Fall 2021

Daihai He
Hong Kong Polytechnic University

Title:  Combining COVID-19 death data and serological survey results to estimate COVID-19 attack rate

The COVID-19 pandemic has caused a huge impact globally. One usually estimates the COVID-19 attack rate (the proportion of the population infected) in a country or region based on daily new or cumulative cases. However, the daily new cases are affected by the detection capacity and detection rate. Therefore, the estimates could be biased. While, the daily new deaths are relatively accurate. If the infection fatality rate (IFR) is known. Then the attack rate can be calculated based on cumulative deaths. On the other hand, COVID-19 patients have specific antibodies in their serum for a period of time. Through serological antibody surveys, one can also roughly estimate the attack rate. However, due to the sample size and sampling method, the estimates of AR through this method could also be biased. In this report, I will introduce how our team recently combined the death data and the serological survey to estimate the attack rate. I will use Mumbai, India and Manaus, Brazil as examples.

Audrey Fu
University of Idaho

Title:  Types of cis- and trans-gene regulation of expression quantitative trait loci across human tissues

The Genotype-Tissue Expression (GTEx) Consortium has identified expression quantitative trait loci (eQTLs) for most genes in the human genome across nearly 50 tissues or cell types. While most of the eQTLs are near the associated genes, some can be far away or on different chromosomes. We are interested in the regulatory relationship in a trio consisting of an eQTL, its cis- and trans-genes. Using MRPC, the causal network inference method we have developed, we have identified multiple types of regulatory net- works for trios across all the tissues. Across the tissues, more than half of the trios are inferred to be conditionally independent, where the two genes are conditionally independent given the genotype of the eQTL: cis-gene ← eQTL → target. Around 1.5% of the trios are inferred to be the mediation type, where one of the genes is the mediator, and the other the target: eQTL → mediator → target. Unexpectedly, across the tissues, on average more than half of the mediation trios have the trans-gene as the mediator. Furthermore, cis- gene mediators are enriched for protein-coding genes compared with the genome average, whereas trans-gene mediators are enriched for pseudogenes.

Glenn Webb
Vanderbilt University

Title:  Modelling the Aqueous Transport of an Infectious Pathogen in Regional Communities. Application to the Cholera Outbreak in Haiti in 2010

A mathematical model is developed to describe the dynamics of the spread of a waterborne disease among communities located along a flowing waterway. The model is formulated as a system of reaction-diffusion-advection partial differential equations in this spatial setting. The compartments of the model consist of susceptible, infected, and recovered individuals in the communities along the waterway, together with a term representing the pathogen load in each community, and a term representing the spatial concentration of pathogens flowing along the waterway. The model is applied to the cholera outbreak in Haiti in 2010.

Reference:  Fitzgibbon William E., Morgan Jeffrey J., Webb Glenn F. and Wu Yixiang , Modelling the aqueous transport of an infectious pathogen in regional communities: application to the cholera outbreak in Haiti.  J. R. Soc. Interface. 1720200429.

Dan Wilson

University of Tennessee

Title:  Model Order Reduction of Limit Cycle Oscillators Far Beyond the Weakly Perturbed Limit

Self-sustaining oscillatory behaviors are widely observed in the physical, chemical, and biological sciences.  Art Winfree made great strides in characterizing the perturbed behavior of nonlinear limit cycles by representing these oscillatory dynamics in terms of an asymptotic phase.  This so-called phase reduction has been used extensively in recent decades to successfully and elegantly characterize complicated patterns that emerge in groups of weakly interacting oscillators.
While standard phase reduction is useful in many situations, its applicability degrades as coupling strength increases often resulting in incorrect predictions about dynamical behavior.  Currently, very few general reduction techniques exist that can be used to analyze oscillatory dynamics in response to arbitrary, large magnitude coupling and other large magnitude inputs.
In this presentation, I will discuss two recently developed reduced order modeling frameworks that can be used to understand the aggregate behaviors of coupled oscillators in regimes that extend far beyond the weakly perturbed paradigm.  Both frameworks leverage the properties of isostable coordinates, which characterize level sets of the slowest decaying eigenmodes of the Koopman operator.  Numerical illustrations show that the proposed methods accurately reflect synchronization and entrainment dynamics of coupled oscillators in situations where several other phase-amplitude reduction strategies fail.  Applications related to neurological function and circadian physiology will be highlighted.

Eric Shea-Brown
University of Washington

Title:  Dimensionality in neural networks

There is an avalanche of new data on activity in neural networks and the biological brain, revealing the collective dynamics of vast numbers of neurons.  In principle, these collective dynamics can be of almost arbitrarily high dimension, with many independent degrees of freedom — and this may reflect powerful capacities for general computing or information.  In practice, neural datasets reveal a range of outcomes, including collective dynamics of much lower dimension — and this may reflect other desiderata for neural codes.  For what networks does each case occur?  We begin by exploring bottom-up mechanistic ideas that link tractable statistical properties of network connectivity with the dimension of the activity that they produce.  We then cover “top-down” ideas that describe how features of connectivity and dynamics that impact dimension arise as networks learn to perform fundamental computational tasks.

Stacey Smith?

Univeristy of Ottawa

Title:  Modelling the daily risk of Ebola in the presence and absence of a potential vaccine

Ebola virus – one of the deadliest viral diseases, with a mortality rate around 90% – damages the immune system and organs, with symptoms including episodic fever, chills, malaise and myalgia. The Recombinant Vesicular Stomatitis Virus-based candidate vaccine (rVSV-ZEBOV) has demonstrated clinical efficacy against Ebola in ring-vaccination clinical trials. In order to evaluate the potential effect of this candidate vaccine, we developed risk equations for the daily risk of Ebola infection both currently and after vaccination. The risk equations account for the basic transmission probability of Ebola and the lowered risk due to various protection protocols: vaccination, hazmat suits, reduced contact with the infected living and dead bodies. Parameter space was sampled using Latin Hypercube Sampling, a statistical method for generating a near-random sample of parameter values. We found that at a high transmission rate of Ebola (i.e., if the transmission rate is greater than 90%), a large fraction of the population must be vaccinated (>80%) to achieve a 50% decrease in the daily risk of infection. If a vaccine is introduced, it must have at least 50% efficacy, and almost everyone in the affected areas must receive it to effectively control outbreaks of Ebola. These results indicate that a low-efficacy Ebola vaccine runs the risk of having vaccinated people be overconfident in a weak vaccine and hence the possibility that the vaccine could make the situation worse, unless the population can be sufficiently educated about the necessity for high vaccine uptake.

Veronica Ciocanel
Duke University

Title:  Modeling and data analysis for filament organization in cell

Actin filaments are polymers that interact with myosin motor
proteins inside cells and play important roles in cell motility, shape,
and development. Depending on its function, this dynamic network of
interacting proteins reshapes and organizes in a variety of structures,
including bundles, clusters, and contractile rings. Motivated by
observations from the reproductive system of the roundworm C. elegans,
we use an agent-based modeling framework to simulate interactions
between actin filaments and myosin motor proteins inside cells. We also
develop tools based on topological data analysis to understand
time-series data extracted from these filamentous network interactions.
We use these tools to compare the filament organization resulting from
myosin motors with different properties. We have recently been
interested in gaining insights into myosin motor regulation and the
resulting actin network architectures during cell cycle progression.
This work also raises questions about how to assess the significance of
topological features in common topological summary visualizations.

Joe Tien
Ohio State University

Title:  From prisons to universities to misinformation: some experiences modeling SARS-CoV-2 dynamics

The SARS-CoV-2 pandemic has disrupted societal functioning at all levels.  I will describe some modeling efforts that my collaborators and I have been involved with for understanding SARS-CoV-2 dynamics.  This includes examination of an outbreak in an Ohio prison, utilizing repeat testing data to study incidence and prevalence on a college campus, and prospects for suppressing transmission in the context of SARS-CoV-2 misinformation.  I will discuss some of the mathematical and statistical approaches we have used and developed, which include survival analysis integrated into transmission models, dynamical systems analysis, and a 'sentinel node' network-based approach for monitoring online content and assessing breadth and depth of misinformation penetration.  This is joint work with many collaborators.

Yixiang Wu
Middle Tennessee State University

Title:  Global dynamics of a Lotka-Volterra competition patch model

The global dynamics of the two-species Lotka-Volterra competition patch model with asymmetric dispersal is classified under the assumptions of weak competition and  the weighted digraph of the connection matrix is strongly connected and cycle-balanced. It is shown that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs use the theory of monotone dynamical system and a graph-theoretic approach based on the Tree-Cycle identity. This talk is based on joint works with Shanshan Chen (Harbin Institute of Tech at Weihai), Junping Shi (William&Mary), and Zhisheng Shuai (UCF).

Cengiz Pehlevan
Harvard University

Title:  Inductive bias of neural networks
Abstract:  A learner's performance depends crucially on how its internal assumptions, or inductive biases, align with the task at hand. I will present a theory that describes the inductive biases of neural networks in the infinite width limit using kernel methods and statistical mechanics. This theory elucidates an inductive bias to explain data with “simple functions” which are identified by solving a related kernel eigenfunction problem on the data distribution. This notion of simplicity allows us to characterize whether a network is compatible with a learning task, facilitating good generalization performance from a small number of training examples. I will present applications of the theory to deep networks (at finite width) trained on synthetic and real datasets, and demonstrate that it explains generalization performance and inductive biases of deep networks very well. Then I will present applications of the theory to recordings from the mouse primary visual cortex. Finally, I will briefly present an extension of the theory to out-of-distribution generalization.

Naveen Vaidya
San Diego State University

Title:  Data-driven Modeling of COVID-19 from Within-Host Infection to Between-Host Transmission

The COVID-19 pandemic still poses a continuous threat due to the emergence of multiple strains. Insights into the virus infection dynamics within a host and virus transmission dynamics between hosts are crucial for effective control strategies. In the first part of this talk, I will present mathematical models describing how viruses spread within a single host. In particular, we implement viral load data from ferrets (animal model) to estimate key parameters related to SARS-CoV-2 infection. In the second part, I will discuss between-hosts models for the transmission dynamics of COVID-19 in a community. We apply our models to the unique data-sets from Nepal, which shares an open-border with India, one of the most COVID-19 impacted countries in the world. I will demonstrate how our data-driven models can provide vital information for developing control strategies for both within-host and between-host scales.    

Mason Porter
University of California, Los Angeles

Title:  Topological Data Analysis of Spatial Complex Systems

From the venation patterns of leaves to spider webs, roads in
cities, social networks, and the spread of COVID-19 infections and
vaccinations, the structure of many systems is influenced significantly
by space. In this talk, I'll discuss the application of topological data
analysis (specifically, persistent homology) to spatial systems. I'll
discuss a few examples, such as voting in presidential elections, city
street networks, spatiotemporal dynamics of COVID-19 infections and
vaccinations, and webs that were spun by spiders under the influence of
various drugs.

Mark Lewis
University of Alberta

Title:  Mathematical models for the neutral genetics of changing populations

In this talk I will discuss the genetic structure of populations subject to climate change and undergoing range expansion. The models and analyses are based on reaction diffusion and integrodifference equations for the asymptotic neutral genetic structure of populations. We decompose solutions into neutral genetic components called neutral fractions. The "inside dynamics" then describe the spatiotemporal evolution of these fractions and can be used to predict changes in genetic diversity. Extensions are made to include stage-structure in the population dynamics and mutations in the genetic fractions. Results are compared with small-scale experimental systems that have been developed to test the mathematical theory. This work is joint with Nathan Marculis, Roger Lui and Jimmy Garnier.

Patrick De Leenheer
Oregon State University

Title:  The mathematics behind the basic reproduction number R0

Abstract:  We review some mathematical results that are part of the folklore of the basic reproduction number, a concept that is prevalent in epidemiology and population biology. The basic reproduction number is commonly used in applications because it is often easier to calculate than the spectral radius of the non-negative matrix to which it is associated. Moreover, its value helps to establish the stability or instability of the linear recursion defined by the matrix, because, as the saying goes, "The spectral radius of a non-negative matrix, and its associated basic reproduction number, lie on the same side of 1". Consequently, controlling an infectious disease amounts to making the basic reproduction number less than 1. Perhaps not as well-known is that these results had already been obtained by Vandergraft in 1968 and are applicable to the more general class of linear maps that preserve a cone in R^n and not just to linear maps described by a non-negative matrix. Vandergraft's work was carried out decades before the notion of the basic reproduction number became popular in mathematical biology, yet interestingly, Vandergraft attributes the ideas to even earlier work in numerical analysis by Varga in 1963. We strengthen one of Vandergraft's results, albeit very slightly, using an idea of Li and Schneider that was proposed for linear maps which preserve the non-negative orthant cone. Looming in the background, and grounding all the proofs of these results, is the celebrated Perron-Frobenius Theorem for linear maps that preserve a cone, which is presented in a concise, yet comprehensive way in a relatively recent book by Lemmens and Nussbaum.  

Priscilla (Cindy) Greenwood
University of British Columbia

Title:  Building stochastic dynamical neural circuits for cortical control

We have data and successful models for several types of single
neurons. There are detailed diagrams showing the connectivity patterns of
parts of the brain. What we would like to understand, though is how the
brain "works", how we think, how we are aware of ourselves, how we
associate and remember. Even the comparatively simple matter of how our
attention is directed to a particular input is a topic of speculation.

In this talk I will explain a stochastic dynamic model of communication in
the brain through coherence. A message moves from one piece of cortex to a
nearby piece of cortex with help from a part of the thalamus called the
pulvinar. The cortices operate primarily using gamma (say 40 Hz)
oscillations, whereas the pulvinar operates using primarily alpha (say
10Hz) oscillations. The pulvinar controls cortical communication through
its input to the cortices. We show, mathematically, that this control may
happen through an optimal phase-offset between the two cortices in each of
the two oscillation frequencies, alpha and gamma. Using alpha input, the
pulvinar may control the direction of attention by choosing the cortical
connections to be made coherent. This is joint work with Lawrence Ward, UBC