Mathematical Modeling
A course in building, analyzing and testing mathematical models of real systems, organized around three running case studies: the walleye fishery on a northern lake, an influenza season in a city of half a million, and bunching on a busy bus route. It covers the modeling cycle from question and assumptions to validation, dimensional analysis and the Buckingham pi theorem, scaling, nondimensionalization and perturbation methods for small parameters, difference equations with cobweb diagrams, stability, period doubling and chaos, the logistic equation, phase lines and the saddle-node bifurcation behind maximum sustainable yield, phase plane analysis of predator–prey and competition models, SIR compartment models with their threshold, peak and final size, optimization with the square-root law, Lagrange multipliers, linear programming and shadow prices, Markov chains, branching processes and Monte Carlo simulation, linear and nonlinear least squares with standard errors, identifiability and model selection, conservation laws for diffusion and traffic flow, and a capstone on verification, validation, sensitivity analysis, uncertainty propagation and writing the modeling report, with runnable Python examples throughout.
MATH-424 Year 4 3 credits 12 lessons · 4 h 36 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
This is the American version of this course. See the Canadian version.
What you'll learn
- Describe the modeling cycle, turn a practical question into a precise mathematical problem with named variables, parameters, controls, units and stated assumptions, and classify models as deterministic or stochastic, discrete or continuous, lumped or distributed, mechanistic or empirical.
- Check dimensional homogeneity, find dimensionless groups as the null space of a dimension matrix, and apply the Buckingham pi theorem to pendulums, water waves, drag and model testing.
- Nondimensionalize differential equation models by choosing characteristic scales, identify the dimensionless parameters that remain, construct regular perturbation expansions for small parameters, and recognize singular problems with boundary layers.
- Solve linear and affine difference equations, find fixed points of nonlinear maps and test their stability, draw cobweb diagrams, and explain period doubling and chaos in the Ricker map and the instability of bus headways.
- Solve the logistic equation, analyze autonomous equations with the phase line, explain the saddle-node bifurcation and the maximum sustainable yield of a harvested population, and compare the robustness of quota and effort policies.
- Analyze two-dimensional autonomous systems with nullclines and Jacobian linearization, classify equilibria by trace and determinant, and apply these tools to predator–prey and competition models.
- Formulate compartment models from flow diagrams, derive the epidemic threshold, peak and final size of the SIR model, and evaluate vaccination and the timing of temporary interventions.
- Formulate optimization models, derive the square-root law for headways, use Lagrange multipliers and shadow prices, solve small linear programs at the vertices of the feasible region, and analyze the economics of open-access fisheries.
- Build and analyze Markov chains and their stationary distributions, compute outbreak probabilities with branching processes, and design Monte Carlo simulations with stated accuracy.
- Estimate parameters by linear and nonlinear least squares, compute approximate standard errors, interpret residual plots, and address identifiability and overfitting with held-out data and the Akaike information criterion.
- Derive conservation laws with constitutive relations, solve and simulate the diffusion equation with attention to numerical stability, and analyze traffic flow with characteristics and shocks.
- Distinguish verification from validation, carry out local and global sensitivity analysis and uncertainty propagation, assess structural uncertainty, and write a complete modeling report with a defensible recommendation.
Course content
12 lessons · 4 h 36 min total · final exam
- 1 The modeling cycle: questions, assumptions and variables 27 min · quiz
- 2 Dimensional analysis and the Buckingham pi theorem 25 min · quiz
- 3 Scaling, nondimensionalization and small parameters 23 min · quiz
- 4 Discrete-time models: difference equations and cobwebs 24 min · quiz
- 5 Continuous population models: logistic growth and harvesting 20 min · quiz
- 6 Phase plane analysis: predator–prey and competition models 23 min · quiz
- 7 Compartment models and the spread of epidemics 22 min · quiz
- 8 Optimization models: calculus, constraints and linear programming 22 min · quiz
- 9 Stochastic models: Markov chains and Monte Carlo simulation 22 min · quiz
- 10 Fitting models to data: least squares and parameter estimation 23 min · quiz
- 11 Conservation laws: diffusion and traffic flow 24 min · quiz
- 12 Capstone: validation, sensitivity analysis and the modeling report 21 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A course in building, analyzing and testing mathematical models of real systems, organized around three running case studies: the walleye fishery on a northern lake, an influenza season in a city of half a million, and bunching on a busy bus route. It covers the modeling cycle from question and assumptions to validation, dimensional analysis and the Buckingham pi theorem, scaling, nondimensionalization and perturbation methods for small parameters, difference equations with cobweb diagrams, stability, period doubling and chaos, the logistic equation, phase lines and the saddle-node bifurcation behind maximum sustainable yield, phase plane analysis of predator–prey and competition models, SIR compartment models with their threshold, peak and final size, optimization with the square-root law, Lagrange multipliers, linear programming and shadow prices, Markov chains, branching processes and Monte Carlo simulation, linear and nonlinear least squares with standard errors, identifiability and model selection, conservation laws for diffusion and traffic flow, and a capstone on verification, validation, sensitivity analysis, uncertainty propagation and writing the modeling report, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 4.