31,657 research outputs found
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
Hamiltonian analysis of subcritical stochastic epidemic dynamics
We extend a technique of approximation of the long-term behavior of a
supercritical stochastic epidemic model, using the WKB approximation and a
Hamiltonian phase space, to the subcritical case. The limiting behavior of the
model and approximation are qualitatively different in the subcritical case,
requiring a novel analysis of the limiting behavior of the Hamiltonian system
away from its deterministic subsystem. This yields a novel, general technique
of approximation of the quasistationary distribution of stochastic epidemic and
birth-death models, and may lead to techniques for analysis of these models
beyond the quasistationary distribution. For a classic SIS model, the
approximation found for the quasistationary distribution is very similar to
published approximations but not identical. For a birth-death process without
depletion of susceptibles, the approximation is exact. Dynamics on the phase
plane similar to those predicted by the Hamiltonian analysis are demonstrated
in cross-sectional data from trachoma treatment trials in Ethiopia, in which
declining prevalences are consistent with subcritical epidemic dynamics
Stochastic epidemics in a homogeneous community
These notes describe stochastic epidemics in a homogenous community. Our main
concern is stochastic compartmental models (i.e. models where each individual
belongs to a compartment, which stands for its status regarding the epidemic
under study : S for susceptible, E for exposed, I for infectious, R for
recovered) for the spread of an infectious disease. In the present notes we
restrict ourselves to homogeneously mixed communities. We present our general
model and study the early stage of the epidemic in chapter 1. Chapter 2 studies
the particular case of Markov models, especially in the asymptotic of a large
population, which leads to a law of large numbers and a central limit theorem.
Chapter 3 considers the case of a closed population, and describes the final
size of the epidemic (i.e. the total number of individuals who ever get
infected). Chapter 4 considers models with a constant influx of susceptibles
(either by birth, immigration of loss of immunity of recovered individuals),
and exploits the CLT and Large Deviations to study how long it takes for the
stochastic disturbances to stop an endemic situation which is stable for the
deterministic epidemic model. The document ends with an Appendix which presents
several mathematical notions which are used in these notes, as well as
solutions to many of the exercises which are proposed in the various chapters.Comment: Part I of "Stochastic Epidemic Models with Inference", T. Britton &
E. Pardoux eds., Lecture Notes in Mathematics 2255, Springer 201
Timing of Pathogen Adaptation to a Multicomponent Treatment
The sustainable use of multicomponent treatments such as combination
therapies, combination vaccines/chemicals, and plants carrying multigenic
resistance requires an understanding of how their population-wide deployment
affects the speed of the pathogen adaptation. Here, we develop a stochastic
model describing the emergence of a mutant pathogen and its dynamics in a
heterogeneous host population split into various types by the management
strategy. Based on a multi-type Markov birth and death process, the model can
be used to provide a basic understanding of how the life-cycle parameters of
the pathogen population, and the controllable parameters of a management
strategy affect the speed at which a pathogen adapts to a multicomponent
treatment. Our results reveal the importance of coupling stochastic mutation
and migration processes, and illustrate how their stochasticity can alter our
view of the principles of managing pathogen adaptive dynamics at the population
level. In particular, we identify the growth and migration rates that allow
pathogens to adapt to a multicomponent treatment even if it is deployed on only
small proportions of the host. In contrast to the accepted view, our model
suggests that treatment durability should not systematically be identified with
mutation cost. We show also that associating a multicomponent treatment with
defeated monocomponent treatments can be more durable than associating it with
intermediate treatments including only some of the components. We conclude that
the explicit modelling of stochastic processes underlying evolutionary dynamics
could help to elucidate the principles of the sustainable use of multicomponent
treatments in population-wide management strategies intended to impede the
evolution of harmful populations.Comment: 3 figure
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
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