Mathematical Modeling – Complete Reference Guide

1. Modeling Process

Steps

Problem Definition

Variable Selection

Assumptions & Constraints

Mathematical Formulation

Parameter Estimation

Validation & Testing

Model Validation

Compare to real data

Residual analysis

Sensitivity testing

2. Dimensional Analysis

Buckingham Pi Theorem

If $n$ variables with $k$ dimensions

Then $\pi = n - k$ dimensionless groups

$\Pi_i = \prod_{j=1}^n x_j^{a_{ij}}$

Example

Pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$

Variables: $L, g, m, T$

$\Pi_1 = \frac{T}{\sqrt{L/g}}$

3. Scaling & Non-dimensionalization

Non-dimensional Variables

$\tilde{x} = \frac{x}{x_0}, \tilde{t} = \frac{t}{t_0}$

Benefits

Reveal parameter regimes

Reduce parameters

Improve numerics

Logistic Example

$\frac{d\tilde{N}}{d\tilde{t}} = \tilde{N}(1-\tilde{N})$

One dimensionless param

4. Population Models

Exponential

$P' = rP$

$P(t) = P_0 e^{rt}$

Logistic

$P' = rP(1 - P/K)$

$P(t) = \frac{K}{1+Ae^{-rt}}$

$r$: growth rate, $K$: carrying capacity

5. Lotka-Volterra (Predator-Prey)

Equations

$\frac{dx}{dt} = \alpha x - \beta xy$

$\frac{dy}{dt} = \delta xy - \gamma y$

Parameters

$x$: prey, $y$: predators

$\alpha$: prey growth

$\beta$: predation rate

$\gamma$: predator death

Equilibrium

$(x^*, y^*) = (\gamma/\delta, \alpha/\beta)$

6. Epidemic Models (SIR)

SIR System

$S' = -\beta SI$

$I' = \beta SI - \gamma I$

$R' = \gamma I$

Key Metrics

$R_0 = \beta S_0 / \gamma$

$R_0 > 1$: epidemic spreads

$\beta$: transmission, $\gamma$: recovery

7. Epidemic Models (SIS)

SIS Dynamics

$S' = -\beta SI + \gamma I$

$I' = \beta SI - \gamma I$

Endemic Equilibrium

$I^* = 1 - 1/R_0$ (when $R_0 > 1$)

No immunity/recovery to immunity

Used for influenza models

8. Linear Models & Systems

Linear System

$\vec{x}' = A\vec{x}$

Eigenvalue Analysis

Find eigenvalues $\lambda$ of $A$

$\lambda_i > 0$: growing

$\lambda_i < 0$: decaying

Equilibria

$A\vec{x}^* = 0$

Stability: signs of $\lambda_i$

9. Nonlinear Dynamics

Phase Portraits

Trajectories in state space

Nullclines: $\dot{x} = 0, \dot{y} = 0$

$\frac{dy}{dx} = \frac{y'}{x'}$

Stability

Asymptotically stable

Lyapunov stable

Unstable (saddle)

10. Bifurcations

Saddle-Node

$x' = r + x^2$

Two fixed points collide & disappear

Transcritical

$x' = rx - x^2$

Two fixed points exchange stability

Pitchfork

$x' = rx - x^3$

Symmetry breaking bifurcation

11. Optimization Models

Unconstrained

$\min_x f(x)$

$\nabla f = 0$ at optimum

Constrained

Lagrange: $\nabla f = \lambda \nabla g$

Applications

Resource allocation

Cost minimization

Profit maximization

12. Linear Programming

Standard Form

$\max c^Tx$ s.t. $Ax \le b, x \ge 0$

Simplex Method

Move along vertices

Improves objective each step

Applications

Diet problem

Shipping logistics

Portfolio optimization

13. Network Models

Graph Basics

Nodes (vertices), Edges

Degree: edges connected to node

$\sum_i d_i = 2|E|$

Network Flow

Conservation of flow

Capacity constraints

Centrality

Betweenness, closeness

14. Markov Chains & Stochastic

Transition Matrix

$P_{ij} = P(X_{n+1} = j | X_n = i)$

$\vec{p}(n+1) = P^T \vec{p}(n)$

Steady State

$\vec{p}^* = P^T \vec{p}^*$

Long-run distribution

Ergodic Property

Converges to unique steady state

15. Monte Carlo Methods

Basic Idea

Use random sampling to estimate

Pi Estimation

$\pi \approx 4 \cdot \frac{\text{points in circle}}{\text{total points}}$

Integration

$\int f \approx V \cdot \bar{f}$

Volume × average value

Error: $O(1/\sqrt{N})$ independent of dimension!

16. Regression & Data Fitting

Linear Regression

$\min \sum (y_i - \hat{y}_i)^2$

$\hat{\beta} = (X^TX)^{-1}X^Ty$

Least Squares Fit

Minimize residuals

$R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$

Nonlinear

Curve fitting, exponential

MATHEMATICAL MODELING Complete Formula Sheet