In modeling biological systems and other applications, an important recurring question is whether those systems maintain healthy regimes not just at dynamic attractors but also during transient excursions away from those attractors. For ODE models, these excursions are not due only to nonlinearities; they can also occur in the linearization. Since transient excursions in linearized systems are not preserved by diagonalization, we develop a new framework for analyzing linear systems through the lens of reactivity (the maximum instantaneous rate of radial amplification). We shift attention to a radial and tangential decomposition of the vector field and conjugate by rotation to generate equivalent matrices that preserve reactivity. In particular, we use our new framework to pick four standard matrices that give more direct access to reactivity and highlight interesting parallels between reactivity and eigenvectors. We finish by exploring applications of this framework to examples in ecology.

Noise-induced transitions occur when the presence of noise prompts the state of the system to transition from one (meta)stable state to another, potentially as a rare event. Understanding a system’s behavior through this transition is a significant area of current interest. In this talk, we will describe recent analytical and computational tools to identify Most Probable Escape Paths (MPEPs) in Stochastic Differential Equations (SDEs) over periodic boundaries. The study of noise-induced tipping has been dominated by Freidlin-Wentzell (FW) theory. One limitation of the FW theory is that it necessitates vanishingly small noise. However, for several applications of interest, particularly in environmental, biological, or social contexts, intermediate noise is of more relevance. In focusing on the intermediate noise regime, the transient behavior of the underlying deterministic system will play a key role. We will thus use a dynamical system approach to identify MPEPs in SDEs. The Maslov index will help us distinguish which critical points of the FW functional are minimizers and help explain the effect of the interaction of noise and transient dynamics. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out a specific MPEP.  Subsequently, our computations will be compared with Monte Carlo simulations to verify theoretical predictions in the Inverted Van der Pol system.  

Antibody tests are routinely used to identify past infection, with examples including Lyme disease and, of course, COVID-19. An accurate classification strategy is crucial to interpreting diagnostic test results. However, classification of samples as positive or negative is difficult when the corresponding measurement values overlap. In the first part of this talk, I will discuss a data-driven approach to separating positive and negative populations. We accomplish this by using more available measurements per person to build probabilistic models that capture structural characteristics of the data. These models are inputs to an optimal decision theory framework for classification. I will then extend to the situation with more than two classes. Such a multiclass classification is complicated when the relative fraction of the population in each class is unknown. To address this, we develop a prevalence estimation method that is independent of classification and an associated classification scheme that minimizes false classifications. Our methods are applied to SARS-CoV-2 antibody data as an illustration.

Hydrologic and ecological processes are tightly coupled in drylands – semiarid and arid regions that comprise some 40% of the earth's land surface and support over 2 billion people.  In these water-limited regions, the redistribution of rainfall runoff from bare soil to vegetated patches is critical to plant survival and the maintenance of ecosystem function, sustaining vegetation in regions where annual rainfall inputs alone are insufficient. Observed ecohydrological feedbacks between surface runoff and ecosystem function have motivated ongoing efforts to infer ecosystem vulnerability from spatial patterns of vegetation.  At the heart of such efforts is the premise that dryland vegetation self-organizes to promote the capture of surface runoff and that spatial pattern can thus be used to assess proximity to ‘tipping points’. While these efforts hinge on the ‘runoff run-on’ conceptual model, progress has been hindered by challenges involved in translating this conceptual understanding to a predictive one.  This talk addresses three such challenges: (1) resolving uncertainties in the physics of very shallow flows traversing heterogeneous terrain, (2) bridging patch- and hillslope-scale runoff processes in dryland ecosystems, and (3) relating runoff at storm timescales to vegetation growth and dispersal on annual to decadal timescales.  The results provide a blueprint for future research developing dryland monitoring tools, restoration strategies, and agricultural techniques maximizing the harvesting of water through the spatial organization of crops.

Many models encode a switch in dynamics to account for switches in the behavior of the system modeled: neurons fire and relax, predators switch between prey, water melts and freezes. When noise causes a system to transition to another state across such a switch, we call it a noise-induced transition. Transition paths are typically derived as functions that minimize the Freidlin-Wentzell functional. However, this requires differentiability, which is often lost across a switch. We explore recent efforts to study noise-induced transitions in systems with switches, particularly in the case of a piecewise-smooth vector field where switching occurs across an (n-1)-dimensional manifold. We will apply this to a conceptual case study model for Arctic energy balance.

To understand the fluid dynamics of marine phenomena, fluid-structure interaction problems must be solved. Challenges exist in developing numerical methods to solve these  flow problems with boundary conditions at fluid-structure interfaces. I will present details of two  problems where these challenges are addressed: (1) modeling and simulations of pulsating soft corals and (2) accurate evaluation of layer potentials near interfaces. The first problem of pulsating soft corals will be motivated by field and experimental work in the marine sciences. I will discuss these related data and provide comparisons with the modeling. For the second problem of accurate evaluation of layer potentials, I will show how classical numerical methods are problematic for evaluations close to boundaries and how newly developed numerical methods can be used to improve accuracy.

Motivated by the study of complex social behavior, we introduce a novel nonlinear model of belief formation dynamics. According to our model, belief updates of individuals are informed by the complex interplay of external factors, i.e. social network effects, and internal factors, i.e. internal biases, networked relationships between an individual's belief representations, and nonlinear processing of social information. We rigorously show how groups overcome deadlock to form strong beliefs when it is urgent to do so, how the structure of social relationships and of the underlying belief system shapes social decisions in the group, and how group-level beliefs can be highly sensitive to the personal biases of a small number of individuals. This work provides novel insights into the dynamics of complex social systems in nature and society and motivates a new approach for the design of distributed behavior in engineered networks of social agents.

One of the most fundamental yet elusive questions in Neuroscience is how our brains perceive important sensory signals among a muddled pool of extraneous information (that is often thought of as 'noise'). We present a biologically plausible and viable solution based on our recent mathematical results. We test our new theory, that is akin to an unsupervised learning algorithm, on 2 vastly different data sets (visual and olfactory) with promising results. Our theory may generalize to neural systems beyond sensory systems, provided certain conditions (based on the math) are met.

Physiologically-based pharmacokinetic (PBPK) modeling is a technique for compartment modeling of biological entities. This talk discusses an application of PBPK modeling, and in particular, dermal modeling techniques.  Suitable for all audiences, this talk will introduce both the biology and the mathematics behind PBPK models in context.

Under-reporting and, thus, uncertainty around the true incidence of health events is common in all public health reporting systems. While the problem of under-reporting is acknowledged in epidemiology, the guidance and methods available for assessing and correcting the resulting bias are obscure. We present a simple method for the Susceptible – Infected – Removed (SIR) model for estimating the fraction or proportion of reported infection cases. The proposed method is illustrated using simulated data with known disease cases and applied to two empirical reported data sets to estimate the fraction of reported cases in Missoula County, Montana, USA, using: (1) flu data for 2016 – 2017 and (2) COVID-19 data for fall of 2020.

In vertebrates, primary photoreceptor cells consisting of rods and cones undergo daily renewal of their outer segment through the addition of disc-like structures at their base and shedding of these discs at their distal end.  During retinal detachment (RD), the neural layer of the eye is removed from the retinal pigmented epithelium, which prevents the supply of nutrients to these cells, causing the outer segment of these cells to shorten. If surgical procedures for retinal reattachment are not done in a timely manner, the cells outer segment will shorten to zero, leading to cell death and potential blindness. The precise effect of RD on the renewal process is not well understood. Additionally, a time frame within which reattachment of the retina can restore proper photoreceptor cell function is unclear. Focusing on rod cells, we propose a mathematical model to clarify the influence of retinal detachment on the renewal process. Our model simulation and analysis suggests that RD stops or significantly reduces the formation of new discs at the base, a theory that is disputed in the literature.

Polymerization, or aggregation, is essential for many physiological systems. For example, the emergence of a fibrin polymer mesh during the formation of a blood clot is required for a stable clot and long-term, sustained intracellular transport in neurons rely on persistent yet dynamic polymers that comprise the microtubule cytoskeleton. In this talk, I will discuss efforts I have made towards understanding polymerization processes in physiology, as well as new research directions. In the first part, I will discuss mathematical models that represent the formation of a fibrin polymer mesh with interactions with its precursor molecule, fibrinogen. We investigate how these interactions can impact gel structure and gel time. In the second part, I will introduce a stochastic mathematical model of microtubule growth in the dendrite of a neuron. Using parameters informed by experimental data, we explore what mechanisms could control the equilibrium microtubule length and validate these mechanisms using fluorescence microscopy data.

I work as a collaborative Biostatistician at The Ohio State University Wexner Medical Center. As part of my role, I assist oncologist and pharmacy residents with study design, data management, statistical analyses, and manuscript preparation on various research projects. Most of the studies that I work on are observational studies where we use secondary data sources such as the National Cancer Database (NCDB) and the Surveillance and Epidemiology End Results (SEER) registries. Today I will talk about three of the studies that I have or am currently working on, where all three studies are oncological retrospective cohort studies using three different data sources. The in the first study, I will talk about how we used data from the electronic medical record to assess the association between racial segregation and survival outcomes among patients diagnosed with breast cancer. The second study will focus on using SEER to assess the association of race and survival outcomes among patients diagnosed with ductal carcinoma in-situ with microinvasion. In the third study, I will talk about how we are using the NCDB to assess the role of systemic therapy in patients with metastatic bladder cancer.

Here on earth we live in the “Goldilocks zone” where conditions are "just right", temperatures are neither too hot nor too cold, providing the perfect cradle for Earth to sustain liquid water and nurture life as we know it. Our study aims to identify the optimal immune recovery in children after transplantation, where patients can thrive with reduced chances of leukemia relapse and adverse transplant outcomes.
Stem cell transplant provides a possible cure for leukemia; however, cancer recurrence still occurs in many and remains the primary cause of treatment failure in children. Transplant provides a unique treatment option as donor immune T cells, can recognize, and kill leukemia cells, a process called graft vs. leukemia (GVL). T cells recognize altered proteins present on cancer cells through their T cell receptors. They also demonstrate functional activation and exhaustion patterns which tell us how effectively they will kill cancer cells. Currently doctors have no way of understanding or predicting who will benefit from GVL.  Graft vs. leukemia, along with T cell activation and diversity, have predominantly been researched within the adult population, yet children are fundamentally different, not merely miniature adults. In children undergoing transplantation, an operational thymus exists, the thymus is an organ to help train the immune system to recognize self from non-self which we believe will influence their T cell activation and diversity when compared to adults. Gaining deeper insights into these distinctions could enable us to refine transplant protocols for our pediatric patients, which are traditionally derived from adult-centric methodologies.
 
Our goal is to identify clinical parameters that can forecast the course of T cell reconstitution before any relapse occurs. Ultimately, we strive to enhance our ability to anticipate the immune trajectory of pediatric patients following transplantation. By studying immune recovery and developmental patterns, we might detect certain trends that suggest an increased risk of disease relapse and adverse transplantation outcomes specifically within the pediatric population. Such knowledge will enable us to tailor therapies, fine-tuning them to hit the therapeutic "Goldilocks zone," and thus minimize the likelihood of unwanted outcomes post-transplant. With this enriched comprehension, we aim to devise subsequent interventional clinical trials, personalizing patients' immunosuppression, and employing immunomodulatory drugs to enhance post-transplant outcomes.

Filamentous material may exhibit structure dependent material properties and function that depends on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links.  In this talk we will introduce a novel framework in knot theory that can characterize the complexity of (collections of) open curves in 3-space in general. This leads to novel metrics of entanglement of open curves in 3-space that generalize classical topological invariants, like, for example, the Jones polynomial and Vassiliev invariants. For open curves, these are continuous functions of the curve coordinates and tend to topological invariants of classical knots and links when the endpoints of the curves tend to coincide. We will apply our methods to proteins and we will show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data.  When applied to the SARS-CoV-2 spike protein, we see that the local native topological landscape can predict residues where mutations can have an important impact on protein structure and possibly in viral transmissibility.  These methods thus not only open a new mathematical direction in knot theory, but can also help us understand polymer and biopolymer function and material properties in many contexts with the goal of their prediction and design.

Barrier islands found along the coast are constantly changing in shape and size due to sea level rise and intense weather.  Because these islands are essential in protecting communities near the coast, it is critical to understand their evolution.  We created a cellular model to approximate the evolution of these islands.  This model utilizes data from some Virginia barrier islands, and incorporates both biotic and abiotic processes, which include vegetation, beach dynamics, wind erosion, and overwash processes for various storm strengths and sea level rise estimates to investigate island evolution over time.