📚 Mathematical Modeling: Construction and Application of Models Across Disciplines | 数学建模：跨领域问题的模型构建与应用
Mathematical modeling is the process of translating real-world problems into mathematical expressions, equations, or computational algorithms. It allows us to describe, predict, and control the behavior of complex systems using mathematical tools.
数学建模是将现实中的问题转化为数学表达式、方程或计算方法的过程。它帮助我们运用数学工具来描述、预测并控制复杂系统的行为。
In this article, we explore how mathematical models are constructed and applied across physics, biology, chemistry, finance, medicine, and other fields. We will also discuss the general workflow, calibration, and validation of models, as well as their strengths and limitations.
本文将探讨数学模型如何在物理、生物、化学、金融、医学等领域中构建与应用，同时介绍建模的一般流程、参数校准、模型验证，以及模型的优点与局限性。
1. What Is a Mathematical Model? | 什么是数学模型？
A mathematical model is a simplified representation of a real system, expressed in terms of variables, parameters, and equations.
数学模型是对现实系统的一种简化表示，通常用变量、参数和方程来描述。
For example, the linear relationship between distance and time for an object moving at constant speed can be written as d = vt, where d is distance, v is speed, and t is time.
例如，一个物体以恒定速度运动时，距离与时间的关系可以写为 d = vt，其中 d 表示距离，v 表示速度，t 表示时间。
Models may be deterministic or stochastic, discrete or continuous, linear or nonlinear, depending on the nature of the problem and the assumptions made.
根据问题本质与假设，模型可以是确定性的或随机性的，离散的或连续的，线性的或非线性的。
2. The General Process of Mathematical Modeling | 数学建模的基本步骤
A typical modeling cycle includes the following steps: identifying the real problem, making assumptions, building the model, solving the mathematical problem, comparing the solutions with real data, and refining the model.
典型的建模循环包括以下步骤：识别实际问题、作出假设、建立模型、求解数学问题、将结果与真实数据比较，并不断修正模型。
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Identify the key variables and the relationships between them.
识别关键变量以及它们之间的相互关系。
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State the assumptions clearly so that the model remains tractable yet realistic.
清晰陈述假设条件，使模型既易于处理又具有真实性。
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Formulate the model using functions or differential equations.
利用函数或微分方程来表达模型。
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Solve the model, often with the help of numerical methods or software.
求解模型，通常可以借助数值方法或软件工具。
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Validate the model against independent observations and repeat the cycle if necessary.
用独立观测数据验证模型，必要时重复上述过程。
3. Newton’s Law of Cooling | 牛顿冷却定律：物理中的经典模型
Newton’s law of cooling states that the rate of heat loss of a body is proportional to the difference between its own temperature and the surrounding temperature.
牛顿冷却定律指出，物体散热的速率与其自身温度和环境温度之差成正比。
Let T(t) be the temperature of the body at time t, and let T_s be the surrounding temperature. Then the model is given by the differential equation:
设 T(t) 为物体在时刻 t 的温度，T_s 为环境温度，则该模型的微分方程为：
dT/dt = −k(T − T_s),
where k is a positive constant that depends on the material and surface area.
其中 k 为正的常数，它与物体材料及表面积有关。
Solving this equation gives the temperature as a function of time:
解此方程，可得到温度随时间变化的函数：
T(t) = T_s + (T0 − T_s)e−kt,
where T0 is the initial temperature. This model is widely used in forensic science to estimate time of death.
其中 T0 是初始温度。该模型广泛应用于法医学中推断死亡时间。
4. Population Growth in Biology | 生物学中的种群增长模型
In the simplest exponential growth model, a population grows at a rate proportional to its current size, so the equation is:
在最简单的指数增长模型中，种群增长率与当前种群大小成正比，方程为：
dN/dt = rN,
where N is the number of individuals and r is the intrinsic growth rate.
其中 N 是个体数量，r 是内禀增长率。
The solution is N(t) = N0ert, which implies unbounded growth when r > 0. However, real populations cannot grow forever because resources are limited.
其解为 N(t) = N0ert，说明当 r > 0 时种群会无限增长。然而实际种群由于资源有限不可能一直增长。
To improve the model, the logistic model adds a carrying capacity K:
为改进模型，逻辑斯谛模型引入了环境容纳量 K：
dN/dt = rN(1 − N/K).
This produces an S-shaped curve that stabilizes at N = K, making it more realistic for many ecological systems.
该模型产生一条 S 形曲线，并在 N = K 处趋于稳定，因此对许多生态系统更加贴近现实。
5. Modelling Reaction Rates in Chemistry | 化学反应速率的建模
In chemical kinetics, the rate of a first-order reaction can be modelled using the concentration of a reactant A. The reaction rate is proportional to the concentration:
在化学动力学中，一级反应的速率可以由反应物 A 的浓度来建模。反应速率与浓度成正比：
rate = −d[A]/dt = k[A],
where k is the rate constant and [A] is the molar concentration.
其中 k 是速率常数，[A] 是摩尔浓度。
Integration gives the first-order integrated rate law:
积分后得到一级反应的积分速率方程：
[A](t) = [A]0e−kt,
with [A]0 the initial concentration. This exponential decay equation is also used to model radioactive decay and drug elimination from the body.
这里 [A]0 是初始浓度。这个指数衰减方程同样用于描述放射性衰变与药物在体内的消除过程。
6. Compound Interest in Financial Mathematics | 金融数学中的复利模型
Compound interest is a classic application of exponential growth in finance. If a principal P is invested at an annual interest rate r, compounded n times per year, the amount after t years is:
复利是金融中指数增长的典型应用。若本金为 P，年利率为 r，每年复利 n 次，则 t 年后的金额为：
A(t) = P(1 + r/n)nt.
As n tends to infinity, the formula approaches continuous compounding:
当 n 趋向无穷大时，该公式趋向连续复利：
A(t) = Pert.
This continuous model underpins option pricing, e.g. the Black-Scholes equation, which is fundamental to modern financial mathematics.
这一连续模型是现代金融数学中期权定价的基础，例如 Black-Scholes 方程。
7. Epidemic Modelling: The SIR Model | 传染病建模：SIR 模型
One of the most important biomedical models is the SIR model, which classifies a fixed population into three compartments: susceptible (S), infectious (I), and recovered (R).
最重要的生物医学模型之一是 SIR 模型，它将固定人口划分为三类：易感者（S）、感染者（I）和康复者（R）。
If the population is normalized to size 1, the model is described by the following system:
若将总人口归一化为 1，该模型可用以下方程组表示：
dS/dt = −βSI, dI/dt = βSI − γI, dR/dt = γI,
where β is the transmission rate and γ is the recovery rate.
其中 β 是传播速率，γ 是恢复速率。
A key threshold parameter is the basic reproduction number R0 = β/γ. If R0 > 1, the disease can spread; if R0 < 1, it will eventually die out.
关键阈值参数是基本再生数 R0 = β/γ。若 R0 > 1，疾病能够传播；若 R0 < 1，疫情最终将会消退。
8. Model Calibration and Data Fitting | 模型校准与数据拟合
Model parameters are often unknown and need to be estimated from observed data. This process is called calibration.
模型参数往往是未知的，需要通过观测数据进行估计，这个过程称为校准。
For example, to fit an exponential growth curve N(t) = N0ert to data, we can take logarithms:
例如，要将指数增长曲线 N(t) = N0ert 拟合到数据，可以取对数：
ln N = ln N0 + rt.
This linearizes the model, so linear regression can be used to estimate ln N0 and r.
这使模型线性化，从而可用线性回归来估计 ln N0 和 r。
For more complex nonlinear models, numerical optimization methods such as least squares minimize the sum of squared residuals between observations and predictions.
对于更复杂的非线性模型，通常使用最小二乘等数值优化方法，使观测值与模型预测值之间的残差平方和最小。
9. Model Validation and Sensitivity Analysis | 模型验证与敏感性分析
Validation checks whether a model actually describes reality. This is done by comparing model predictions with data that were not used for calibration.
验证用于检查模型是否真实反映实际，通常将模型预测与未用于校准的数据进行比较。
A common measure in epidemiology modelling is the mean squared error:
在流行病学建模中，常用均方误差作为指标：
MSE = (1/n) Σ (yi − ŷi)2,
where yi are observed values and ŷi are predicted values. A smaller MSE indicates better predictive power.
这里 yi 是观测值，ŷi 是预测值。MSE 越小，说明预测能力越好。
Sensitivity analysis studies how the output of a model is affected by changes in parameters, helping researchers identify which factors most influence the system.
敏感性分析研究参数变化如何影响模型输出，帮助研究者识别对系统影响最大的关键因素。
10. Strengths, Limitations, and Ethical Considerations | 模型的优点、局限与伦理思考
Mathematical models help us make predictions, design experiments, and inform policy decisions. They are often cheaper and safer than physical experiments.
数学模型有助于预测、实验设计以及政策决策。它们通常比实物实验更廉价、更安全。
However, every model is a simplification. If the assumptions are poor, the model may lead to misleading conclusions.
然而，每个模型都具有简化性。如果假设不准确，模型可能产生误导性的结论。
In areas such as epidemiology and finance, an unreliable model may cause serious harm to individuals or society. Therefore, model results must be interpreted carefully and communicated with honesty about uncertainty.
在流行病学和金融等领域，不可靠的模型可能对个人或社会造成严重危害。因此，必须谨慎解读模型结果，并诚实地向公众传达其中的不确定性。
Good mathematical modeling requires not only technical skill, but also critical thinking and ethical responsibility.
优秀的数学建模不仅需要技术技能，还需要批判性思维和伦理责任感。
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