887 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
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
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
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
A Framework for Inferring Unobserved Multistrain Epidemic Subpopulations Using Synchronization Dynamics
A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations
Escape Rates in a Stochastic Environment with Multiple Scales
We consider a stochastic environment with two time scales and outline a
general theory that compares two methods to reduce the dimension of the
original system. The first method involves the computation of the underlying
deterministic center manifold followed by a naive replacement of the stochastic
term. The second method allows one to more accurately describe the stochastic
effects and involves the derivation of a normal form coordinate transform that
is used to find the stochastic center manifold. The results of both methods are
used along with the path integral formalism of large fluctuation theory to
predict the escape rate from one basin of attraction to another. The general
theory is applied to the example of a surface flow described by a generic,
singularly perturbed, damped, nonlinear oscillator with additive, Gaussian
noise. We show how both nonlinear reduction methods compare in escape rate
scaling. Additionally, the center manifolds are shown to predict high
pre-history probability regions of escape. The theoretical results are
confirmed using numerical computation of the mean escape time and escape
prehistory, and we briefly discuss the extension of the theory to stochastic
control.Comment: 32 pages, 8 figures, Final revision to appear in SIAM Journal on
Applied Dynamical System
Converging towards the optimal path to extinction
Extinction appears ubiquitously in many fields, including chemical reactions,
population biology, evolution, and epidemiology. Even though extinction as a
random process is a rare event, its occurrence is observed in large finite
populations. Extinction occurs when fluctuations due to random transitions act
as an effective force which drives one or more components or species to vanish.
Although there are many random paths to an extinct state, there is an optimal
path that maximizes the probability to extinction. In this article, we show
that the optimal path is associated with the dynamical systems idea of having
maximum sensitive dependence to initial conditions. Using the equivalence
between the sensitive dependence and the path to extinction, we show that the
dynamical systems picture of extinction evolves naturally toward the optimal
path in several stochastic models of epidemics.Comment: 18 pages, 5 figures, Final revision in Journal of the Royal Society
Interface. arXiv admin note: substantial text overlap with arXiv:1003.091
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