mathematical-modeling

Mathematical Modeling

Mathematical modeling is the art and science of translating real-world phenomena into mathematical language, analyzing the resulting mathematical system, and translating the conclusions back into practical insights. It is applied mathematics at its most consequential — the bridge between abstract theory and physical, biological, economic, and engineering reality. This skill covers the complete modeling cycle, standard model types, optimization, probabilistic models, and the critical practice of model criticism.

Agent affinity: euler (analysis, differential equations, optimization), polya (problem-solving strategy, heuristic reasoning)

Concept IDs: math-functions, math-correlation-causation, math-descriptive-statistics, math-probability-foundations

Part I — The Modeling Cycle

The Six-Stage Framework

Mathematical modeling is iterative, not linear. The cycle repeats until the model is fit for purpose.

1. PROBLEM IDENTIFICATION
   What question are we answering? What decisions depend on the answer?
   
2. ASSUMPTIONS
   What simplifications make the problem tractable? What is ignored?
   
3. FORMULATION
   Translate assumptions into equations, inequalities, or algorithms.
   
4. ANALYSIS
   Solve the mathematical system (exactly or numerically).
   
5. VALIDATION
   Compare model predictions against real data. Quantify error.
   
6. INTERPRETATION
   Translate mathematical conclusions into actionable real-world answers.
   Report limitations and uncertainty.

The most common failure mode is skipping stage 2 (assumptions) and stage 5 (validation). A model without explicit assumptions is untestable; a model without validation is fiction.

Polya's Framework Adapted for Modeling

George Polya's How to Solve It (1945) provides four phases that map directly to modeling:

1. Understand the problem. What is known? What is unknown? What are the constraints? Draw a picture. Introduce notation. Can you restate the problem in your own words?

2. Devise a plan. Have you seen a similar problem? Can you solve a simpler version first? What model type fits the phenomenon? What variables matter most?

3. Carry out the plan. Execute the mathematical analysis. Check each step. If stuck, go back to step 2.

4. Look back. Is the answer reasonable? Can you verify it by a different method? Can you generalize? What are the limitations?

Polya's emphasis on looking back corresponds directly to the validation and interpretation stages. Most student modelers and many professional ones skip this step, producing models that are mathematically correct but practically useless.

Part II — Common Model Types

Linear Models

y = mx + b. The simplest model, appropriate when the rate of change is constant.

When to use. Short-term predictions, small perturbations, situations where the relationship is approximately proportional. Linear models are the first approximation to any smooth function (Taylor's theorem guarantees this locally).

Worked example. A company's revenue has been $2M in 2020, $2.4M in 2021, $2.8M in 2022, $3.2M in 2023. Project revenue for 2025.

The data increases by $0.4M/year. Model: R(t) = 2 + 0.4(t - 2020) million dollars. For 2025: R(2025) = 2 + 0.4(5) = $4M.

Limitation. Linear growth cannot continue indefinitely in bounded systems (market size, resource constraints). Always state the range of validity.

Exponential Models

y = A * e^(kt). Appropriate when the rate of change is proportional to the current value: dy/dt = ky.

When to use. Unrestricted growth or decay: radioactive decay (k < 0), compound interest, early-stage population growth, viral spread in early phases.

Worked example. A bacterial culture doubles every 3 hours starting from 500 cells. How many cells after 12 hours?

Doubling time T = 3 hours means k = ln(2)/T = ln(2)/3. Model: N(t) = 500 * e^(t*ln(2)/3) = 500 2^(t/3). At t = 12: N(12) = 500 2^4 = 8000 cells.

Logistic Models

y = K / (1 + A * e^(-rt)), where K is the carrying capacity.

When to use. Growth that saturates: population with limited resources, technology adoption (S-curve), disease spread with herd immunity. The logistic model transitions from exponential growth (small y) to saturation (y near K).

Derivation. The logistic ODE is dy/dt = r*y*(1 - y/K). When y << K, this is approximately dy/dt = ry (exponential). As y approaches K, the growth rate drops to zero.

Worked example. A rumor spreads in a school of 1000 students. Initially 10 know it, and after 2 days 100 know it. Model the spread.

Logistic model: N(t) = 1000 / (1 + 99*e^(-rt)). At t = 0: N(0) = 1000/100 = 10. Check. At t = 2: 100 = 1000/(1 + 99*e^(-2r)). Solve: 1 + 99*e^(-2r) = 10, so e^(-2r) = 9/99 = 1/11. Thus r = ln(11)/2 approximately 1.20.

At t = 5: N(5) = 1000/(1 + 99*e^(-6.0)) approximately 1000/(1 + 99*0.00248) approximately 1000/1.245 approximately 803 students.

Periodic Models

y = A*sin(omega*t + phi) + C. Appropriate for oscillating phenomena.

When to use. Seasonal patterns (temperature, sales cycles), wave phenomena, circadian rhythms, alternating current.

Parameters. A = amplitude, omega = angular frequency (omega = 2*pi/period), phi = phase shift, C = vertical offset (baseline).

Power-Law Models

y = C * x^alpha. On a log-log plot, power laws appear as straight lines with slope alpha.

When to use. Scaling relationships in physics (gravity: F proportional to r^(-2)), biology (metabolic rate scales as mass^(3/4), Kleiber's law), network science (degree distributions), linguistics (Zipf's law: word frequency proportional to rank^(-1)).

Detecting power laws. Plot log(y) vs. log(x). A linear relationship indicates a power law. However, many claimed power laws in empirical data do not survive rigorous statistical testing (Clauset, Shalizi, and Newman, 2009).

Part III — Dimensional Analysis

The Buckingham Pi Theorem

If a physical relationship involves n variables with k independent dimensions (length, mass, time, etc.), then the relationship can be expressed in terms of n - k dimensionless groups (Pi groups).

Worked example. Find the period T of a simple pendulum.

Variables: T (period), L (length), m (mass), g (gravitational acceleration). Dimensions: T = [T], L = [L], m = [M], g = [L*T^(-2)]. Number of variables: n = 4. Independent dimensions: k = 3 (M, L, T). Number of Pi groups: n - k = 1.

The single dimensionless group must be: Pi = T sqrt(g/L) (verify: [T] [L^(1/2) T^(-1) L^(-1/2)] = [T * T^(-1)] = dimensionless).

Since Pi must be a constant: T = C * sqrt(L/g). Detailed analysis (or experiment) gives C = 2*pi, yielding the familiar T = 2*pi*sqrt(L/g).

Power of the method. Dimensional analysis determined the functional form T proportional to sqrt(L/g) without solving any differential equation. The mass m dropped out entirely, predicting (correctly) that the period is independent of mass.

Dimensional Homogeneity

Every valid physical equation must be dimensionally homogeneous — every term must have the same dimensions. This provides a powerful error check: if the dimensions don't match, the equation is wrong.

Part IV — Optimization

Linear Programming (LP)

Standard form. Maximize c^T * x subject to Ax <= b, x >= 0.

The feasible region is a convex polytope. The optimal solution (if it exists) occurs at a vertex. The simplex method (Dantzig, 1947) traverses vertices until optimality is reached.

Worked example. A factory makes chairs ($20 profit) and tables ($30 profit). Each chair needs 2 hours of carpentry and 1 hour of finishing. Each table needs 3 hours of carpentry and 2 hours of finishing. Available: 120 hours carpentry, 80 hours finishing. Maximize profit.

Variables: x = chairs, y = tables. Maximize: 20x + 30y Subject to: 2x + 3y <= 120, x + 2y <= 80, x >= 0, y >= 0.

Corner points: (0, 0) -> $0, (60, 0) -> $1200, (0, 40) -> $1200, (24,