Theory Of Multitype Branching Processes Research Articles

Multitype branching processes in random environment

A survey of results in the theory of multitype branching processes evolving in a random environment is presented. Bibliography: 104 titles.

Disease clearance of tuberculosis infection: An in-host continuous-time Markov chain model

The clearance of tuberculosis infection shows an elimination of infectious Mycobacterium tuberculosis (Mtb) pathogens and infected macrophage cells. The evidence shows the existence of individuals, who are still tested negative in tuberculin skin test after living with people with active tuberculosis for up to six months. Since the Mtb pathogen is spread from person to person through airborne particles, we build a continuous-time Markov chain (CTMC) model to describe the initial infection with small amount of inhaled bacteria. The CTMC model successfully simulates sample paths presenting disease clearance. We apply the theory of multitype branching processes to analytically approximate the probability of disease clearance. We also estimate the disease clearance time, which is as less than a month for R0∈[1,1.5]. Our results demonstrate that the host immune factors affect both the probability and the time of the disease clearance. These relationships are linked by the basic reproduction number R0. Our findings provide new mechanisms for disease prevention and therapy developments.

Evolutionary Rescue and Drug Resistance on Multicopy Plasmids.

Bacteria often carry "extra DNA" in the form of plasmids in addition to their chromosome. Many plasmids have a copy number greater than one such that the genes encoded on these plasmids are present in multiple copies per cell. This has evolutionary consequences by increasing the mutational target size, by prompting the (transitory) co-occurrence of mutant and wild-type alleles within the same cell, and by allowing for gene dosage effects. We develop and analyze a mathematical model for bacterial adaptation to harsh environmental change if adaptation is driven by beneficial alleles on multicopy plasmids. Successful adaptation depends on the availability of advantageous alleles and on their establishment probability. The establishment process involves the segregation of mutant and wild-type plasmids to the two daughter cells, allowing for the emergence of mutant homozygous cells over the course of several generations. To model this process, we use the theory of multitype branching processes, where a type is defined by the genetic composition of the cell. Both factors-the availability of advantageous alleles and their establishment probability-depend on the plasmid copy number, and they often do so antagonistically. We find that in the interplay of various effects, a lower or higher copy number may maximize the probability of evolutionary rescue. The decisive factor is the dominance relationship between mutant and wild-type plasmids and potential gene dosage effects. Results from a simple model of antibiotic degradation indicate that the optimal plasmid copy number may depend on the specific environment encountered by the population.

Modeling Y-Linked Pedigrees through Branching Processes

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

Stochastic Modeling and Simulation of Viral Evolution.

RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simulations and the theoretical analysis of the model. In order to perform the simulations, we have implemented the mathematical model into a computational platform capable of running simulations and presenting the results in a graphical format in real time. Among other things, we show that the establishment of virus populations may display four distinct regimes from its introduction into new hosts until achieving equilibrium or undergoing extinction. Also, we were able to simulate different fitness distributions representing distinct environments within a host which could either be favorable or hostile to the viral success. We addressed the most used mechanisms for explaining the extinction of RNA virus populations called lethal mutagenesis and mutational meltdown. We were able to demonstrate a correspondence between these two mechanisms implying the existence of a unifying principle leading to the extinction of RNA viruses.

Individual‐based integral projection models: the role of size‐structure on extinction risk and establishment success

Summary Matrix models or integral projection models (IPMs) are commonly used to study the dynamics of structured populations, where discrete or continuous traits influence survival, growth or reproduction. When a population's size is small, as is often the case for threatened species or potentially invasive species arriving in novel habitats, extinction risk may be substantial due to demographic stochasticity. Branching processes, which are individual‐based counterparts to matrix models and IPMs, allow one to quantify these risks of extinction. For discretely structured populations, the theory of multitype branching processes provides analytic methods to compute how extinction risk changes over time and how it depends on the size and composition of the population. Building on prior work on continuous‐state branching processes, we extend these analytic methods to individual‐based models accounting for any mixture of discrete and continuous population structure. The individual‐based IPMs are defined by probabilistic update rules at the level of the individual which determine how each individual with a given trait value dies, changes trait values (e.g. grows in size) or produces individuals with the same or other trait values. We show that probabilities of extinction can be analytically determined by probability‐generating functionals associated with the individual‐based IPMs. In particular, we present analytical expressions for how extinction probabilities change over time and depend on the initial abundance and trait distribution of the population. We illustrate how to numerically implement these methods using data from the short‐lived desert shrub species Cryptantha flava and provide a more general discussion of how to implement these methods to other data sets including those involving fluctuating environmental conditions. As most IPM studies have the necessary data to parameterize individual‐based IPMs, these methods provide a computationally efficient means to explore how continuously structured populations differing in their evolutionary history and environmental context may differ in their vulnerability to extinction or ability to colonize new habitats.

POLLING SYSTEMS WITH TWO-PHASE GATED SERVICE

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

The effective population size of some age-structured populations

It was shown in a previous paper that if generations are discrete, then the effective population size of a large population can be derived from the theory of multitype branching processes. It turns out to be proportional to the reciprocal of a term that appears in the denominator of expressions for survival probabilities when there is a supercritical positively regular branching process for which the dominant positive eigenvalue of the first moment matrix is slightly larger than 1. If there is an age-structured population with unchanging proportions among sexes and age groups, then the effective population size is shown to be also obtainable from the theory of multitype branching processes. The expression for this parameter has the same form as in the corresponding model for discrete generations, multiplied by an appropriate measure of the average length of a generation. Results are obtained for dioecious random mating populations, populations reproducing partly by selfing, and populations reproducing partly by full-sib mating.

Estimates of transition densities for Brownian motion on nested fractals

We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d J, related to the “shortest path metric” and “chemical exponent” on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.

Polynucleotide evolution and branching processes

The theory of multitype branching processes is applied to the kinetics of polynucleotide replication. The results obtained are compared with the solutions of the deterministic differential equations of conventional chemical kinetics.

Some stochastic models for plasmid copy number

Some stochastic models for the copy number of plasmids in a cell line are studied. When considering the behavior of copy number in the whole cell line, the theory of multitype branching processes is appropriate. Attention is paid to the cure rate in the cell line, and the asymptotic fractions of cells containing a given number of plasmids. These quantities are used to compare the models numerically.

FIXATION PROBABILITIES AND FIXATION TIMES IN A SUBDIVIDED POPULATION.

The rate of genetic evolution depends in part on the probabilities of fixation and times to fixation of alleles of different types. Using a diffusion approximation to the Wright-Fisher model of selection and genetic drift in a single panmictic population, Kimura (1962) found a general formula for the probability of fixation at a diallelic locus with arbitrary selection coefficients. That greatly extended the results obtained for haploid models or for diploid models with no dominance which were solved using branching processes (Fisher, 1922) and by the direct analysis of a Markov chain (Moran, 1962). Kimura and Ohta (1969) used the same type of diffusion approximation to find the expected time to fixation in a diallelic system. These results are all for models of a single population. Since many natural populations are widely distributed in space, the assumption of panmixis may not be valid. It is important, then, to investigate the effects of geographic subdivision of the probabilites of fixation and times to fixation in a subdivided population. Pollak (1966) and Maruyama (1970a, 1972, 1974) have obtained several results for the case of a diallelic locus with no dominance. Pollak used the theory of multi-type branching processes to find the fixation probabilities of mutants subject to different selection pressures in different local populations. Maruyama showed that if the same selection pressure is acting in each local population and if the migration process has the property that the average allele frequencies do not change during the migration stage, then the fixation probabilities are the same as in a single panmictic population of the same total size. Maruyama's results were obtained using both a diffusion approximation and a direct analysis of a Markov chain. The same result could be obtained by using one of Pollak's formulae but he did not note that fact. None of these results are applicable to the case with other degrees of dominance and no information is obtained on fixation times. More recently Lande (1979) found the fixation probability of an underdominant mutant with equal homozygote fitnesses in the limit of low migration rates. Maruyama (1980) has independently obtained the same results as presented below for the fixation probabilities of mutants with different degrees of dominance. The only attempt at determining fixation times in a subdivided population that I am aware of is my heuristic analysis of a linear stepping stone model (Slatkin, 1976). The approach taken in this paper to the problem of finding the probabilities of fixation and times to fixation in a subdivided population is necessarily indirect. There is no way to solve the complete multi-dimensional diffusion equation and obtain the required quantities. Instead, I will obtain bounds on these quantities by considering certain limiting cases for which approximate analytic results can be found. This indirect approach will be complemented by computer simulations.

The dispersion of a neutral allele considered as a branching process

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data fromDrosophilapopulations are used to show the improbability of the same neutral allele occurring at widely separated localities.

The dispersion of a neutral allele considered as a branching process

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data fromDrosophilapopulations are used to show the improbability of the same neutral allele occurring at widely separated localities.