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In this article, a multitype branching process is proposed for the behavior of populations of linear clusters of cells. The main results concern the asymptotic cluster size distribution.
"Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections...are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations."
A discrete-time multitype branching process model is presented for the evolution of transposable elements in haploid populations. An individual is classified as type i if it possesses i copies of the TE, i greater than or equal to 0. The general model incorporates copy-dependent selection and transposition, and recursion relations are derived for the distribution of the number of individuals of the various types. The asymptotic relative proportions of individuals of the different types is studied in the neutral case. The behavior of this equilibrium distribution is examined for various patterns of regulated transposition and deletion.
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The purpose of this paper is to describe a model of megakaryocytopoiesis as a branching process with stochastic processes regulating critical control points of differentiation along the stem cell megakaryocyte platelet axis. Progress of cells through these critical control points are regulated by transitional probabilities, which in turn are regulated by influences such as growth factors. The critical control points include transition of resting megakaryocytic stem cells (CFU-meg) into proliferating stem cells, the cessation of cytokinesis, and the cessation of DNA synthesis. A computerized computational method has been developed for directly fitting the stochastic branching model to colony growth data. The computational model has allowed transitional probabilities to be derived from colony size data. The model provides a unifying explanation for much of the heterogeneity of stages of maturation within populations of megakaryocytes and is fully compatible with historical data supporting the stochastic nature of hematopoietic stem cell regulation and with modern molecular concepts about control of the cell cycle.
A deleterious gene achieves a population balance between the opposing forces of selection and mutation. In this paper we explore the nature of this stochastic balance when the surrounding normal population is not at equilibrium. Assuming that new mutations occur according to a Poisson process and thereafter evolve by the rules of a continuous time branching process, we derive explicit formulas and recurrence relations determining the probability distribution of the current number of mutant individuals. In fact, we compute expectations for a variety of interesting random variables for genetic models involving autosomal dominant and X-linked diseases. We can also handle haplotype information on linked markers. This feature will be especially helpful in understanding the linkage disequilibrium strategy of positional cloning in population isolates. In the presence of exponential growth of the normal population, our formulas reduce to the evaluation of certain Laplace transforms.
We study some exact properties of supercritical branching processes. A proper rescaling of the relevant variable allows us to determine the distribution of population sizes after a number of generations have elapsed. Both time-continuous and discrete processes are analysed and compared. The obtained results are of relevance for the growth of populations that are not resource limited (a typical situation in some biological processes that can be modelled by laboratory experiments). Large fluctuations inherent to the process play a main role when bottlenecks occur.
We construct a mathematical model for the polymerase chain reaction and its mutations using the theory of branching processes. Under this model we study the number of mutations in a randomly chosen sequence after n PCR cycles. A method for estimating the mutation is proposed and the variance of this estimator is studied. We also study the distribution of the Hamming distance between two randomly chosen sequences and a method for estimating the mutation rate based on pairwise differences is proposed.
This paper is concerned with the properties of a stochastic integral which arises in the study of a modified Markov branching process. Explicit expressions are found for the mean and the limit distribution of the integral.
Solution of a genetic improvement problem under the model of assortative mating is proposed. This has been achieved through the technique applied to a branching process incorporating a generation-dependent emigration component. The model explains a population subject to culling against genes governing undesirable characters and this work has been tried for a diploid population with two alleles at a single locus, which may be extended to the multilocus case.
The link between the multi-type contact birth process and the spatial deterministic S-->I epidemic is exploited to give convergence results for a multi-type Markovian contact branching process. Let U(t) be the position of furthest spread by time t in a specific direction. The asymptotic speed of translation of the distribution function of U(t) is derived and U(t)/t is shown to converge in probability to this speed.
We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods according to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctuations on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson.
We consider the bisexual Galton-Watson branching process with population-size dependent mating as a mathematical model adequate for the description of some natural phenomena. More specifically we are interested in studying some questions about the problem of populating an environmental with new animal species or re-populating it with species which have previously disappeared.
We study the spreading dynamics on graphs with a power law degree distribution pk approximately k-gamma, with 2<gamma<3, as an example of a branching process with a diverging reproductive number. We provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth. First, the population growth is extensive, meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit. Second, the temporal evolution is governed by a polynomial growth, with a degree determined by the characteristic distance between vertices in the graph. These results open a path to further investigation on the dynamics on networks.
The transmission of HIV in a monogamous heterosexual population structured by the ordinal number of the current partnership is considered. The sexual carreer of a man (woman) is thought to be a succession of k( m) partnerships, and a multitype Galton-Watson process is defined, in which the objects are infections and the types are related to the ordinal number of the partnership during which a person has acquired the infection. Contrary to multitype models in which the types are not age-related in some sense, this process contains at least two singular types, namely infections acquired in the last partnership of a man or a woman. The criticality parameter of this branching process is the epidemic threshold parameter R(0). In the case k= m an epidemic is impossible, however large k may be, if the difference between the ordinal numbers of the partners in a pair is never > 1. When the frequency of pairs in which this difference is >or= 2 increases, then R(0) increases. This is demonstrated for the cases k= m=3 and k=4, m=3. The formulae obtained show also the joint influence of the mixing pattern and of variable infectivity. The result for the case of uniform mixing implies that a formula of May and Anderson (1987) is an approximation for k and m large.
At the offset of a (stochastic) epidemic, it is of importance to have a mathematical model that will assist in the making of an informed judgement on whether the epidemic will explode, or will be minor and die out. In this paper, we consider probabilistic inferences related to the event of extinction of a discrete time branching process when this cannot be directly observed. Instead, we are able to observe only a random "trace" of the process, which not only trails the latter, but also directly affects it (in terms of interventions). A simple model is proposed that provides tractability, preserves a marginal branching property, and gives reasonable closed form expressions.
The present work is aimed at developing the mathematical tools by which the dynamics of gene amplification (GA) can be described in detail. Some discrete compartmental models of GA by disproportionate replication and a general model for other putative GA mechanisms are presented and analyzed. The dynamical distribution of gene copy number in the cell population is calculated with the loss of cells taken either as constant or as copy-number-dependent. Our analysis shows that for a one-copy GA process with constant loss of cells, the relative frequency of single-gene-copy cells (sensitive cells) converges to zero, with the rate of convergence depending on the amplification probability. In contrast, for a one-copy GA process with copy-number-dependent loss of cells, the relative frequency of single-copy cells is bounded, implying a bounded compartment of many-gene-copy cells. Using branching processes theory we calculate the dynamical distribution of the single-gene-copy compartment as well as its extinction probability. Our models are used for estimating treatment prognosis as affected by drug resistance due to GA, showing significant differences in prognosis resulting from small changes in drug dose.
Mass vaccination programmes aim to maintain the effective reproduction number R of an infection below unity. We describe methods for monitoring the value of R using surveillance data. The models are based on branching processes in which R is identified with the offspring mean. We derive unconditional likelihoods for the offspring mean using data on outbreak size and outbreak duration. We also discuss Bayesian methods, implemented by Metropolis-Hastings sampling. We investigate by simulation the validity of the models with respect to depletion of susceptibles and under-ascertainment of cases. The methods are illustrated using surveillance data on measles in the USA.