Mathematical Modelling of Immune and Inflammatory Responses

This repository contains the code developed for my MMath Mathematics dissertation on mathematical biology, specifically modeling immune and inflammatory responses.

Project Overview

The project analyses a data set supplied by Imperial College London's Bioengineering department of dendritic cell trajectories in a chemokine gradient. Next we produce PDE models (three-variable PDE and a five-variable PDE) to model cell density and flux. Then we produce a hybrid model (stochastic cell movement, continuous chemokine concentration) as a different way to calculate cell density and flux. We conclude by fitting the data to the model by parameter estimation.

Code Structure

Data supplied by Imperial College London's Bioengineering department is given in 'dataset'. Ensure all files are in the same folder as the remaining code when running.

Code to display cell trajectory data:

• 'cell_trajectory_data.py'
• 'kde.py'
• 'visualising_density.py'
• 'visualising_flux.py'

Code for PDE models

• 'three_pde_solve.py'
• 'three_pde_plot.py'
• 'five_pde_solve.py'
• 'five_pde_plot.py'
• 'pde_parameter_fitting.py'

Code for hybrid model

• 'hybrid_model.py'
• 'hybrid_model_fitting.py'

Dependencies

This project requires the following Python libraries:

• numpy
• scipy
• matplotlib
• pandas

Author: Benjamin Stephens
Dissertation Submission Date: April 11, 2025
University: University of Nottingham, School of Mathematics