Module 1: Low-Dimensional Dynamics

Part I of the syllabus, Sessions 1 to 9

This module covers models with one or two state variables described by ordinary differential equations. You start with a single equation for an insect population and learn to find equilibria, decide their stability and follow them as parameters change. You then apply the same tools to epidemics, fit a model to data, and move to the plane, where oscillations, limit cycles, Hopf bifurcations and excitable systems appear (Strogatz 2024). Lectures and laboratories alternate: each lecture is followed by a session in which you implement the model in Python. Reusable code is in amlab/odes_1d/, amlab/odes_2d/ and amlab/epidemics/.

Session Structure

Session 1: Course overview. What is mathematical modeling?

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

Session 2: A first model: the spruce budworm

The Spruce Budworm Model Fixed Points and Stability Bistability

Session 3: Laboratory: numerical tools for one-dimensional 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

Session 4: The budworm outbreak: hysteresis and its consequences

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

Session 5: Modelling the spread of epidemics: SIR and SIS

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

Session 6: Laboratory: epidemics and real data

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

Session 7: Two-dimensional systems: modelling oscillations

Linear Systems in the Plane The Phase Plane Hopf Bifurcation

Session 8: Laboratory: 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

Session 9: Excitable systems and models of the neuron

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

Extra (optional): the Lorenz system

Lorenz Attractor Sensitivity and Chaos Practice: Lorenz System

Pages marked “optional extension” and the practice pages (grey buttons) are ungraded. The graded assignments of the course are in Modules 2 and 4.

Learning Goals

Formulate a model as an ODE, and remove units to reduce the number of parameters.
Find equilibria and decide their stability from the linearization.
Recognize saddle-node and Hopf bifurcations, bistability and hysteresis.
Integrate ODEs with scipy.integrate.solve_ivp, and validate the result against known solutions.
Build interactive tools (Streamlit apps, Matplotlib animations and clickable parameter maps).
Use SIR and SIS models to compute reproduction numbers, herd immunity thresholds and final sizes.
Fit a model to incidence data and explain why the peak of an epidemic is hard to predict.
Read a phase plane: nullclines, fixed points, limit cycles, excitability and relaxation oscillations.

References

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "Module 1: Low-Dimensional Dynamics" subtitle: "Part I of the syllabus, Sessions 1 to 9" format: html --- This module covers models with one or two state variables described by ordinary differential equations. You start with a single equation for an insect population and learn to find equilibria, decide their stability and follow them as parameters change. You then apply the same tools to epidemics, fit a model to data, and move to the plane, where oscillations, limit cycles, Hopf bifurcations and excitable systems appear [@strogatz2024nonlinear]. Lectures and laboratories alternate: each lecture is followed by a session in which you implement the model in Python. Reusable code is in `amlab/odes_1d/`, `amlab/odes_2d/` and `amlab/epidemics/`. # Session Structure **Session 1: Course overview. What is mathematical modeling?** [What Is a Mathematical Model?](what-is-a-model.qmd){.btn .btn-primary} [The Modelling Workflow](modeling-workflow.qmd){.btn .btn-primary} [How the Course Works](how-the-course-works.qmd){.btn .btn-primary} **Session 2: A first model: the spruce budworm** [The Spruce Budworm Model](budworm-model.qmd){.btn .btn-primary} [Fixed Points and Stability](fixed-points-stability.qmd){.btn .btn-primary} [Bistability](bistability.qmd){.btn .btn-primary} **Session 3: Laboratory: numerical tools for one-dimensional models** [Solving Initial Value Problems](solve-ivp.qmd){.btn .btn-primary} [Budworm: the Model in Code](budworm-implementation.qmd){.btn .btn-primary} [Budworm: Integration and Time Series](budworm-integration.qmd){.btn .btn-primary} [Budworm: a Streamlit App](budworm-streamlit.qmd){.btn .btn-primary} [Michaelis-Menten Kinetics](michaelis-menten.qmd){.btn .btn-primary} [Practice: Budworm App](practice-budworm-app.qmd){.btn .btn-secondary} [Reproducible Code](reproducibility.qmd){.btn .btn-primary} **Session 4: The budworm outbreak: hysteresis and its consequences** [Saddle-Node Bifurcation](saddle-node-bifurcation.qmd){.btn .btn-primary} [Hysteresis](hysteresis.qmd){.btn .btn-primary} [Budworm and Forest: Fast and Slow](budworm-forest-fast-slow.qmd){.btn .btn-primary} **Session 5: Modelling the spread of epidemics: SIR and SIS** [Compartmental Models: SIR and SIS](compartmental-models.qmd){.btn .btn-primary} [The Reproduction Number](reproduction-number.qmd){.btn .btn-primary} [The Final Size of an Epidemic](final-size.qmd){.btn .btn-primary} **Session 6: Laboratory: epidemics and real data** [Simulating the SIR Model](sir-simulation.qmd){.btn .btn-primary} [Simulating the SIS Model](sis-simulation.qmd){.btn .btn-primary} [Fitting an Epidemic to Data](fitting-epidemic-data.qmd){.btn .btn-primary} [Predicting the Peak](peak-prediction.qmd){.btn .btn-primary} **Session 7: Two-dimensional systems: modelling oscillations** [Linear Systems in the Plane](linear-systems-2d.qmd){.btn .btn-primary} [The Phase Plane](phase-plane.qmd){.btn .btn-primary} [Hopf Bifurcation](hopf-bifurcation.qmd){.btn .btn-primary} **Session 8: Laboratory: two-dimensional systems** [CDIMA: Phase Plane](cdima-phase-plane.qmd){.btn .btn-primary} [CDIMA: Nullclines and Fixed Points](cdima-nullclines.qmd){.btn .btn-primary} [CDIMA: Stability and the Hopf Point](cdima-stability.qmd){.btn .btn-primary} [Animations and Interactivity](animation.qmd){.btn .btn-primary} [Van der Pol Oscillator](van-der-pol.qmd){.btn .btn-primary} [Practice: Planar Systems](practice-2d.qmd){.btn .btn-secondary} [CDIMA: Parameter Map](parameter-map.qmd){.btn .btn-primary} **Session 9: Excitable systems and models of the neuron** [Excitability](excitability.qmd){.btn .btn-primary} [FitzHugh-Nagumo Model](fitzhugh-nagumo.qmd){.btn .btn-primary} [Relaxation Oscillations](relaxation-oscillations.qmd){.btn .btn-primary} [Morris-Lecar Model](morris-lecar.qmd){.btn .btn-primary} [Bistable Switches](bistable-switches.qmd){.btn .btn-primary} **Extra (optional): the Lorenz system** [Lorenz Attractor](lorenz.qmd){.btn .btn-primary} [Sensitivity and Chaos](lorenz-sensitivity.qmd){.btn .btn-primary} [Practice: Lorenz System](practice-lorenz.qmd){.btn .btn-secondary} Pages marked "optional extension" and the practice pages (grey buttons) are ungraded. The graded assignments of the course are in Modules 2 and 4. # Learning Goals - Formulate a model as an ODE, and remove units to reduce the number of parameters. - Find equilibria and decide their stability from the linearization. - Recognize saddle-node and Hopf bifurcations, bistability and hysteresis. - Integrate ODEs with `scipy.integrate.solve_ivp`, and validate the result against known solutions. - Build interactive tools (Streamlit apps, Matplotlib animations and clickable parameter maps). - Use SIR and SIS models to compute reproduction numbers, herd immunity thresholds and final sizes. - Fit a model to incidence data and explain why the peak of an epidemic is hard to predict. - Read a phase plane: nullclines, fixed points, limit cycles, excitability and relaxation oscillations.