Prosiectau Ffiseg ar gyfer myfyrwyr bl.3 a bl.4

Gerbils, batteries and infection clusters - modelling percolation

(supervisor: Rudi Winter)

Nature of project: theory, software

Available to students on full-time physics degree schemes or joint students.

Project description and methodology

When the world started to come to terms with the Covid epidemic in the spring of 2020, scientists were called on to explain what was happening and advise governments and the public on how to respond to limit the spread of the disease. Strikingly, apart from medics of various flavours, most of them were physicists. They use mathematical models and simulation techniques which are routinely used in various branches of our subject (such as nuclear fission or ionic conduction) and adapt them to understand and predict the progression of the disease under various scenarios, with constraints for their models provided by medics and sociologists.

In this project, we will investigate percolation theory as a means to understand processes which occur in a non-linear fashion, with jumps and bumps connecting several localised systems. This has been used to understand ionic conduction in solid battery electrolytes, population dynamics in zoology and is now considered as a model explaining and predicting the sudden spread of an infection wave in an epidemic that has been in a stationary state for a while. Percolation models split a population into two sub-populations (e.g. infected vs. healthy, or mobile vs. stationary) and treat clusters of each species as a thermodynamic phase, with phase boundaries separating them.

The core part of this project will code a simple 2D percolation model and visualisation (agnostic of any specific application) based on literature approaches. This should then be refined in several ways as outlined below.

A successful project will develop beyond the above in one/some of the following directions:
(1) Investigate dimensionality. In a 2D percolation model, it is impossible to have continuous pathways of both phases since a continuous path of one phase would cut off one of the other phase. However, in 3D it is possible to have intertwining continuous pathways. This means that conduction is possible within both phases. Is there an epidemiological analogy of this observation?

(2) Investigate the effect of the number of phases. Epidemiological models usually have more than two sub-population. For example, the commonly used SEIR model distinguishes susceptible, exposed, infected and recovered sub-populations. Can this concept be used to refine a 2D percolation model for use in epidemic modelling?

(3) Investigate the effect of interventions intended to limit percolation. Are these simply reducing the likelihood of phase switches linearly or can the system be kept below the percolation threshold?

(4) Once a working simulation has been developed and validated, it could be interesting to turn it into an app (e.g. an interactive web page) to enable users to explore the various parameters. This should include documentation suitable for dissemination to the general public.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

Additional scope or challenge if taken as a Year-4 project: A Y4 student should work the aspect of compliance with interventions into the model. There is no nuclear analogy to this as neutrons are not generally considered to have a free will. Does a reduced compliance rate simply reduce the effectiveness of interventions linearly or has it more complex effects?

Please speak to Rudi Winter (ruw) if you consider doing this project.

Initial literature for students:

1. A Bunde et al.; Diffusion and Conduction in Percolation Systems; in: P Heitjans et al., Diffusion in Condensed Matter; Berlin: Springer (2005) 895
2. S Davis et al.; Nature 454 (2008) 634
3. A Bonasera et al.; Front Phys 8 (2020) 171

Novelty, degree of difficulty and amount of assistance required

Simple 2D percolation models are a staple of theoretical physics, and it will be straightforward to code a simple simulation and visualisation. The model is quite abstract, and care has to be taken when mapping its parameters to features of a particular real problem, whether it is ionic conduction, population dynamics or disease spread. The model, including any interventions intended to keep the system below the percolation threshold, must be validated against publicly available data. The student is free in their choice of programming language; help can be provided in python or C. Of the two epidemic modelling projects, this is the more open-ended one, which can make it both more challenging and rewarding.

Project milestones and deliverables (including timescale)

milestoneto be completed by
Identification of base model and a selection of interventions/extensions to be modelled.end of November
Outline of pseudo-code for the percolation model software.Christmas
Core percolation model coded, tested and validated.end of February
Interventions/extensions of the model coded, tested and validated.Easter