50 research outputs found
Predicting unobserved exposures from seasonal epidemic data
We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR)
epidemiological model with a contact rate that fluctuates seasonally. Through
the use of a nonlinear, stochastic projection, we are able to analytically
determine the lower dimensional manifold on which the deterministic and
stochastic dynamics correctly interact. Our method produces a low dimensional
stochastic model that captures the same timing of disease outbreak and the same
amplitude and phase of recurrent behavior seen in the high dimensional model.
Given seasonal epidemic data consisting of the number of infectious
individuals, our method enables a data-based model prediction of the number of
unobserved exposed individuals over very long times.Comment: 24 pages, 6 figures; Final version in Bulletin of Mathematical
Biolog
A primer on noise-induced transitions in applied dynamical systems
Noise plays a fundamental role in a wide variety of physical and biological
dynamical systems. It can arise from an external forcing or due to random
dynamics internal to the system. It is well established that even weak noise
can result in large behavioral changes such as transitions between or escapes
from quasi-stable states. These transitions can correspond to critical events
such as failures or extinctions that make them essential phenomena to
understand and quantify, despite the fact that their occurrence is rare. This
article will provide an overview of the theory underlying the dynamics of rare
events for stochastic models along with some example applications
Quantifying Transport of Passive Tracers and Inertial Particles in Geophysical Flows
There has been a steady increase in the deployment of autonomous underwater and surface vehicles for applications such as ocean monitoring, tracking of marine processes, and forecasting contaminant transport. The underwater environment poses unique challenges since robots must operate in a communication and localization-limited environment where their dynamics are tightly coupled with the environmental dynamics. This work presents current efforts in understanding the impact of geophysical fluid dynamics on underwater vehicle control and autonomy. The focus of the first part of the talk is on the control of collaborative vehicles to track Lagrangian coherent structures and to localize contaminant spills. In the second part of the talk, the focus is on the investigation of the dynamics of inertial particles in geophysical flows which include the Coriolis force
Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction
Extinction of an epidemic or a species is a rare event that occurs due to a
large, rare stochastic fluctuation. Although the extinction process is
dynamically unstable, it follows an optimal path that maximizes the probability
of extinction. We show that the optimal path is also directly related to the
finite-time Lyapunov exponents of the underlying dynamical system in that the
optimal path displays maximum sensitivity to initial conditions. We consider
several stochastic epidemic models, and examine the extinction process in a
dynamical systems framework. Using the dynamics of the finite-time Lyapunov
exponents as a constructive tool, we demonstrate that the dynamical systems
viewpoint of extinction evolves naturally toward the optimal path.Comment: 21 pages, 5 figures, Final revision to appear in Bulletin of
Mathematical Biolog
Noise, Bifurcations, and Modeling of Interacting Particle Systems
We consider the stochastic patterns of a system of communicating, or coupled,
self-propelled particles in the presence of noise and communication time delay.
For sufficiently large environmental noise, there exists a transition between a
translating state and a rotating state with stationary center of mass. Time
delayed communication creates a bifurcation pattern dependent on the coupling
amplitude between particles. Using a mean field model in the large number
limit, we show how the complete bifurcation unfolds in the presence of
communication delay and coupling amplitude. Relative to the center of mass, the
patterns can then be described as transitions between translation, rotation
about a stationary point, or a rotating swarm, where the center of mass
undergoes a Hopf bifurcation from steady state to a limit cycle. Examples of
some of the stochastic patterns will be given for large numbers of particles
Distributed allocation of mobile sensing swarms in gyre flows
We address the synthesis of distributed control policies to enable a swarm of
homogeneous mobile sensors to maintain a desired spatial distribution in a
geophysical flow environment, or workspace. In this article, we assume the
mobile sensors (or robots) have a "map" of the environment denoting the
locations of the Lagrangian coherent structures or LCS boundaries. Based on
this information, we design agent-level hybrid control policies that leverage
the surrounding fluid dynamics and inherent environmental noise to enable the
team to maintain a desired distribution in the workspace. We establish the
stability properties of the ensemble dynamics of the distributed control
policies. Since realistic quasi-geostrophic ocean models predict double-gyre
flow solutions, we use a wind-driven multi-gyre flow model to verify the
feasibility of the proposed distributed control strategy and compare the
proposed control strategy with a baseline deterministic allocation strategy.
Lastly, we validate the control strategy using actual flow data obtained by our
coherent structure experimental testbed.Comment: 10 pages, 14 Figures, added reference
Computing the optimal path in stochastic dynamical systems
In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high- dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly simple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higher- dimensional spaces
Distributed Blowing and Suction for the Purpose of Streak Control in a Boundary Layer Subjected to a Favorable Pressure Gradient
An analysis of the optimal control by blowing and suction in order to generate stream- wise velocity streaks is presented. The problem is examined using an iterative process that employs the Parabolized Stability Equations for an incompressible uid along with its adjoint equations. In particular, distributions of blowing and suction are computed for both the normal and tangential velocity perturbations for various choices of parameters