# Read PDF Workshop on Branching Processes and Their Applications

In this paper we extend the two-parameter linear-fractional family to a much richer four-parameter family of reproduction laws. The corresponding Galton-Watson processes also allow for explicit calculations, now with possibility for infinite mean, or even infinite number of offspring. We study the properties of this special family of branching processes, and show, in particular, that in some explosive cases the time to explosion can be approximated by the Gumbel distribution. This paper is concerned with the use of vaccination schemes to control an epidemic in terms of the total number of individuals infected.

In particular, monotonicity and continuity properties of total progeny of Crump-Mode-Jagers branching processes are derived depending on vaccination level. Furthermore, optimal vaccination policies based on the mean and quantiles of the total number of infected individuals are proposed.

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Finally, how to apply the proposed methodology in real situations is shown through a simulated example motivated by an outbreak of influenza virus in humans, in Indonesia. It is assumed that an estimate of the early exponential growth rate r of the epidemic is available, together with more detailed household-level data from a sample of households. A basic method, which uses the total size distribution of single-household epidemics, is usually biased owing to the emerging nature of an epidemic.

An alternative method, which uses the asymptotic theory of continuous-time, multitype Markov branching processes to account for the emerging nature of an epidemic, is developed and shown by simulations to be feasible for realistic population sizes. A modified single-household epidemic process is used to show that the basic method is approximately unbiased when r is small.

Some defective alleles of certain genes can cause severe diseases or serious disorders in the organisms that carry them.

Some of these genes, as could be the responsible for hemophilia, are linked to X chromosome. If the alleles causing the disorder are dominant, all the carriers are affected and most of them do not reach breeding age so they are rarely detected in a population. However, recessive pernicious alleles can survive since they only affect to carrier males and homozygous carrier females the last ones must be daughters of a carrier male, so they rarely exist.

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Hence, heterozygous carrier females are not affected but can pass the allele onto offspring. In this work, we introduce a multitype two-sex branching process for describing the evolution of the number of individuals carrying the alleles, R and r , of a gene linked to X chromosome.

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The R allele is considered dominant and the r allele is supposed to be recessive and defective, responsible of a disorder. In this model we study the fate of the recessive allele in the population, i.

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We also investigate conditions for the fixation of the dominant allele and the extinction of the population. Finally, conditions for the coexistence of both alleles are conjectured and their proofs are proposed as open problems. This work deals with stochastic modeling in biological populations. We develop a two-sex branching process as an appropriate mathematical model to describe the demographic dynamics of biological populations with sexual reproduction. We assume several mating and reproduction strategies. Moreover, unlike other classes of two-sex branching processes where mating and reproduction are influenced by the number of couples in the population, we now consider the most realistic case where both mating and reproduction are affected by the numbers of females and males in the population.

The main goal of this paper is to consider branching processes with two types and in continuous time to model the dynamics of the number of different types of cells, which due to a small reproductive ratio are fated to become extinct. However, mutations occurring during the reproduction process may lead to the appearance of a new type of cells that may escape extinction.

This is a typical real world situation with the emergence of scatters after local eradication of a certain type of cancer during the chemotherapy. Mathematically, we are deriving the numbers of mutations of the escape type and their moments. In general, our results aim to prove the limits of expanding the methods used by Serra and Haccou Theor. Titel Branching Processes and Their Applications. Verlag Springer International Publishing. Print ISBN Electronic ISBN Since then, branching processes have been regarded as appropriate probability models for the description of the behaviour of systems whose components cells, particles, individuals in general reproduce, are transformed, or die.

This theory has developed from simple models to increasing realism.

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Added to the theoretical interest in these processes there was therefore a major practical dimension due to their potential applications in such diverse fields as biology, epidemiology, genetics, medicine, nuclear physics, demography, actuarial mathematics, algorithm and data structures, etc. Nowadays, this theory goes on being an area of active and interesting research. Conference proceedings. Papers Table of contents 20 papers About About these proceedings Table of contents Search within book.

## IV Workshop on Branching Processes and their Applications. April , Badajoz. Spain

Front Matter Pages i-xix. Front Matter Pages A refinement of limit theorems for the critical branching processes in random environment. Pages Branching processes in stationary random environment: The extinction problem revisited. Environmental versus demographic stochasticity in population growth. Stationary distributions of the alternating branching processes.

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Approximations in population-dependent branching processes. Extension of the problem of extinction on Galtonâ€”Watson family trees. Limit theorems for critical randomly indexed branching processes. Kosto V. Mitov, Georgi K.