Mathematical Modeling and Applications
Mathematical-modeling practice develops the ability to translate a described situation into equations, solve the resulting problem, and judge how much the answer can be trusted.
The modelling cycle
- State assumptions explicitly, name variables and units, formulate, solve, interpret, validate against data, then revise.
- Sensitivity analysis asks how the output moves when a parameter moves; a conclusion that flips under a small change is fragile.
- Distinguish parameters (fixed within a scenario) from variables (solved for).
Standard model families
- Exponential growth/decay: dP/dt = kP → P = P₀e^{kt}; a constant proportional-rate model valid only over an interval where its assumptions are appropriate.
- Logistic: for r > 0 and K > 0, dP/dt = rP(1 − P/K) has equilibria 0 (unstable) and K (stable on the nonnegative population state space); growth from 0 < P₀ < K follows the familiar S-shaped trajectory.
- Linear programming: maximise cᵀx subject to Ax ≤ b, x ≥ 0; for a nonempty bounded feasible polytope, a linear objective attains an optimum at a vertex; unbounded or infeasible cases require separate checks.
- Markov chains: state vector times a transition matrix; the steady state solves πP = π with Σπᵢ = 1.
Worked example
Maximise 3x + 2y subject to x + y ≤ 4, x ≤ 2, y ≤ 3, x, y ≥ 0. Vertices of the feasible region: (0, 0), (2, 0), (2, 2), (1, 3), (0, 3). Objective values: 0, 6, 10, 9, 6. Maximum 10 at (2, 2), where x + y = 4 and x = 2 are both binding while y ≤ 3 has slack 1. Any interior point is a convex combination of vertices, so no interior point can beat the best vertex.
Validation and interpretation
- Inspect residuals for systematic pattern, not just size; a curved residual plot signals a missing term.
- Report units and round to a precision the data supports.
- Correlation is not causation; a confounder can create or mask an association.
- Distinguish interpolation from extrapolation - extrapolated predictions carry model risk.
Traps that cost marks
- Optimising over the wrong region, or missing a vertex created by two intersecting constraints.
- Confusing the logistic growth rate r with the carrying capacity K.
- Fitting a high-degree polynomial to noisy data and calling the wiggle a finding.
- Quoting output with more precision than the inputs justify.
- Forgetting that a steady state must satisfy the normalisation Σπᵢ = 1.