Other projects in dynamical systems or mathematical biology:

To branch out into new areas, I have worked on diverse projects outside of my main research program. Two of these projects are broadly related to stochastic processes: applying a dynamical-systems approach to noise-induced tipping and reframing pattern formation using piecewise-deterministic Markov processes. My other projects involve building network, agent-based, and hybrid models to investigate disease spread, intracellular transport, and tumor growth, respectively. My work on modeling intracellular transport is ongoing.

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Noise-induced tipping in periodically forced systems

Together with Yuxin Chen, John Gemmer, and Mary Silber, I am interested in how additive noise impacts periodically-forced systems with a double-well potential structure. While our original motivation was based on seasonal fluctuations in the depth of Arctic sea ice (in this setting, tipping could represent jumping from a stable ice-covered ocean to a stable ice-free Arctic), we have focused on a more general system that allows us to explore different time scales in the problem:

dXt = (1/ε) ( Xt - Xt3 + α + A cos(2πt) )dt + σdWt
= (1/ε) f(x,t) + σdWt,
where σ is the noise strength and ε measures how quickly the flow relaxes to its steady states relative to the external forcing period. The case when ε << 1 has been studied extensively, so we instead focus on ε = O(1). We use a path integral formulation to identify most likely tipping paths between stable periodic orbits, and my role has been to explore bifurcations in the problem. Interestingly, the associated Euler-Lagrange equations have a Hamiltonian system, namely:

x' = ψ + (1/ε) f(x,t)
ψ' = (1/ε) fx(x,t) ψ + ( σ2/(2ε) ) fxx(x,t),
where ψ = x' - (1/ε) f(x,t) is the "momentum" and measures the deviation from the deterministic dynamics introduced by noise. The Hamiltonian system admits several periodic orbits, some of which correspond to the solutions of the underlying deterministic problem when σ = 0. I have been using AUTO-07p (numerical continuation) to explore bifurcations in the Hamiltonian equations, which may be related to regions with qualitatively different noise-induced tipping behavior.

Yuxin Chen (Vanguard), John Gemmer (Wake Forest University), Mary Silber (University of Chicago)


Agent-based modeling of intracellular transport in neurons

Neurons depend on a system of intracellular roadways, called microtubules, to transport organelle "cargoes" throughout the cell. Together with Chuan Xue, I have been developing a stochastic agent-based model of intracellular transport in 3-D axons. By specifying a series of fixed microtubule arrangements, we explore how the organization of these roadways affects cargo transport.

Chuan Xue (University of Minnesota)

Network construction from diary-based data

This project focused on lifting diary-based data from a college social community to a larger extended network. We constructed separate networks for home, social, and work interactions and tested how dynamic changes in these networks affect disease spread. Simulating an influenza outbreak using an SIR (susceptible-infected-recovered) model, we found that reducing encounters at work after infection is an effective way of decreasing flu season severity.

Joshua Rubin Abrams* (University of Arizona), Anne Schwartz* (Amazon), Maria-Veronica Ciocanel (Duke University), Kristina Mallory (Brown University), Björn Sandstede (Brown University)

*Asterisk denotes undergraduate students mentored


Hybrid modeling of drug resistance and tumor dynamics

This project on tumor growth arose from the WhAM! Workshop for Women in Applied Math at the IMA in the fall of 2014. We developed a hybrid model to explore how different mechanisms of drug resistance impact tumor behavior. Our model specifies PDEs for drug and oxygen diffusion from stationary blood vessels, and allows discrete cancer cells to reproduce in a 2-D domain under two conditions: acquired (drug-induced) and pre-existing resistance. Our results suggest that the tumor micro-environment has a strong impact on cancer dynamics when drug resistance is acquired.

Zahra Aminzare (University of Iowa), Jana L. Gevertz (The College of New Jersey), Kerri-Ann Norton (Bard College), Judith Perez-Velazquez (Helmholtz Zentrum München), Katarzyna A. Rejniak (Moffit Cancer Center)

Toy models of pattern formation using piecewise-deterministic Markov processes

Agent-based models arise naturally in many different settings: pedestrians in a crowded room, shoaling fish, and cars on a road can all be studied as systems of moving agents. Often it is appropriate to consider a set number of agents (e.g., no off-ramp), but some systems couple movement with random fluctuations in population size. This summer REU project was motivated by zebrafish patterning, since our modeling work suggests that the time-scales for cell migration, birth, and death are similar in this setting. We are interested in understanding the stability of zebrafish stripes directly from an agent-based perspective, and, more generally, exploring the long-term behavior of agent-based models that couple deterministic movement with random fluctuations in population size. Our work focused first on a toy model of patterning in one dimension and made use of piecewise-deterministic Markov processes (PDMPs) to describe agent dynamics.

Cassandra Cole* (Brown University), Philip Doldo* (Cornell University), Claire Qing Fan* (University of Chicago), Maria-Veronica Ciocanel (Duke University), Björn Sandstede (Brown University)

*Asterisk denotes undergraduate students mentored