Applied Math Modeling (Python)

Applied Math Lab: course hub

Welcome to Applied Math Lab. This site brings together the lecture notes, computational labs, slides, and supporting materials for the course.

Note

Quick links:

• Use the navigation bar at the left to explore sessions and modules.
• Syllabus / program overview
• Slide library

Course at a glance

• Format: 6 modules organized into 32 sessions
• Tools: Python, NumPy, SciPy, matplotlib, Streamlit, NetworkX
• Main goal: learn mathematical modeling through analysis, simulation, and interpretation

Learning Approach

• Start from a mathematical model and identify its main variables, parameters, and assumptions.
• Move from qualitative reasoning to numerical simulation and visual interpretation.
• Compare related models across sessions so each new topic extends a familiar workflow.
• Use the assignments to turn lecture examples into independent investigations.

Course map

Low-Dimensional Dynamics

S1 · Course overview

• What Is a Mathematical Model?
• The Modelling Workflow
• How the Course Works

S2 · A first model: the spruce budworm

• The Spruce Budworm Model
• Fixed Points and Stability
• Bistability

S3 · Lab: numerical tools for 1D models

• Solving Initial Value Problems
• Budworm: the Model in Code
• Budworm: Integration and Time Series
• Budworm: a Streamlit App
• Michaelis-Menten Kinetics
• Practice: Budworm App
• Reproducible Code

S4 · The budworm outbreak: hysteresis

• Saddle-Node Bifurcation
• Hysteresis
• Budworm and Forest: Fast and Slow

S5 · Epidemics: SIR and SIS

• Compartmental Models: SIR and SIS
• The Reproduction Number
• The Final Size of an Epidemic

S6 · Lab: epidemics and real data

• Simulating the SIR Model
• Simulating the SIS Model
• Fitting an Epidemic to Data
• Predicting the Peak

S7 · Two-dimensional systems: oscillations

• Linear Systems in the Plane
• The Phase Plane
• Hopf Bifurcation

S8 · Lab: two-dimensional systems

• CDIMA: Phase Plane
• CDIMA: Nullclines and Fixed Points
• CDIMA: Stability and the Hopf Point
• Animations and Interactivity
• Van der Pol Oscillator
• Practice: Planar Systems
• CDIMA: Parameter Map

S9 · Excitable systems and neurons

• Excitability
• FitzHugh-Nagumo Model
• Relaxation Oscillations
• Morris-Lecar Model
• Bistable Switches

Extra: the Lorenz system

• Lorenz Attractor
• Sensitivity and Chaos
• Practice: Lorenz System

Synchronization and Collective Behaviour

S10 · Synchronization: from one oscillator to many

• Synchronization in Nature
• Flows on the Circle
• Phase Locking

S11 · The Kuramoto model

• The Kuramoto Model
• The Order Parameter
• Critical Coupling
• The Millennium Bridge

S12 · Lab: Kuramoto and the Millennium Bridge

• Initialization
• The ODE
• Pairwise vs Mean Field (optional)
• Animation
• Bifurcation Diagram
• Locating the Threshold
• Full Simulation (optional)
• Finite-Size Effects
• Kuramoto Assignment (graded)

S13 · Collective behaviour: Vicsek and Couzin

• The Vicsek Model
• The Couzin Model
• The Social Force Model

S14 · Lab: collective motion

• Initialization
• Update Rule
• Vectorized Update (optional)
• Order Parameter
• Validation
• Animation
• Animation Loop
• Interactive Animation (optional)
• Predator Avoidance (optional)
• Practice: Collective Motion

Modelling with Randomness

S15 · From random walks to diffusion

• The Random Walk
• Mean Squared Displacement
• The Diffusion Equation
• Diffusion in Biology

S16 · Fat tails and anomalous diffusion

• The Central Limit Theorem
• Power Laws
• Stable Distributions
• Lévy Flights

S17 · Modelling financial markets

• Returns and Efficient Markets
• Stylized Facts of Returns
• Scaling of Returns

S18 · Lab: financial time series

• Loading Prices and Computing Returns
• Normality Tests
• Calibrated KS Test
• Tail Exponent
• Scaling of Returns in Data

Space and Pattern Formation

S19 · Reaction-diffusion equations

• Reaction-Diffusion Equations
• Linearization
• Laplacian Eigenmodes

S20 · Turing instability

• Turing Conditions
• Dispersion Relation
• Turing Space

S21 · Lab: pattern formation in 1D

• Gierer-Meinhardt Model in 1D
• Turing Numerics
• Unstable Modes in 1D
• Laplacian in 1D
• Boundary Conditions
• Gierer-Meinhardt Solver in 1D
• Time-Step Stability
• Animation in 1D
• Animal Coat Patterns
• Assignment: Pattern Formation (graded)

S22 · Lab: pattern formation in 2D

• Laplacian in 2D
• Gierer-Meinhardt Solver in 2D
• Animation in 2D
• Unstable Modes in 2D
• Gray-Scott Model
• Gray-Scott Animation
• Gray-Scott Parameters
• Gray-Scott: Interaction
• Gray-Scott: Sliders

Complex Networks

S23 · Networks: fundamental concepts

• Network Concepts
• Robustness and Percolation
• NetworkX Basics
• Edges and Weights
• Drawing Graphs
• Drawing Attributes
• Graphs from Edge Lists
• Paths and Components
• Paths in Directed Graphs
• Degree and Clustering
• Centrality in NetworkX

S24 · Lab: cascading failures in power grids

• The Iberian Blackout
• The Motter-Lai Model
• Cascade Simulation
• Swing Equation

S25 · Models of network growth

• Erdős-Rényi Random Graphs
• Small-World Networks
• Preferential Attachment
• Comparing Graph Models

S26 · Lab: spreading processes on networks

• SIS Model on Networks
• SIS Initial State
• SIS Simulation
• Comparing Networks under SIS
• SIR on Networks
• SIR Simulation
• Epidemic Threshold
• Vaccination Strategies
• Practice: Real Networks

Explorables and Discrete Models

S27-28 · Explorables

• Explorables Presentations
• Preparation Checklist

S29 · Modelling with cellular automata

• Cellular Automata
• Wolfram Classes
• Special Rules and Theorems

S30 · Lab: cellular automata

• 1D Cellular Automata
• Space-Time Diagrams
• Rule Exploration
• Boundary Conditions
• Game of Life
• Random Soups
• Practice: Cellular Automata

S31 · Traffic flow and congestion

• The Nagel-Schreckenberg Model
• The Fundamental Diagram

S32 · Lab: traffic models

• Traffic Model
• Flow and Congestion
• Traffic Extensions
• Practice: Traffic

What you will build

Across the course you will implement and experiment with:

• Low-dimensional dynamical systems (spruce budworm, SIR/SIS, CDIMA, Van der Pol, FitzHugh-Nagumo)
• Coupled oscillator simulations (Kuramoto synchronization and the Millennium Bridge)
• Collective motion models (Vicsek, Couzin, social force, predator response)
• Random walks and statistical modelling (diffusion, heavy tails, financial returns)
• Network analysis and spreading simulations (metrics, graph models, SIS/SIR)
• Reaction-diffusion solvers and Turing analysis (Gierer-Meinhardt, Gray-Scott)
• Discrete and agent-based explorables (cellular automata, traffic flow, congestion)

How to Use the Material

• Read each module overview first so the mathematical question and modeling goal are clear.
• Use the lecture slides as a compact guide to the main ideas and calculations.
• Work through the case studies in order before starting the assignment.
• Return to earlier modules when a later topic reuses the same analytical or numerical pattern.