375 research outputs found
Stochastic epidemic models: a survey
This paper is a survey paper on stochastic epidemic models. A simple
stochastic epidemic model is defined and exact and asymptotic model properties
(relying on a large community) are presented. The purpose of modelling is
illustrated by studying effects of vaccination and also in terms of inference
procedures for important parameters, such as the basic reproduction number and
the critical vaccination coverage. Several generalizations towards realism,
e.g. multitype and household epidemic models, are also presented, as is a model
for endemic diseases.Comment: 26 pages, 4 figure
The Configuration Model for Partially Directed Graphs
The configuration model was originally defined for undirected networks and
has recently been extended to directed networks. Many empirical networks are
however neither undirected nor completely directed, but instead usually
partially directed meaning that certain edges are directed and others are
undirected. In the paper we define a configuration model for such networks
where nodes have in-, out-, and undirected degrees that may be dependent. We
prove conditions under which the resulting degree distributions converge to the
intended degree distributions. The new model is shown to better approximate
several empirical networks compared to undirected and completely directed
networks.Comment: 19 pages, 3 figures, 2 table
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
Stochastic epidemics in growing populations
Consider a uniformly mixing population which grows as a super-critical linear
birth and death process. At some time an infectious disease (of SIR or SEIR
type) is introduced by one individual being infected from outside. It is shown
that three different scenarios may occur: 1) an epidemic never takes off, 2) an
epidemic gets going and grows but at a slower rate than the community thus
still being negligible in terms of population fractions, or 3) an epidemic
takes off and grows quicker than the community eventually leading to an endemic
equilibrium. Depending on the parameter values, either scenario 1 is the only
possibility, both scenario 1 and 2 are possible, or scenario 1 and 3 are
possible.Comment: 11 page
Maximizing the size of the giant
We consider two classes of random graphs: Poissonian random graphs in
which the vertices in the graph have i.i.d.\ weights distributed as ,
where . Edges are added according to a product measure and
the probability that a vertex of weight shares and edge with a vertex of
weight is given by . A thinned configuration model
in which we create a ground-graph in which the vertices have i.i.d.\
ground-degrees, distributed as , with . The graph of
interest is obtained by deleting edges independently with probability .
In both models the fraction of vertices in the largest connected component
converges in probability to a constant , where depends on or
and .
We investigate for which distributions and with given and ,
is maximized. We show that in the class of Poissonian random graphs,
should have all its mass at 0 and one other real, which can be explicitly
determined. For the thinned configuration model should have all its mass at
0 and two subsequent positive integers
A network epidemic model with preventive rewiring: comparative analysis of the initial phase
This paper is concerned with stochastic SIR and SEIR epidemic models on
random networks in which individuals may rewire away from infected neighbors at
some rate (and reconnect to non-infectious individuals with
probability or else simply drop the edge if ), so-called
preventive rewiring. The models are denoted SIR- and SEIR-, and
we focus attention on the early stages of an outbreak, where we derive
expression for the basic reproduction number and the expected degree of
the infectious nodes using two different approximation approaches. The
first approach approximates the early spread of an epidemic by a branching
process, whereas the second one uses pair approximation. The expressions are
compared with the corresponding empirical means obtained from stochastic
simulations of SIR- and SEIR- epidemics on Poisson and
scale-free networks. Without rewiring of exposed nodes, the two approaches
predict the same epidemic threshold and the same for both types of
epidemics, the latter being very close to the mean degree obtained from
simulated epidemics over Poisson networks. Above the epidemic threshold,
pairwise models overestimate the value of computed from simulations,
which turns out to be very close to the one predicted by the branching process
approximation. When exposed individuals also rewire with (perhaps
unaware of being infected), the two approaches give different epidemic
thresholds, with the branching process approximation being more in agreement
with simulations.Comment: 25 pages, 7 figure
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