20,505 research outputs found
Approximating time to extinction for endemic infection models
Approximating the time to extinction of infection is an important problem in
infection modelling. A variety of different approaches have been proposed in
the literature. We study the performance of a number of such methods, and
characterize their performance in terms of simplicity, accuracy, and
generality. To this end, we consider first the classic stochastic
susceptible-infected-susceptible (SIS) model, and then a multi-dimensional
generalization of this which allows for Erlang distributed infectious periods.
We find that (i) for a below-threshold infection initiated by a small number of
infected individuals, approximation via a linear branching process works well;
(ii) for an above-threshold infection initiated at endemic equilibrium, methods
from Hamiltonian statistical mechanics yield correct asymptotic behaviour as
population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck
diffusion approximation gives a very poor approximation, but may retain some
value for qualitative comparisons in certain cases; (iv) a more detailed
diffusion approximation can give good numerical approximation in certain
circumstances, but does not provide correct large population asymptotic
behaviour, and cannot be relied upon without some form of external validation
(eg simulation studies)
The snowball effect of customer slowdown in critical many-server systems
Customer slowdown describes the phenomenon that a customer's service
requirement increases with experienced delay. In healthcare settings, there is
substantial empirical evidence for slowdown, particularly when a patient's
delay exceeds a certain threshold. For such threshold slowdown situations, we
design and analyze a many-server system that leads to a two-dimensional Markov
process. Analysis of this system leads to insights into the potentially
detrimental effects of slowdown, especially in heavy-traffic conditions. We
quantify the consequences of underprovisioning due to neglecting slowdown,
demonstrate the presence of a subtle bistable system behavior, and discuss in
detail the snowball effect: A delayed customer has an increased service
requirement, causing longer delays for other customers, who in turn due to
slowdown might require longer service times.Comment: 23 pages, 8 figures -- version 3 fixes a typo in an equation. in
Stochastic Models, 201
Extinction times in the subcritical stochastic SIS logistic epidemic
Many real epidemics of an infectious disease are not straightforwardly super-
or sub-critical, and the understanding of epidemic models that exhibit such
complexity has been identified as a priority for theoretical work. We provide
insights into the near-critical regime by considering the stochastic SIS
logistic epidemic, a well-known birth-and-death chain used to model the spread
of an epidemic within a population of a given size . We study the behaviour
of the process as the population size tends to infinity. Our results cover
the entire subcritical regime, including the "barely subcritical" regime, where
the recovery rate exceeds the infection rate by an amount that tends to 0 as but more slowly than . We derive precise asymptotics for
the distribution of the extinction time and the total number of cases
throughout the subcritical regime, give a detailed description of the course of
the epidemic, and compare to numerical results for a range of parameter values.
We hypothesise that features of the course of the epidemic will be seen in a
wide class of other epidemic models, and we use real data to provide some
tentative and preliminary support for this theory.Comment: Revised; 34 pages; 6 figure
Circular Stochastic Fluctuations in SIS Epidemics with Heterogeneous Contacts Among Sub-populations
The conceptual difference between equilibrium and non-equilibrium steady
state (NESS) is well established in physics and chemistry. This distinction,
however, is not widely appreciated in dynamical descriptions of biological
populations in terms of differential equations in which fixed point, steady
state, and equilibrium are all synonymous. We study NESS in a stochastic SIS
(susceptible-infectious-susceptible) system with heterogeneous individuals in
their contact behavior represented in terms of subgroups. In the infinite
population limit, the stochastic dynamics yields a system of deterministic
evolution equations for population densities; and for very large but finite
system a diffusion process is obtained. We report the emergence of a circular
dynamics in the diffusion process, with an intrinsic frequency, near the
endemic steady state. The endemic steady state is represented by a stable node
in the deterministic dynamics; As a NESS phenomenon, the circular motion is
caused by the intrinsic heterogeneity within the subgroups, leading to a broken
symmetry and time irreversibility.Comment: 29 pages, 5 figure
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
Matrix-geometric solution of infinite stochastic Petri nets
We characterize a class of stochastic Petri nets that can be solved using matrix geometric techniques. Advantages of such on approach are that very efficient mathematical technique become available for practical usage, as well as that the problem of large state spaces can be circumvented. We first characterize the class of stochastic Petri nets of interest by formally defining a number of constraints that have to be fulfilled. We then discuss the matrix geometric solution technique that can be employed and present some boundary conditions on tool support. We illustrate the practical usage of the class of stochastic Petri nets with two examples: a queueing system with delayed service and a model of connection management in ATM network
- âŠ