# UVic Frontiers in Mathematical Modeling

## Details

In recognition of Rod Edwards’ remarkable contributions to the field of Mathematical Modeling, we are inviting Rod’s current and former students, postdocs, close collaborators, friends and colleagues to attend this event, scheduled to take place on campus on February 17th and 18th, 2024.

**Talks**

February 17

### 9:00–9:30 AM: Leon Glass (in person)

Department of Physiology, McGill University

**Title: **Dynamics in Large Model Genetic Networks: Combinatorics, Inverse Problem, Robustness

Abstract

Genetic activity is partially regulated by a complicated network of proteins called transcription factors. I will describe a mathematical framework to analyze the structure and dynamics of these genetic networks. The underlying idea is to capture the topology and logic of the network interactions by a logical network, and to then embed the logical network into continuous piecewise linear differential equations. The equations can be analyzed using methods from discrete mathematics and nonlinear dynamics. The underlying discrete structure of the phase space enables one to establish a classification of the dynamics and also to perform the inverse problem - determining the structure of the equations based on the observed dynamics. This talk is honor of Rod Edwards with thanks for many fruitful collaborations.

### 9:30–10:00 AM: Pabel Shahrear (online)

Department of Mathematics, Shahjalal University of Science and Technology

**Title: **A decade-long philosophical expedition of genetic networks with relish fellow Dr. Rod Edward and mentor Dr. Leon Glass

Abstract

Activities of genes are controlled in a combinatorial fashion by the concentrations of chemicals called transcription factors. We generate continuous nonlinear equations by replacing the step function discontinuities in the piecewise linear equations with sigmoidal control functions. As the sigmoidal functions become steep, the continuous equations approach piecewise linear differential equations. Moreover, we examine dynamics in a particular 4-dimensional equation of this class. In the equation, two variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system amenable to detailed analysis. The numerical calculation of Poincare ́ maps address new evidence of dynamics, pointing to a new research direction. Finally, we have explored relations of Boolean states on hypercube for higher dimensions. We developed matrix algorithm such that the connections of higher dimensional genetic networks are understandable on the hypercube. Before this work, at best five-dimensional system can be addressed in such a way that two four-dimensional hypercube was connected together. We have obtained the resultant output matrix and tested it for 24th dimensional system

### 10:00–10:30 AM: Mike Jeffrey (online)

School of Engineering Mathematics and Technology, University of Bristol

**Title: **The ambiguity of nonsmooth dynamics, and why it matters

Abstract: It is a basic fact of dynamical systems that if they aren’t differentiable, then their solutions are not unique. Models with discontinuities turn up across modern biology and engineering wherever there are regulatory, control, or decision processes. Much of the literature in these “nonsmooth systems” seeks to resolve non-uniqueness, but here’s the simple fact: you cannot banish non-uniqueness altogether. And you don’t want to. It gives insight into the sensitivity and complexity of switching processes, and opens up entire new levels of modeling. I’ll discuss the basic reason that non-uniqueness cannot be got rid of, and how we’re starting to tackle it.

### 11:00–11:30 AM: Stephanie Willerth (in person)

Department of Mechanical Engineering, University of Victorai

**Title: **Mathematical modeling cell behavior

Abstract: Mathematical models can provide insight into understanding how cells behave. Here I will present collaborative work conducted with Dr. Rod Edwards discussing how we implemented a variety of mathematical models to study cancer biology and biomaterial scaffolds for promoting stem cell differentiation. I will also discuss the potential applications of machine learning for similar applications.

### 11:30–12:00 PM: Alessia Ando (online)

Area of Mathematics, Gran Sasso Science Institute

**Title: **Nonuniqueness phenomena in discontinuous dynamical systems and their regularizations

Piecewise smooth ordinary differential equations (ODEs) find applications in numerous fields, such as gene regulatory networks, where they are employed to replace prohibitively stiff continuous systems whose numerical integration would otherwise be unfeasible. Recently, an analysis in the ε → 0 limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains but, in a few cases, the existence of a unique limit solution was not proved, leaving the question of whether or not the ambiguity could be resolved. In this talk, we will show that the answer is negative, in that the limit solution turns out to be dependent on the form of the switching function. Then, we will observe that a range of possible behaviors can be revealed in the bifurcation analysis of a parameter-dependent model in the “ambiguous” class of discontinuous ODEs. Finally, we will comment on the possible nonuniqueness in the sense of sensitivity of solutions arising from periodic fast dynamics, when transitioning from codimension-2 to codimension-3 switching domains.

### 12:00–12:30 PM: Michael Tsatsomeros (online)

Department of Mathematics & Statistics, Washington State University

**Title: **Sharing space

Abstract:

The existence of common invariant subspaces and common invariant proper cones among two matrices is revisited. This is a problem that we considered with Rod almost 20 years ago and is motivated by a model for gene and neural networks.

**February 18**

### 9:00–9:30 AM: Tomas Gedeon (online)

Department of Mathematical Sciences, Montana State University

**Title: **Combinatorial structure of continuous dynamics in gene regulatory networks

Abstract

We first discuss two paradigms of models of gene regulatory networks: ordinary differential equations and monotone Boolean models. We show that switching (Glass) models connect the world of continuous dynamics of ODEs and discrete dynamics of Boolean models by describing a precise connection between all parameterizations of switching systems and all collections of monotone Boolean maps compatible with the network structure. This connection allows on one hand a combinatorial (i.e. finite) description of ODE dynamics over parameter space and, on the other hand, to make sense of bifurcation theory of Boolean maps.

### 9:30–10:00 AM: Hidde de Jong (online)

Inria - Université Grenoble Alpes

**Title: **Modeling gene regulatory networks: from transcription factors to global cell physiology

Regulatory networks controlling the adaptation of gene expression in bacteria involve transcription factors that sense environmental and metabolic signals and specifically activate or inhibit target genes. In addition to such specific factors, gene expression also responds to changes in a variety of physiological parameters that modulate the rate of transcription and translation, such as the concentrations of (free) RNA polymerase and ribosome, and the size of amino acid and nucleotide pools. Contrary to specific regulators, these so-called global physiological effects affect the expression of all genes. Recent experimental work has emphasized the important role played by global physiological effects in the adaptation of bacterial cells to changes in their environment. These results have given rise to coarse-grained mathematical models of bacterial growth, which I will briefly discuss in my presentation.

### 10:00–10:30 AM: Etienne Farcot (online)

School of Mathematical Sciences, University of Nottingham

**Title: **Analysis of continuous-time switching networks’ – 24 years later

Abstract

In 2000 Rod Edwards published a landmark paper on continuous-time switching networks, also known as Glass networks. This paper gave a series of new insights on the dynamics of these networks, as well as a clear formalism encompassing many previous papers. The tools from this paper have been of great use to researchers working on these and related systems, including myself. In this talk, I will aim to give an overview of the current landscape, highlighting solved problems and some open questions.

### 11:00–11:30 AM Nicola Guglielmi (online)

****Division of Mathematics, Gran Sasso Science Institute

Title: **Solving integro-differential equations modeling pharmacodynamics**.

This is a joint work with Ernst Hairer (Geneva).

Abstract:

There exist excellent codes for an efficient numerical treatment of stiff and differential-algebraic problems. Let us mention {\sc Radau5} which is based on the $3$-stage Radau IIA collocation method, and its extension to problems with discrete delays {\sc Radar5}. The aim of the present work is to present a technique that permits a direct application of these codes to problems having a right-hand side with an additional distributed delay term (which is a special case of an integro-differential equation). Models with distributed delays are of increasing importance in pharmacodynamics and pharmacokinetics for the study of the interaction between drugs and the body. The main idea is to approximate the distribution kernel of the integral term by a sum of exponential functions or by a quasi-polynomial expansion, and then to transform the distributed (integral) delay term into a set of ordinary differential equations. This set is typically stiff and, for some distribution kernels (e.g., Pareto distribution), it contains discrete delay terms with constant delay.The original equations augmented by this set of ordinary differential equations can have a very large dimension, and a careful treatment of the solution of the raising linear systems is necessary. The use of the codes {\sc Radau5} and {\sc Radar5} is illustrated with three examples (two test equations and one problem taken from pharmacodynamics). The driver programs for these examples are publicly available from the homepages of the authors.

### 11:30–12:00 PM: Aude Maignan (online)

Maître de conférences, Université Grenoble Alpes

**Title: **Bio-Inspired Graph-Rewriting Automata

Abstract : DEM-systems are an extension of Cellular Automata to a dynamic structure using local graph-rewriting rules.

We first consider the one-dimensional case : There sites are arranged into a ring with rules governing dynamics in which new site can be created. We will study the behavior of these DEM-systems in terms of growth rate and prove that the growth rate can be sublinear, linear, quadratic, cubic, or exponential.

In terms of pattern complexity, we’ll see that a finite initial sequence can produce positive spatial entropy over time, which is not the case for cellular automata. We will also study behaviors of a sub-class called fragmentable DEM-systems in which self-reproducing patterns appear.

In the second time, we consider regular 3-D DEM-systems. Here again a reproduction rule allows the graph to expand. We will classify these new DEM-systems according to their growth rate, and present some complex structures. Some of them imitate organic structures.

### 12:00–12:30 PM Patrick Gill (online)

Apple Inc.

Title: Thanks to Chaos, Our Quantum Multiverse Disrupts the Philosophy of Mind

Abstract

The mind–body problem ponders how there can be one physical collection of atoms, a body, that seems to follow the commands of something that seems to be nonphysical, a mind. I argue that the mind–body problem changes were the cosmos big enough to contain at least one instance of every plausible body and its actions: there would no longer be a pressing question of how a mind can cause a body to act in a particular way as out there somewhere there are bodies acting in every plausible way. For this picture to be feasible, the cosmos would have to be massively huge. This talk explores how chaos and the most parsimonious picture of quantum theory populates a staggeringly massive collection of every plausible body, thus contributing a new spin on a classic problem in the philosophy of mind.

### 1:30–2:00 PM: Mark Lewis (in person)

Department of Mathematics and Statistics, University of Victoria

**Title: **Nonlocal Multispecies Advection-Diffusion Models

**Abstract**

Nonlocal advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modeling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modeling the nonlocality. One common formalism is the aggregation-diffusion equation, a class of advection-diffusion models with nonlocal advection. This was originally used to model a single population but has recently been extended to the multispecies case to model the way organisms may alter their movement in the presence of coexistent species. Here we prove existence theorems for a class of nonlocal multispecies advection-diffusion models with an arbitrary number of coexisting species. We give methods for determining the qualitative structure of local minimum energy states and analyze the pattern formation potential using weakly nonlinear analysis and numerical methods. This is joint work with Valeria Giunta, Thomas Hillen and Jonathan Potts.

### 2:00–2:30 PM: Frithjof Lutscher (in person)

Department of Mathematics and Statistics, University of Ottawa

**Title: A hybrid model accounting for seasonal switches in food-web composition on a small island**

**Abstract:**

Seasonality and migration lead to relatively abrupt changes in food-web composition on islands in the Canadian North. While the food-web in the winter consists of relatively few mammalian species that reside on the island year round (lemmings, arctic foxes, ermines), additional avian predators may choose the island as nesting grounds if they perceive sufficient prey on the island. Under many simplifying assumptions, we derive a particularly simple model for the main prey species, the lemmings, as a hybrid model that can be reduced to a simple, non-smooth discrete-time model. We completely classify the behavior of the resulting model and present some rough comparison with data collected on Bylot island.

### 2:30–3:00PM: Geoffrey McGregor (in person)

Department of Mathematics, University of Toronto

**Title:** From a Mathematical Model for COVID-19 to Conservative Hamiltonian Monte Carlo

In this talk I will begin by briefly discussing a collaborative project between colleagues at the University of Northern British Columbia and the University of Waterloo, where we studied the difference between a regional and provincial-wide model for COVID-19 with physical distancing (McGregor et al., 2022). Intrigued by the methods utilized for parameter estimation in this work, Dr. Andy Wan and I began studying Bayesian statistics and learning about Markov Chain Monte Carlo and Hamiltonian Monte Carlo (HMC). For the remainder of this talk, I will discuss the strengths and weaknesses of HMC and introduce our work on Conservative Hamiltonian Monte Carlo (CHMC), where we tweak the algorithm employed by HMC to improve convergence and acceptance rates in high-dimensional problems.

### 3:30–3:30PM: Slim Ibrahim (in person)

Department of Mathematics and Statistics, University of Victoria

**Title: **TBA

**Abstract: **TBA

**Scientific, Seminar**

**February 17–18, 2024**

**-**

3800 Finnerty Road

Victoria, BC V8P 5C2