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  • 1.
    Andersson, Patrik
    Stockholm University, Faculty of Science, Department of Mathematics.
    CARD COUNTING IN CONTINUOUS TIME2012In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 49, no 1, p. 184-198Article in journal (Refereed)
    Abstract [en]

    We consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. The sampling without replacement makes it possible to take advantage of the changes in the expected value as the deck is depleted, making large bets when the game is advantageous. Using such a strategy, which is easy to implement, is known as card counting. We consider the case of a large number of decks, making an approximation to continuous time possible. A limit law of the return process is found and the optimal card counting strategy is derived. This continuous-time strategy is shown to be a natural analog of the discrete-time strategy where the so-called effects of removal are replaced by the infinitesimal generator of the card process.

  • 2. Ball, Frank
    et al.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Neal, Peter
    ON EXPECTED DURATIONS OF BIRTH-DEATH PROCESSES, WITH APPLICATIONS TO BRANCHING PROCESSES AND SIS EPIDEMICS2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 1, p. 203-215Article in journal (Refereed)
    Abstract [en]

    We study continuous-time birth-death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is alpha(n). We focus on two important examples, namely alpha(n) = lambda n being a branching process, and alpha(n) = lambda n(N-n)/N which corresponds to an SIS (susceptible -> infective -> susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i. e. in state 1. Let T, A(n), C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth-death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible -> infective -> recovered) epidemic, its threshold parameter R-* is insensitive to the distribution of Q.

  • 3. Björnberg, J. E.
    et al.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Broman, E. I.
    Natan, E.
    A Stochastic Model for Virus Growth in a Cell Population2014In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 3, p. 599-612Article in journal (Refereed)
    Abstract [en]

    In this work we introduce a stochastic model for the spread of a virus in a cell population where the virus has two ways of spreading: either by allowing its host cell to live and duplicate, or by multiplying in large numbers within the host cell, causing the host cell to burst and thereby let the virus enter new uninfected cells. The model is a kind of interacting Markov branching process. We focus in particular on the probability that the virus population survives and how this depends on a certain parameter A which quantifies the 'aggressiveness' of the virus. Our main goal is to determine the optimal balance between aggressive growth and long-term success. Our analysis shows that the optimal strategy of the virus (in terms of survival) is obtained when the virus has no effect on the host cell's life cycle, corresponding to lambda = 0. This is in agreement with experimental data about real viruses.

  • 4.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindholm, Mathias
    The early stage behaviour of a stochastic SIR epidemic with term-time forcing2009In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 46, no 4, p. 975-992Article in journal (Refereed)
    Abstract [en]

    The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An 'environment' in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here 'term-time forcing' refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {A(t)}<sub>t≥0</sub>. Threshold conditions in terms of R<sub>⋆</sub> are obtained, such that R<sub>⋆</sub> > 1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.

  • 5.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lindholm, Mathias
    Uppsala universitet.
    Turova, Tatyana
    Lunds universitet.
    A dynamic network in a dynamic population: asymptotic properties2011In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 48, p. 1163-1178Article in journal (Refereed)
    Abstract [en]

    We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model, we derive a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation rho (of neighbouring nodes) which shows that rho is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.

  • 6.
    Britton, Tom
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Maximizing the size of the giant2012In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 49, no 4, p. 1156-1165Article in journal (Refereed)
    Abstract [en]

    Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring 'on' the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X-i with E(X-i) = mu, and where there is an edge between i and j with probability 1 - e(-XiXj/(mu n)), independently of other edges. The second model is the thinned configuration model in which then vertices of the ground graph have i.i.d. ground degrees, distributed as D, with 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.

  • 7.
    Deijfen, Maria
    Stockholm University, Faculty of Science, Department of Mathematics.
    Random networks with preferential growth and vertex death2010In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 47, no 4, p. 1150-1163Article in journal (Refereed)
    Abstract [en]

    A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then pkk-(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then pk ∼ (1 + β)-k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logpk ∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

  • 8.
    Deijfen, Maria
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gantert, Nina
    Routeing on trees2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 475-488Article in journal (Refereed)
    Abstract [en]

    We consider three different schemes for signal routeing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with independent and identically distributed weights representing the strength of the transceivers. The edges of the tree are also equipped with independent and identically distributed weights, representing the costs for passing the edges. For each one of our schemes, we derive sharp conditions on the distributions of the vertex weights and the edge weights that determine when the root can transmit a signal over arbitrarily large distances.

  • 9.
    Hammarlid, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tools to estimate the first passage time to a convex barrier2005In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 42, no 1, p. 61-81Article in journal (Refereed)
    Abstract [en]

    he first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear bather is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

  • 10.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    A SPATIO-TEMPORAL POINT PROCESS MODEL FOR PARTICLE GROWTH2019In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 56, no 1, p. 23-38Article in journal (Refereed)
    Abstract [en]

    A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson-Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

  • 11.
    Lopes, Fabio
    Stockholm University, Faculty of Science, Department of Mathematics.
    Invariant Bipartite Random Graphs on Rd2014In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 3, p. 769-779Article in journal (Refereed)
    Abstract [en]

    Suppose that red and blue points occur in R d according to two simple point processes with finite intensities λ R and λ B , respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λ R ν̅ = λ B μ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

  • 12.
    Malmros, Jens
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Liljeros, Fredrik
    Stockholm University, Faculty of Social Sciences, Department of Sociology.
    Britton, Tom
    Stockholm University, Faculty of Science, Department of Mathematics.
    Respondent-driven sampling and an unusual epidemic2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 2, p. 518-540Article in journal (Refereed)
    Abstract [en]

    Respondent-driven sampling (RDS) is frequently used when sampling from hidden populations. In RDS, sampled individuals pass on participation coupons to at most c of their acquaintances in the community (c = 3 being a common choice). If these individuals choose to participate, they in turn pass coupons on to their acquaintances, and so on. The process of recruiting is shown to behave like a new Reed-Frost-type network epidemic, in which `becoming infected' corresponds to study participation. We calculate R-0, the probability of a major `outbreak', and the relative size of a major outbreak for c < infinity in the limit of infinite population size and compare to the standard Reed-Frost epidemic. Our results indicate that c should often be chosen larger than in current practice.

  • 13.
    Trapman, Pieter
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lambert, Amaury
    UPMC Univ Paris 06.
    Splitting trees stopped when the first clock rings and Vervaat's transformation2013In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 50, no 1, p. 208-227Article in journal (Refereed)
    Abstract [en]

    We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T)is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}. HereM+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y'M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T<a}, the ages and residual lifetimes of the n alive individuals at timeT 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

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