209 research outputs found
Dangerous connections: on binding site models of infectious disease dynamics
We formulate models for the spread of infection on networks that are amenable
to analysis in the large population limit. We distinguish three different
levels: (1) binding sites, (2) individuals, and (3) the population. In the
tradition of Physiologically Structured Population Models, the formulation
starts on the individual level. Influences from the `outside world' on an
individual are captured by environmental variables. These environmental
variables are population level quantities. A key characteristic of the network
models is that individuals can be decomposed into a number of conditionally
independent components: each individual has a fixed number of `binding sites'
for partners. The Markov chain dynamics of binding sites are described by only
a few equations. In particular, individual-level probabilities are obtained
from binding-site-level probabilities by combinatorics while population-level
quantities are obtained by averaging over individuals in the population. Thus
we are able to characterize population-level epidemiological quantities, such
as , , the final size, and the endemic equilibrium, in terms of the
corresponding variables
Some aspects of non-uniform convergence in an elliptic singular perturbation problem : (prepublication)
Mean field at distance one
To be able to understand how infectious diseases spread on networks, it is
important to understand the network structure itself in the absence of
infection. In this text we consider dynamic network models that are inspired by
the (static) configuration network. The networks are described by
population-level averages such as the fraction of the population with
partners, This means that the bookkeeping contains information
about individuals and their partners, but no information about partners of
partners. Can we average over the population to obtain information about
partners of partners? The answer is `it depends', and this is where the mean
field at distance one assumption comes into play. In this text we explain that,
yes, we may average over the population (in the right way) in the static
network. Moreover, we provide evidence in support of a positive answer for the
network model that is dynamic due to partnership changes. If, however, we
additionally allow for demographic changes, dependencies between partners
arise. In earlier work we used the slogan `mean field at distance one' as a
justification of simply ignoring the dependencies. Here we discuss the
subtleties that come with the mean field at distance one assumption, especially
when demography is involved. Particular attention is given to the accuracy of
the approximation in the setting with demography. Next, the mean field at
distance one assumption is discussed in the context of an infection
superimposed on the network. We end with the conjecture that an extension of
the bookkeeping leads to an exact description of the network structure.Comment: revised versio
Structured populations with distributed recruitment: from PDE to delay formulation
In this work first we consider a physiologically structured population model
with a distributed recruitment process. That is, our model allows newly
recruited individuals to enter the population at all possible individual
states, in principle. The model can be naturally formulated as a first order
partial integro-differential equation, and it has been studied extensively. In
particular, it is well-posed on the biologically relevant state space of
Lebesgue integrable functions. We also formulate a delayed integral equation
(renewal equation) for the distributed birth rate of the population. We aim to
illustrate the connection between the partial integro-differential and the
delayed integral equation formulation of the model utilising a recent spectral
theoretic result. In particular, we consider the equivalence of the steady
state problems in the two different formulations, which then leads us to
characterise irreducibility of the semigroup governing the linear partial
integro-differential equation. Furthermore, using the method of
characteristics, we investigate the connection between the time dependent
problems. In particular, we prove that any (non-negative) solution of the
delayed integral equation determines a (non-negative) solution of the partial
differential equation and vice versa. The results obtained for the particular
distributed states at birth model then lead us to present some very general
results, which establish the equivalence between a general class of partial
differential and delay equation, modelling physiologically structured
populations.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
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