65 research outputs found

### 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

### Splitting trees stopped when the first clock rings and Vervaat's transformation

We consider a branching population where individuals have i.i.d.\ life
lengths (not necessarily exponential) and constant birth rate. We let $N_t$
denote the population size at time $t$. %(called homogeneous, binary
Crump--Mode--Jagers process). We further assume that all individuals, at birth
time, are equipped with independent exponential clocks with parameter $\delta$.
We are interested in the genealogical tree stopped at the first time $T$ when
one of those clocks rings. This question has applications in epidemiology, in
population genetics, in ecology and in queuing theory.
We show that conditional on $\{T<\infty\}$, the joint law of $(N_T, T,
X^{(T)})$, where $X^{(T)}$ is the jumping contour process of the tree truncated
at time $T$, is equal to that of $(M, -I_M, Y_M')$ conditional on
$\{M\not=0\}$, where : $M+1$ is the number of visits of 0, before some single
independent exponential clock $\mathbf{e}$ with parameter $\delta$ rings, by
some specified L{\'e}vy process $Y$ without negative jumps reflected below its
supremum; $I_M$ is the infimum of the path $Y_M$ defined as $Y$ killed at its
last 0 before $\mathbf{e}$; $Y_M'$ is the Vervaat transform of $Y_M$.
This identity yields an explanation for the geometric distribution of $N_T$
\cite{K,T} and has numerous other applications. In particular, conditional on
$\{N_T=n\}$, and also on $\{N_T=n, T<a\}$, the ages and residual lifetimes of
the $n$ alive individuals at time $T$ are i.i.d.\ and independent of $n$. We
provide explicit formulae for this distribution and give a more general
application to outbreaks of antibiotic-resistant bacteria in the hospital

### Maximizing the size of the giant

We consider two classes of random graphs: $(a)$ Poissonian random graphs in
which the $n$ vertices in the graph have i.i.d.\ weights distributed as $X$,
where $\mathbb{E}(X) = \mu$. Edges are added according to a product measure and
the probability that a vertex of weight $x$ shares and edge with a vertex of
weight $y$ is given by $1-e^{-xy/(\mu n)}$. $(b)$ A thinned configuration model
in which we create a ground-graph in which the $n$ vertices have i.i.d.\
ground-degrees, distributed as $D$, with $\mathbb{E}(D) = \mu$. The graph of
interest is obtained by deleting edges independently with probability $1-p$.
In both models the fraction of vertices in the largest connected component
converges in probability to a constant $1-q$, where $q$ depends on $X$ or $D$
and $p$.
We investigate for which distributions $X$ and $D$ with given $\mu$ and $p$,
$1-q$ is maximized. We show that in the class of Poissonian random graphs, $X$
should have all its mass at 0 and one other real, which can be explicitly
determined. For the thinned configuration model $D$ should have all its mass at
0 and two subsequent positive integers

### Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new percolation model to describe and analyze the spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in such a way
that the weights of different vertices are i.i.d., but the two entries of the
vector assigned to a vertex need not be independent. The probability for an
edge to be open depends on the weights of its end vertices, but conditionally
on the weights, the states of the edges are independent of each other. In an
epidemiological setting, the vertices of a graph represent the individuals in a
(social) network and the edges represent the connections in the network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and below by the
corresponding quantities for respectively independent bond and site percolation
with certain densities; this generalizes a result of Kuulasmaa. Many models in
the literature are special cases of our general model.Comment: 15 page

### Disease management in organic apple orchards is more than applying the right product at the correct time

The relative importance of diseases on apple is varying with cultivar, management, time,
and climate. Many aspects of the cropping system influence the development of diseases.
The choice of the variety determines the disease management during the lifetime of the
orchard. Cultural practices improve the growth and nutrial status of the tree, and therewith
influence the susceptibility of the plant and fruits to diseases directly. Prolonged growth
can also have an indirect effect by causing a microclimate and growing pattern that
favours infection of tree, leafs and fruits by various diseases. Sanitation measures are
common practise for most organic fruit growers and help to make other measures more
effective by reducing infection inoculums. Despite all preventive measures, disease control
in organic orchards at an economically feasible level still largely depends on the
application of fungicides. Measures that allow reduction of fungicidal applications on key
diseases, lead to the development of a secondary disease complex that can cause severe
losses when not managed effectively and make a well thought-out control strategy
necessary.
In research, advisory and practical decision making, disease management in organic
orchards should always be seen in the perspective of the management of the total growing
system. With all factors that contribute to disease management in organic orchards
optimized, we are able to successfully implement new materials and methods that may not
be as effective as common fungicides in themselves, but add to the effectiveness of the
disease management system as a whole.
This total system approach makes organic fruit growing what it is

### The growth of the infinite long-range percolation cluster

We consider long-range percolation on $\mathbb{Z}^d$, where the probability
that two vertices at distance $r$ are connected by an edge is given by
$p(r)=1-\exp[-\lambda(r)]\in(0,1)$ and the presence or absence of different
edges are independent. Here, $\lambda(r)$ is a strictly positive,
nonincreasing, regularly varying function. We investigate the asymptotic growth
of the size of the $k$-ball around the origin, $|\mathcal{B}_k|$, that is, the
number of vertices that are within graph-distance $k$ of the origin, for
$k\to\infty$, for different $\lambda(r)$. We show that conditioned on the
origin being in the (unique) infinite cluster, nonempty classes of
nonincreasing regularly varying $\lambda(r)$ exist, for which, respectively:
$\bullet$ $|\mathcal{B}_k|^{1/k}\to\infty$ almost surely; $\bullet$ there exist
$1<a_1<a_2<\infty$ such that $\lim_{k\to
\infty}\mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}<a_2)=1$; $\bullet$
$|\mathcal{B}_k|^{1/k}\to1$ almost surely. This result can be applied to
spatial SIR epidemics. In particular, regimes are identified for which the
basic reproduction number, $R_0$, which is an important quantity for epidemics
in unstructured populations, has a useful counterpart in spatial epidemics.Comment: Published in at http://dx.doi.org/10.1214/09-AOP517 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order $N$,
where any edge of length $k$ is present with probability
$p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed
$\beta$, we show that the critical value $\alpha_c(\beta)$ is non-trivial if
and only if $N < \beta < N^2$. Furthermore, we show uniqueness of the infinite
component and continuity of the percolation probability and of
$\alpha_c(\beta)$ as a function of $\beta$. This means that the phase diagram
of this model is well understood.Comment: 24 page

- …