时代以来，大气中二氧化碳浓度近似呈指数增长。如果浓度 C（单位：ppm）满足以下方程，其中 k 为年增长率，那么 k 的微小变化会造成巨大的长期差异。
C = C₀eᵏᵗ
For example, C₀ = 280 ppm in 1750 and C = 420 ppm in 2023 gives 420 = 280e²⁷³ᵏ. Rearranging, k = (1/273)ln(420/280) ≈ 0.00148 per year. This exponential model explains why delaying emission cuts increases the required future reduction.
例如，1750 年 C₀ = 280 ppm，2023 年 C = 420 ppm，得到 420 = 280e²⁷³ᵏ。整理得 k = (1/273)ln(420/280) ≈ 0.00148 / 年。该指数模型解释了为何推迟减排会增加未来所需的削减幅度。
Edexcel Pure Mathematics includes eˣ and natural logarithms, so you can solve problems exactly like this. The same equation appears in compound interest and radioactive decay, showing that one mathematical structure applies across many global issues.
Edexcel 纯数学涵盖 eˣ 和自然对数，因此你可以求解完全类似的问题。同一方程也出现在复利和放射性衰变中，表明一种数学结构适用于多种全球问题。
3. Differential Equations and Population Dynamics | 微分方程与人口动态
The spread of a disease can be modelled by the logistic differential equation, where P is the infected population, r is the transmission rate and K is the carrying capacity of the healthcare system.
疾病传播可以用逻辑斯蒂微分方程来模拟，其中 P 为感染人口，r 为传播率，K 为医疗系统的承载能力。
dP/dt = rP(1 − P/K)
When P is small, the term (1 − P/K) ≈ 1, so growth is approximately exponential: dP/dt ≈ rP. As P approaches K, growth slows, reflecting hospital saturation and behavioural changes. This is why ‘flattening the curve’ means reducing r so that P stays below K.
当 P 很小时，项 (1 − P/K) ≈ 1，因此增长近似为指数：dP/dt ≈ rP。当 P 接近 K 时，增长放缓，反映了医院饱和和行为改变。这就是为什么“压平曲线”意味着降低 r，使 P 始终低于 K。
In Edexcel A-level, you learn to solve separable differential equations and to interpret the solution curve. Although the full logistic solution is not always required, the qualitative analysis of dP/dt against P is a key skill.
在 Edexcel A-level 中，你学习求解可分离变量微分方程并解释解曲线。虽然完整的逻辑斯蒂解不总是要求，但对 dP/dt 关于 P 的定性分析是一项关键技能。
4. Statistical Hypothesis Testing in Public Health | 公共卫生中的统计假设检验
When a new vaccine is tested, researchers set up a null hypothesis H₀: the vaccine has no effect, against an alternative H₁: the vaccine reduces infection. They collect sample data and calculate a test statistic and p-value.
在测试新疫苗时，研究者设立原假设 H₀：疫苗无效，备择假设 H₁：疫苗降低感染。他们收集样本数据并计算检验统计量和 p 值。
If the p-value is less than the significance level, say 0.05, the result is statistically significant and H₀ is rejected. However, a small p-value does not measure the size of the effect; that requires a confidence interval for the difference in infection rates.
如果 p 值小于显著性水平（如 0.05），则结果在统计上显著，拒绝 H₀。然而，小的 p 值并不衡量效应的大小；这需要感染率差值的置信区间。
Edexcel Statistics includes binomial and normal hypothesis tests, including one-tailed and two-tailed tests. These are exactly the tools used by the European Medicines Agency when approving vaccines and treatments.
Edexcel 统计学包括二项分布和正态分布的假设检验，包括单尾和双尾检验。这些正是欧洲药品管理局在批准疫苗和治疗方法时使用的工具。
5. Probability and Risk Assessment for Natural Disasters | 自然灾害的概率与风险评估
Flood defences are designed using return periods. A ‘1-in-100-year’ flood has probability 0.01 of occurring in any given year. The probability of at least one such flood in 30 years is given below.
防洪设施的设计使用重现期。“百年一遇”洪水在任意一年发生的概率为 0.01。30 年内至少发生一次的概率由下式给出。
1 − 0.99³⁰ ≈ 0.260
So the chance is about 26%. This calculation shows that rare events become likely over long time horizons. Engineers therefore combine probability distributions with cost-benefit analysis to choose a design level that balances construction cost and expected damage.
因此概率约为 26%。这一计算表明，稀有事件在较长时间跨度内会变得相当可能。因此工程师将概率分布与成本效益分析相结合，选择在建设成本和预期损失之间取得平衡的设计标准。
In Edexcel Statistics, the binomial distribution B(n, p) and the geometric distribution are used to model such independent trials. You also learn that the mean of B(30, 0.01) is 0.3, which is the expected number of floods in 30 years.
在 Edexcel 统计学中，二项分布 B(n, p) 和几何分布用于模拟此类独立试验。你还会学到 B(30, 0.01) 的均值为 0.3，即 30 年内的预期洪水次数。
6. Linear Programming for Resource Allocation | 资源分配的线性规划
During a humanitarian crisis, aid agencies must allocate limited funds, food and medical supplies to different regions. Linear programming maximises an objective function, such as lives saved, subject to constraints on budgets, transport capacity and minimum needs.
在人道主义危机中，援助机构必须将有限的资金、食品和医疗物资分配到不同地区。线性规划在预算、运输能力和最低需求等约束下最大化目标函数，例如挽救的生命数。
A typical formulation is:
典型形式是：
Maximise Z = 3x + 2y
subject to 2x + y ≤ 100, x + 2y ≤ 80, x ≥ 0, y ≥ 0
The feasible region is a polygon and the optimal solution occurs at a vertex. Edexcel Decision Mathematics 1 covers formulating constraints, drawing the feasible region and using the objective line method.
可行域是多边形，最优解出现在顶点处。Edexcel 决策数学 1 涵盖建立约束、绘制可行域以及使用目标直线法。
This technique is widely used by the UN World Food Programme to plan deliveries of aid under limited budgets and transport capacity.
联合国世界粮食计划署广泛使用这一技术，在预算和运输能力有限的情况下规划援助物资的投送。
7. Network Flows and Supply Chain Resilience | 网络流与供应链韧性
Global supply chains are networks of suppliers, warehouses and ports. Graph theory and maximum flow algorithms identify bottlenecks and the minimum-cut capacity, which is the weakest point of the network.
全球供应链是由供应商、仓库和港口构成的网络。图论和最大流算法能够识别瓶颈以及最小割容量，即网络中最薄弱的环节。
For example, if the maximum flow from a vaccine factory to distribution hubs is 500 units per day but demand is 800, the algorithm shows which edge must be expanded first. This prevents spending on links that do not increase overall capacity.
例如，如果从疫苗工厂到配送中心的最大流量为每天 500 单位，而需求为 800，算法会显示必须先扩建哪条边。这可以防止在无法提升整体容量的链路上浪费资金。
In Edexcel D1, you learn labelling procedures for maximum flow and how to find a minimum cut. These skills are directly relevant to disaster logistics and global trade resilience.
在 Edexcel D1 中，你学习最大流的标号算法以及如何找到最小割。这些技能与灾害物流和全球贸易韧性直接相关。
8. Game Theory and International Cooperation | 博弈论
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