She also makes her own awesome maths-themed dresses! During #ANZIAM2025 I spotted them so I went up and asked because I'm equally interested in making my own maths-themed clothing 😁
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Congratulations👏 to Centre Chief Investigator Jennifer Flegg (pictured left) - recently awarded the EO Tuck Medal at #ANZIAM2025. The award is ANZIAM's top mid-career honour recognising outstanding contributions & service to applied mathematics. FULL STORY:
macsys.org/jen-flegg-eo...
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Using GSPT, it is possible to identify an explicit path (heteroclinic connection) between two shock points and determine the speed along said path.
That's pretty amazing given the sheer complexity & dimensions of the problem!
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Identifying these conditions requires us to simplify (regularize) the problem by adding/modifying terms in the original equation to better understand of what happens close to these shock points
Geometric singular perturbation theory (GSPT) is one such technique
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However on the actual paper itself we only see the particle 'teleporting' from one place on the paper to another, and thus there's a sudden change in direction, hence the shock.
There are conditions where we can identify where shocks are going to happen.
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What do these shocks look like mathematically?
Bend a piece of paper into an S shape. Imagine there's an extra dimension that 'joins' the paper from one edge to another going through the middle of the S shape. When a shock happens we travel through that middle.
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Shocks and sharp boundaries happen as a result.
Imagine the growth between a maple leaf 🍁and a 4-leaf clover☘️. An analogy of said shocks happening during these plant growth results in the spikes in the maple leaf as opposed to the smoother clover.
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Hopefully you did an osmosis experiment in high school chemistry! But the movement is usually linear (steady).
In our case it's nonlinear and only a bit of backwards diffusivity amongst normal diffusion.
What happens when it flips between positive/negative?
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Presentation slide with title: Simulation of 1-dimensional diffusion with Neumann boundary conditions. The graphs are as described in the post.
I'm skipping over a lot of the finer details but the results are pretty promising when comparing them to examples with known solutions. In the second graph the 'patches' are the individual lines, but you can see how the overall trajectory matches the first graph.
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Presentation slide with title: Couple patches to simulate the macroscale dynamics. The main diagram shows small patches with blue and orange lines joining various edges to indicate the coupling.
So how do we 'join' the patches? Interpolation!
Of course, for this to work well there's a couple of conditions that we should meet so that the modelling remains reasonably accurate. This in turn imposes conditions on corresponding mathematical models.
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Presentation slide with title: Construct a 2-dimensional macroscale solution across space. A diagram of repeated grids indicate the patches, with red axes indicating the larger macroscale domain.
The general sketch of the idea is to generate several 'patches' where the microscale (or individual molecules) behaviour is known, index all of them, and then construct a macroscale solution from the midpoints of each 'patch'.
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Presentation slide with title: Equation free macroscale modelling. Giving a general overview. Github repository for Judy's code can be found at https://github.com/uoa1184615/EquationFreeGit
You might think "Wait, wouldn't existing knowledge of the macroscale system already help us with this?" Imagine if we knew how individual particles behaved, but no clue (therefore equation-free) on how the larger macroscale system functions.
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Presentation slide with title "The need for macroscale modelling". Macroscale modelling describes any method which produces a macroscale solution from a problem which is defined on a microscale, reaching a compromise between accuracy and efficiency.
Imagine a large medium where modelling of particles (microscale) is achievable (e.g. mixture of molecules in air, water). How do we model over a much larger (macroscale) container? Doing all the individual particles is time consuming & needs lots of data!
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Judy is standing on the left with Tony introducing her. The slide contains what looks like a purple-pink crinkled piece of paper sloping downwards.
Two more Thursday #ANZIAM2025 talks to go, one of my supervisors: Judy Bunder (from UniSA), presents "Boundary conditions for multiscale equation-free modelling", joint work with Tony Roberts (also in the photo!) from University of Adelaide.
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Needless to say, it seems like an awful lot of work has gone into this, with a lot of collaboration happening. But no doubt also extremely tricky tying all of this up with statistical inference as well!
I'm really happy Rebecca is still around and kicking ass!😊
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Presentation slide with subtitle: SIRS model of immune response which clarifies some of the details behind the SIR model, including the fact that after recovery there is some infection against future strains i.e. partial cross-strain immunity.
The SIRS model is another where the recovery stage is not the final stage, it circles back to individuals being susceptible, which makes sense for e.g. flu strains as well which I assume mutates much more rapidly than Strep A! Don't forget your flu vax, kids!
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Presentation slide with subtitle: SIRIR model of strain-specific immunity, where the second I term is a repeat infection, and the second R term is resistant/immune. Graphs show that the dynamics are similar to those seen in high-burden settings e.g. remote communities in the NT
I'll start by assuming people know about the SIR (susceptible-infectious-recovered) model thanks to COVID-19. The SIR model is good as a starting point but assumes only one strain of a disease, so when dealing with a multi-strain disease, much more complicated!
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Presentation slide: Multi-strain agent-based model of Strep A transmission. The model itself is a stochastic agent-based model capturing mild infection (no sequelae, no infection site structure). There are different hypotheses for the strain-specific and cross-strain immune response.
Finally starting on Thursday's #ANZIAM2025 talks, with Rebecca Chisholm from La Trobe who presented on "Multi-strain infectious disease dynamics".
Was a tad late, so didn't catch the intro.
Lore Drop: She was my tutor at a UniMelb PDE subject some 13-14+ years ago 😁
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Their initial results are promising but undoubtedly there's more work to be done to refine the model accuracy and adding in other parameters. Nonetheless, any mathematical modelling towards understanding cancer & treatments is always welcome!
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A visual diagram of the three types of cells and interactions between them. In the bottom left is a box with three differential equations.
Their proposed model is a three-ODE model with three states (representing the CAR T-cells and tumour cells) with various transitions between them. They do some testing & simulation to verify whether it's bistable or not (I don't recall if it was, whoops!)
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Presentation slide with a visual of what bistability looks like, in abstract mathematical form. The second dot point says "If you have multiple basins of attraction, then small changes in initial condition causes big changes in outcome."
The mathematical concept that explains this is bi-stability i.e. where a solution has two stable points (results), plus a 'split' somewhere where any diverging from the 'split' means behaviour tends towards one of said results. So how is this modelled?
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Presentation slide showing visual of the details mentioned in the post.
What appears to happen (from Peter Mac Cancer Center observations) is that with low doses, the tumour relapses with high genetic diversity, and with high doses, a few clonal strains survives. However the change happens very suddenly when one varies the dosage.
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Presentation slide showing the process of CAR T-cell therapy, which involves extracting T-cells from cancer patients, growing them in the lab, and then reinjecting them into patients to kill cancer cells.
Last one from me for Wednesday's #ANZIAM2025 is by Tom Cummings & Yang Zhao from the same group as Adriana Zanca: "Clone wars: modelling tumour resistance in CAR T-cell therapy" which looks at a treatment used for leukemia.
Tumours are sometimes resistant to treatment, why?
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Presentation slide showing 5 different cell states with various mathematical symbols and equations on them and the arrows between them.
It's also possible to take a random dynamical systems perspective, where by mapping out cell states and assigning variables & probabilities to the transitions (i.e. arrows), we can convert a seemingly complex cell system into a series of equations.
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Presentation slide with title: What is the Theoretical Systems Biology group doing? There are accompanying images pertaining to what is mentioned in the post.
There has been attempts to model Waddington's metaphors, as well as other vague and ill-defined metaphors. Instead of looking at the overall landscape, we can attempt to define cells based on their properties/state and how they interact with their environment.
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Presentation slide with title: "Cell fate: a history". The last dot point mentions Waddington acknowledges the epigenetic landscape cannot be interpreted rigorously.
Is there a way we can (probabilistically) model the chance that a random stem cell in the body grows into specific types of cells?
Waddington likened the idea to a ball rolling down a hill shaped by pegs: where the ball rolls determines the cell's fate.
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Adriana standing next to her presentation slide with title "One cell to rule them all?" and subtitle "How does the diversity of cell type arise?"
People might know or have heard of stem cells, which can be manipulated to grow into different cell types, and similarly we can chart a network of various cells and their growths.
From the maths side we're more wondering: What kind of model encompasses all that?
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Adriana standing next to her title slide.
Have you ever wondered how to mathematically encompass the sheer variety of cells and their paths from conception to death? Adriana (Age) Zanca from UniMelb gives us some insight in her presentation: "Theoretical and mathematical frameworks of cell fate"
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Presentation slide showing three images of cells in a petri dish, where dead cells are stained a light pink in different scenarios.
The rest was a bit 'woosh!' for me, so I'll round this up by highlighting some of the experiments that have been done on the biology side of things. We can see that cells are stained with Phloxine B dye and presumably imaging/measurements techniques are used.
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Presentation slide with title "2D Numerical Results". There are three diagrams showing similar shapes in terms of a wave coming from the left having a bumpy side, indicating that the petal formation is still consistent.
Anyway despite all that additional effort, petal formation still persists when cell death is considered in the model. This is a good thing because it means the model is continuing to be consistent with observed experiments, and we can carry on.
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