67 research outputs found

    Stochastic epidemics in growing populations

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    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

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    We consider a branching population where individuals have i.i.d.\ life lengths (not necessarily exponential) and constant birth rate. We let NtN_t denote the population size at time tt. %(called homogeneous, binary Crump--Mode--Jagers process). We further assume that all individuals, at birth time, are equipped with independent exponential clocks with parameter δ\delta. We are interested in the genealogical tree stopped at the first time TT 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<}\{T<\infty\}, the joint law of (NT,T,X(T))(N_T, T, X^{(T)}), where X(T)X^{(T)} is the jumping contour process of the tree truncated at time TT, is equal to that of (M,IM,YM)(M, -I_M, Y_M') conditional on {M0}\{M\not=0\}, where : M+1M+1 is the number of visits of 0, before some single independent exponential clock e\mathbf{e} with parameter δ\delta rings, by some specified L{\'e}vy process YY without negative jumps reflected below its supremum; IMI_M is the infimum of the path YMY_M defined as YY killed at its last 0 before e\mathbf{e}; YMY_M' is the Vervaat transform of YMY_M. This identity yields an explanation for the geometric distribution of NTN_T \cite{K,T} and has numerous other applications. In particular, conditional on {NT=n}\{N_T=n\}, and also on {NT=n,T<a}\{N_T=n, T<a\}, the ages and residual lifetimes of the nn alive individuals at time TT are i.i.d.\ and independent of nn. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital

    Bounding basic characteristics of spatial epidemics with a new percolation model

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    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

    Maximizing the size of the giant

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    We consider two classes of random graphs: (a)(a) Poissonian random graphs in which the nn vertices in the graph have i.i.d.\ weights distributed as XX, where E(X)=μ\mathbb{E}(X) = \mu. Edges are added according to a product measure and the probability that a vertex of weight xx shares and edge with a vertex of weight yy is given by 1exy/(μn)1-e^{-xy/(\mu n)}. (b)(b) A thinned configuration model in which we create a ground-graph in which the nn vertices have i.i.d.\ ground-degrees, distributed as DD, with E(D)=μ\mathbb{E}(D) = \mu. The graph of interest is obtained by deleting edges independently with probability 1p1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1q1-q, where qq depends on XX or DD and pp. We investigate for which distributions XX and DD with given μ\mu and pp, 1q1-q is maximized. We show that in the class of Poissonian random graphs, XX should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model DD should have all its mass at 0 and two subsequent positive integers

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

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    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

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    We consider long-range percolation on Zd\mathbb{Z}^d, where the probability that two vertices at distance rr are connected by an edge is given by p(r)=1exp[λ(r)](0,1)p(r)=1-\exp[-\lambda(r)]\in(0,1) and the presence or absence of different edges are independent. Here, λ(r)\lambda(r) is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the kk-ball around the origin, Bk|\mathcal{B}_k|, that is, the number of vertices that are within graph-distance kk of the origin, for kk\to\infty, for different λ(r)\lambda(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r)\lambda(r) exist, for which, respectively: \bullet Bk1/k|\mathcal{B}_k|^{1/k}\to\infty almost surely; \bullet there exist 1<a1<a2<1<a_1<a_2<\infty such that limkP(a1<Bk1/k<a2)=1\lim_{k\to \infty}\mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}<a_2)=1; \bullet Bk1/k1|\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, R0R_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

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    We study long-range percolation on the hierarchical lattice of order NN, where any edge of length kk is present with probability pk=1exp(βkα)p_k=1-\exp(-\beta^{-k} \alpha), independently of all other edges. For fixed β\beta, we show that the critical value αc(β)\alpha_c(\beta) is non-trivial if and only if N<β<N2N < \beta < N^2. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of αc(β)\alpha_c(\beta) as a function of β\beta. This means that the phase diagram of this model is well understood.Comment: 24 page
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