Mathematical Modelling

Introduction

Mathematical modelling is the process of representing a real-world situation with mathematical ideas so that you can describe it, explain it, test hypotheses, make predictions, compare alternatives, or support decisions. A model is not a perfect copy of reality. It is a purposeful simplification that keeps the features that matter for a particular question and leaves out details that are less important.

This aiMOOC is designed for Grades 11–13. You will connect Algebra, Functions, Statistics, Probability, Calculus, differential equations, Numerical analysis, and Optimization with real situations. Some sections use derivatives and differential equations; if your course has not yet covered these topics, you can still work with the graphs, tables, difference equations, and simulations.

Mathematical modelling is used across science, engineering, economics, environmental studies, epidemiology, transport planning, business, and many other fields. The central skill is not simply choosing a formula. It is moving repeatedly between the real situation and mathematics, checking whether the model remains useful for its intended purpose.

Learning Goals

By the end of this aiMOOC, you should be able to:

Model formulation: Translate an open real-world question into variables, parameters, assumptions, constraints, and mathematical relationships.
Model selection: Choose between linear, nonlinear, discrete, continuous, deterministic, stochastic, empirical, and mechanistic approaches.
Data analysis: Fit simple models to data and interpret residuals rather than judging a model only by appearance.
Dynamic system: Explain how difference equations and differential equations describe quantities that change over time.
Simulation: Use numerical or random simulation when an exact symbolic solution is unavailable or unnecessary.
Model validation: Compare predictions with evidence and distinguish calibration from validation.
Sensitivity analysis: Investigate how assumptions and parameter values affect conclusions.
Scientific communication: State limitations, uncertainty, and the range in which a model is credible.

What Is a Mathematical Model?

A mathematical model is an abstract description of a system expressed through quantities and relationships. Those relationships may be equations, inequalities, functions, probability distributions, geometric rules, networks, algorithms, or combinations of these.

A useful model begins with a purpose. For example, a town might ask how changing bus frequency could affect waiting times. A biologist might ask how a population changes when resources are limited. A physicist might predict the path of a projectile. Different purposes can require different models of the same real system.

A strong modelling question is specific enough to guide the mathematics but open enough to require choices. “What is the population?” is mainly a measurement question. “How might the population change during the next five years under two resource scenarios?” is a modelling question because it requires assumptions, relationships, data, and interpretation.

Models as Purposeful Simplifications

Every model omits details. This is not automatically a weakness. Simplification can make the important structure visible. The key question is whether the omitted features matter for the intended use.

For example, an introductory projectile model may ignore air resistance. That model can be useful for moderate speeds and short distances, but it becomes less credible when drag strongly affects the motion. A model should therefore be judged in relation to its domain of validity, not by asking whether it is “completely true”.

Ignoring air resistance and assuming constant gravitational acceleration, a two-dimensional projectile can be described by $x (t) = v_{0} \cos (θ) t$ and $y (t) = y_{0} + v_{0} \sin (θ) t - \frac{1}{2} g t^{2}$ . The variables are position and time; the parameters include the initial speed $v_{0}$ , angle $θ$ , initial height $y_{0}$ , and gravitational acceleration $g$ .

The Modelling Cycle

Mathematical modelling is usually iterative. A first model is often deliberately simple. You test it, learn from its failures, and refine it only when the extra complexity is justified.

A practical modelling cycle is:

Problem definition: State the question, the output you need, and the context in which the answer will be used.
Assumption: Decide which effects to include, which to ignore, and why.
Variable: Define measurable quantities, parameters, units, and constraints.
Mathematical model: Create equations, rules, distributions, or algorithms that connect the quantities.
Mathematical analysis: Solve, approximate, simulate, optimize, or fit the model.
Interpretation: Translate mathematical results back into the original context.
Model validation: Compare outputs with data, known behavior, limiting cases, or independent evidence.
Model refinement: Revise assumptions, variables, parameters, or structure when the model is not adequate.
Communication: Present the model, evidence, uncertainty, limitations, and conclusions clearly.

The arrows in a real modelling cycle go in both directions. If a calculated result is physically impossible, you may return to your assumptions. If a parameter cannot be estimated from available data, you may need a different model. If two models make nearly identical predictions, the simpler one may be preferable when it is easier to explain and test.

From Words to Variables

Before writing equations, define quantities carefully. Distinguish between:

Variable: A quantity that can change within a model, such as time, temperature, or population size.
Parameter: A quantity treated as fixed during one model run but possibly changed between scenarios, such as a growth rate.
Initial condition: A value that specifies the state at the beginning of a dynamic model.
Constraint: A condition that limits possible solutions, such as nonnegative production or a fixed budget.
Output: A quantity the model is intended to estimate, predict, or optimize.

Units are part of the model. If distance is measured in metres and time in seconds, then speed must be expressed consistently in metres per second when used in the same equation. Dimensional analysis is a powerful early error check: terms that are added must have compatible units.

Assumptions Should Be Explicit

Assumptions turn an open real situation into a manageable mathematical problem. Useful assumptions are specific, defensible, and testable where possible. “Traffic is simple” is vague. “During the 15-minute observation interval, the average arrival rate is treated as constant” is a model assumption that can be checked against data.

Assumptions may concern geometry, behavior, rates, independence, measurement quality, boundary conditions, or the time scale of interest. When the conclusion changes strongly after a small change in an assumption, that assumption deserves special attention.

Choosing a Model Type

There is no single best form of model for every problem. The form should match the question, available data, relevant mechanisms, and required precision.

Deterministic and Stochastic Models

A deterministic model produces the same output whenever the inputs and parameters are the same. The projectile equations above are deterministic.

A stochastic model contains randomness explicitly. For example, customer arrivals might be represented as random events rather than a fixed number per minute. Stochastic models are useful when variation itself matters, not merely the average.

Discrete and Continuous Models

A discrete model updates at separate steps, such as daily account balances: $P_{n + 1} = P_{n} (1 + r)$ .

A continuous model treats change as occurring at every instant. Exponential growth can be written $\frac{d P}{d t} = r P$ , whose solution is $P (t) = P_{0} e^{r t}$ when $r$ is constant.

Whether time should be discrete or continuous depends on the system and the required detail. A yearly population model may be adequate for long-term planning, while a motion model may need much smaller time steps.

Empirical and Mechanistic Models

An empirical model is chosen mainly because it matches observed data. A regression line is an example.

A mechanistic model is built from assumptions about how a system works. Newtonian motion and compartment models in epidemiology are examples. Many practical models combine both approaches: a mechanism supplies the structure while data are used to estimate parameters.

Functions, Data, and Model Fitting

Suppose you measure pairs $(x_{i}, y_{i})$ and expect an approximately linear relationship. A simple model is $\hat{y} = a x + b$ . The predicted value is ${\hat{y}}_{i}$ , and the residual is $e_{i} = y_{i} - {\hat{y}}_{i}$ .

A least-squares regression line chooses parameters so that the sum of squared residuals is as small as possible. A small residual is evidence that the fitted value is close to that observation, but a single number such as a fitting score does not prove that the model is appropriate.

Reading Residuals

Residuals should be examined for structure. If positive and negative residuals are scattered without a clear pattern, a linear form may be reasonable. If residuals form a curve, the relationship may be nonlinear. If their spread increases with $x$ , the uncertainty may not be constant.

A model can fit its calibration data well and still predict new data poorly. This is why modelling should include validation with evidence that was not used simply to tune the model whenever such evidence is available.

Growth Models

Growth models show how different assumptions create different long-term behavior.

Exponential Growth

The equation $\frac{d P}{d t} = r P$ assumes that the instantaneous rate of change is proportional to the current population. When $r > 0$ , the population grows exponentially. When $r < 0$ , it decays exponentially.

Exponential growth is useful over intervals where resources or limiting effects are not yet dominant. It should not automatically be extrapolated far beyond the range in which its assumptions are credible.

Logistic Growth

A simple way to include a limiting capacity is the logistic model: $\frac{d P}{d t} = r P (1 - \frac{P}{K})$ , where $K$ is a carrying-capacity parameter. When $P$ is much smaller than $K$ , the model resembles exponential growth. As $P$ approaches $K$ , the growth rate decreases.

Datei:Logistic growth and MSY EN.svg

The parameter $K$ is not necessarily a permanent biological constant. In a real ecosystem, resource availability and environmental conditions can change, so the effective carrying capacity may also change.

Compartment Models and Epidemics

A compartmental model divides a population into groups and describes flows between them. In the classic SIR model, individuals are placed in susceptible, infectious, and removed compartments.

Datei:Simple SIR model.svg

A common continuous SIR model is $\frac{d S}{d t} = - β \frac{S I}{N}$ , $\frac{d I}{d t} = β \frac{S I}{N} - γ I$ , and $\frac{d R}{d t} = γ I$ , with $N = S + I + R$ under the assumption of a closed population.

The parameters $β$ and $γ$ summarize transmission and removal processes. This model assumes homogeneous mixing and simplified disease progression. Real epidemic modelling often requires additional compartments, changing contact patterns, uncertainty, age structure, or spatial effects. The value of the simple SIR model is that it makes a mechanism visible and gives you a foundation for asking which assumptions matter.

Numerical Modelling

Many useful models cannot be solved neatly by algebra or calculus. Numerical methods approximate their behavior with repeated calculations.

Euler's Method

For a differential equation $\frac{d y}{d t} = f (t, y)$ , Euler's method uses the update $y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$ , where $h$ is the step size.

Datei:Euler method.svg

A smaller step size often reduces discretization error, but it increases the number of calculations. A numerical result should therefore be checked for convergence: if reducing the step size changes the answer very little, confidence in the numerical approximation increases.

Simulation and Monte Carlo Methods

A computer simulation runs a model repeatedly or over many time steps so that you can explore behavior that is difficult to obtain analytically.

A Monte Carlo method uses random sampling. For example, if random points are placed uniformly in a square containing a quarter-circle of radius one, then $π \approx 4 \frac{N_{inside}}{N_{total}}$ . The estimate fluctuates because the method is stochastic, but the variability generally decreases as the sample size becomes large.

Datei:Monte-Carlo method pi.svg

Simulation is not a substitute for thinking about assumptions. A computer can calculate the consequences of a model very quickly, but it cannot make an unjustified model valid.

Calibration, Validation, and Error

Calibration means estimating or tuning model parameters using data. Validation means checking whether the model performs adequately for its intended purpose using independent observations, known behavior, or tests not used merely to fit the model.

Model discrepancy can come from several sources:

Measurement error: Data are imperfect or noisy.
Parameter uncertainty: Parameter values are not known exactly.
Structural uncertainty: The equations or rules may omit important mechanisms.
Numerical error: Approximation methods introduce computational error.
Scenario uncertainty: Future inputs or external conditions are unknown.

Do not merge these ideas into one vague “error percentage”. Different uncertainties require different responses.

Validation Strategies

Useful checks include:

Residual analysis: Look for systematic patterns in the differences between observations and predictions.
Cross-validation: Fit a model on one subset of data and evaluate it on another when the data structure permits.
Extreme value: Ask whether the model behaves sensibly at very small, very large, or boundary values.
Conservation law: Check whether quantities such as mass, probability, or total population are conserved when the model requires it.
Benchmark: Compare with a simpler model, a known exact result, or an accepted reference case.
Out-of-sample prediction: Test predictions on observations not used for calibration.

Sensitivity and Uncertainty

Sensitivity analysis asks how much model outputs change when inputs, parameters, or assumptions change. A parameter is influential if small plausible changes in it produce important changes in the result.

One simple method is to vary one parameter at a time around a baseline value while keeping the others fixed. More advanced approaches vary several inputs together or assign probability distributions to uncertain parameters.

Sensitivity is especially important for decisions. If two policy choices reverse order after a tiny plausible change in one parameter, the model does not support a robust preference. If the same option remains best across a wide range of credible assumptions, the conclusion is more robust.

Optimization and Decision Models

Some models are built not only to predict but also to choose. An optimization model contains:

Decision variable: Quantities you can control.
Objective function: A quantity to minimize or maximize.
Constraint: Requirements that feasible solutions must satisfy.

For example, a school might minimize the cost of a meal plan while satisfying nutritional and supply constraints. A transport planner might minimize travel time or emissions subject to capacity limits.

Optimization does not decide what society should value. The objective function represents chosen priorities. Changing those priorities can change the mathematical optimum, so the modelling report must make them explicit.

Worked Example: Braking Distance

Consider an idealized model for the distance needed to perceive a hazard, react, and stop a vehicle on a level road. A simple model is $d (v) = v t_{r} + \frac{v^{2}}{2 μ g}$ , where $v$ is speed in metres per second, $t_{r}$ is reaction time, $μ$ is an effective friction coefficient, and $g$ is gravitational acceleration.

The first term $v t_{r}$ models reaction distance. The second term comes from constant-deceleration mechanics under a simplified friction assumption.

A modelling study could proceed as follows:

Question: Estimate how total stopping distance changes with speed under specified dry-road assumptions.
Assumption: Treat the road as level, $μ$ as constant, braking as uniform, and the driver's reaction time as fixed during one scenario.
Parameter estimation: Obtain plausible values or ranges for $t_{r}$ and $μ$ from suitable evidence.
Calculation: Evaluate $d (v)$ across a range of speeds.
Sensitivity analysis: Repeat the calculations for several plausible reaction times and friction values.
Validation: Compare model outputs with trustworthy braking-distance measurements collected under comparable conditions.
Interpretation: Explain that the quadratic braking term causes stopping distance to grow faster than speed.
Limitation: State that gradients, tyres, weather, brake condition, driver behavior, and other factors can make real stopping distances different.

This is an educational model, not a safety calculator. Its value is in showing how assumptions, units, mechanisms, parameter uncertainty, and validation fit together.

Communicating a Model

A high-quality modelling report should allow another reader to reconstruct your reasoning. Include the question, data sources, variable definitions, units, assumptions, equations, parameter choices, solution method, validation, sensitivity analysis, limitations, and conclusion.

Graphs should have labelled axes and units. Tables should state what each column means. Parameters should not appear without definitions. If a result is uncertain, communicate a range or scenario comparison rather than giving unjustified precision.

The strongest conclusion is often conditional: given these assumptions and this evidence, the model suggests... That wording makes clear what the mathematics supports and what still depends on the real-world context.

Responsible Use of Models

Models can influence decisions about health, resources, infrastructure, education, finance, and the environment. You should therefore ask who is represented in the data, who may be affected by the decision, which outcomes are optimized, and which effects are omitted.

A technically correct calculation can still be misleading if the wrong question was modelled, the data are unrepresentative, uncertainty is hidden, or a result is extrapolated far beyond the calibration range. Responsible modelling combines mathematics with transparency, evidence, domain knowledge, and ethical judgement.

Interactive Tasks

Quiz: Test Your Knowledge

What is the main purpose of a mathematical model? (To represent selected features of a real situation for analysis) (!To reproduce every detail of reality exactly) (!To remove the need for assumptions) (!To guarantee a correct prediction)

Which statement best describes a parameter? (A quantity treated as fixed during one model run) (!A quantity that must always be the model output) (!A graph drawn after every calculation) (!A measurement unit with no numerical value)

What is a residual in a fitted model? (The observed value minus the predicted value) (!The predicted value divided by the observed value) (!The largest parameter in the model) (!The difference between two units)

What does carrying capacity represent in the simple logistic model? (A limiting population level built into the model) (!The initial population at time zero) (!The slope of every tangent to the curve) (!The number of simulation runs)

What is validation mainly used for? (To check whether a model performs adequately for its purpose) (!To make every model deterministic) (!To remove all measurement uncertainty) (!To replace data with assumptions)

What makes a Monte Carlo method stochastic? (It uses random sampling) (!It always uses a straight line) (!It avoids repeated calculations) (!It requires an exact symbolic solution)

Why is dimensional consistency useful? (It helps detect equations that combine incompatible units) (!It guarantees that all parameters are known) (!It proves that a model is ethically acceptable) (!It makes validation unnecessary)

What does Euler's method do? (It approximates a differential equation step by step) (!It finds a regression line without data) (!It converts every stochastic model into a deterministic one) (!It removes all numerical error)

What is the goal of sensitivity analysis? (To study how outputs respond to changes in inputs or assumptions) (!To force all model outputs to be identical) (!To choose units after the calculation) (!To hide uncertain parameters)

Why should assumptions be stated explicitly? (So readers can judge and test the basis of the model) (!So the model no longer needs evidence) (!So every prediction becomes exact) (!So parameters cannot be changed)

Memory Game

Variable	Quantity allowed to change within the model
Parameter	Quantity treated as fixed during one run
Residual	Difference between observed and predicted values
Calibration	Process of estimating numerical settings from data
Validation	Check of model performance against relevant evidence
Simulation	Numerical execution used to explore model behavior

Drag and Drop

Match the correct terms.	Topic
Define the problem	State the real-world question and required output
Make assumptions	Select important effects and simplify the situation
Formulate the model	Express relationships with mathematical rules
Validate the result	Compare model output with relevant evidence
Refine the model	Revise assumptions or structure after evaluation

...

Crossword Puzzle

Assumption	What is a stated simplification used to make a real problem manageable?
Variable	What quantity can change within a mathematical model?
Parameter	What quantity controls model behavior and is often estimated from data?
Residual	What is the difference between an observed value and its prediction?
Calibration	What process estimates model parameters using data?
Validation	What process checks whether a model performs adequately for its intended use?

LearningApps

Cloze Text

A mathematical model is a purposeful

of a real situation. The modelling process begins by defining a clear

. Quantities that change within the model are called

. Quantities treated as fixed during one run are often called

. The difference between an observed and predicted value is a

. Estimating parameter values from data is called

. Checking model performance against relevant evidence is called

. Studying how conclusions change when inputs or assumptions change is called

. Random-sampling models are described as

. A modelling report should communicate uncertainty and

.

Open-Ended Tasks

Easy

Household water model: Record or obtain a small set of household water-use data, define one question, build a simple linear model, and explain two assumptions.
Cooling graph: Measure the temperature of a warm drink at regular intervals, plot the data, describe the shape, and propose a function that might model the trend.
Journey estimate: Model the travel time for a familiar route using distance and average speed, then identify at least three reasons why the real time can differ.
Model infographic: Create an image or one-page poster that explains the modelling cycle using your own real-world example and clear arrows between stages.

Standard

Population comparison: Use a spreadsheet or graphing tool to compare exponential and logistic growth for the same initial population and explain when their predictions begin to diverge.
Regression investigation: Collect paired data, fit a linear model, plot residuals, and decide whether a straight-line model is defensible.
Projectile experiment: Record or safely measure a simple projectile trajectory, compare observations with a parabolic model, and discuss the effect of neglected air resistance.
Interview a modeller: Interview someone in science, engineering, economics, data analysis, or another quantitative profession about how assumptions and validation enter their work, then summarize what you learned.

Advanced

SIR simulation: Implement a discrete or numerical SIR-type model, vary transmission and removal parameters, and explain which conclusions are robust and which are sensitive.
Monte Carlo project: Build a Monte Carlo simulation for a probability problem, repeat it with increasing sample sizes, and analyze how the estimate stabilizes.
Optimization study: Formulate a real allocation problem with decision variables, an objective, and constraints, solve it with appropriate software or mathematics, and discuss what the objective leaves out.
Model audit video: Produce a short video that critiques a published or publicly discussed mathematical model by examining its purpose, assumptions, data, uncertainty, validation, and possible social consequences.

Learning Assessment

Model comparison assessment: Given the same data set, construct a linear model and one plausible nonlinear alternative, compare residual behavior, and justify which model you would use for interpolation.
Assumption stress test: Choose a model from this course, identify three important assumptions, vary each assumption in a plausible way, and explain which change matters most for the conclusion.
Validation design: Design a validation plan for a model that predicts daily school-canteen demand, including what data you would hold back, what errors you would examine, and what would count as acceptable performance.
Dynamic model transfer: Adapt an exponential or logistic model to a new context such as battery discharge, rumor spread, resource growth, or product adoption, and justify every variable and parameter.
Numerical reasoning: Use Euler's method with two different step sizes for the same differential equation, compare the approximations, and explain what the comparison suggests about numerical error.
Decision model critique: Analyze an optimization result and show how the recommended decision changes when the objective, a constraint, or an uncertain parameter is changed.

Evidence of Learning

Knowledge: You can explain the modelling cycle, distinguish variables from parameters, compare common model types, and describe calibration, validation, residuals, uncertainty, sensitivity, simulation, and optimization.

Skills: You can translate a real problem into mathematics, check units, fit simple functions, interpret residuals, solve or approximate dynamic models, design numerical experiments, test sensitivity, and evaluate model credibility.

Products: Strong evidence may include a modelling report, annotated graph, spreadsheet, simulation, code notebook, poster, presentation, experimental data set, or explanatory video in which assumptions and limitations are explicit.

Reasoning: You can justify why a chosen model is suitable for a specific purpose, explain why another model might also be reasonable, and support conclusions with calculations and evidence rather than appearance alone.

Transfer: You can apply the modelling process to an unfamiliar problem, revise a model when evidence conflicts with predictions, and communicate when a result should or should not be used for a real decision.

OERs on the Topic

Useful open resources include:

What is Math Modeling? by the SIAM M3 Challenge: A student-oriented overview of the iterative modelling process.
Mathematical Modeling Basics by TU Delft OpenCourseWare: A course that develops modelling through differential equations, numerical methods, validation, and scientific reporting.

The following English Wikipedia article provides a broad reference on mathematical models and their applications:

You can also explore the Mathematical model, Scientific modelling, Computer simulation, Regression analysis, Logistic function, Compartmental models in epidemiology, Euler method, Monte Carlo method, Sensitivity analysis, and Mathematical optimization topics for deeper study.

Linked Learning Areas

Mathematical Modelling

Mathematical modelling links school mathematics with Physics, Biology, Computer science, Economics, Engineering, Environmental science, and Data science. It is especially useful when you must combine quantitative reasoning with evidence, assumptions, technology, and communication.

aiMOOC Projects

Diesen aiMOOC als Battle spielen: Mathematik, Klasse 11-13, G-Niveau

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