Multitype Galton-Watson Process Research Articles

Two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices

AbstractConsider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $\mathbb{P}(\nu=n)$ and $\mathbb{P}(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $\mathbb{P}(\nu=n)$ decays with various speeds such as ${c}/({n^{1/2}\log n)^2}$ , ${c}/{n^\beta}$ , $\beta >1$ , which are very different from those of homogeneous multitype Galton–Watson processes.

The asymptotic behavior of an SIR epidemic model: collective Reed-Frost process

The aim of this research is to provide a historical overview of the mathematical theory of epidemics and to study the asymptotic behavior of the final size of a collective Reed-Frost epidemic process with different types of infected people. This model was introduced by Picard and Lefevre [25] provides an extension of the model of Pettigrew and Weiss [24]. Under certain conditions, we show that when the number of the initial susceptible individuals is large and the number of the initial infected people is finite, the infection process is equivalent to a multitype Galton-Watson process. Our method is simple and based on Bernstein polynomials.

Moments of the stationary distribution of subcritical multitype Galton–Watson processes with immigration

A necessary and sufficient condition for the existence of moments of the stationary distribution of a subcritical multitype GWI process was obtained by Szűcs (2014). In this short note we give a simple proof of this result.

Perron–Frobenius theory for kernels and Crump–Mode–Jagers processes with macro-individuals

AbstractPerron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.

Extinction in lower Hessenberg branching processes with countably many types

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton–Watson processes with typeset $\mathcal{X}=\{0,1,2,\ldots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector $\boldsymbol{\tilde{q}}$.

Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

Multi-type Galton-Watson Processes with Affinity-Dependent Selection Applied to Antibody Affinity Maturation.

We analyze the interactions between division, mutation and selection in a simplified evolutionary model, assuming that the population observed can be classified into fitness levels. The construction of our mathematical framework is motivated by the modeling of antibody affinity maturation of B-cells in germinal centers during an immune response. This is a key process in adaptive immunity leading to the production of high-affinity antibodies against a presented antigen. Our aim is to understand how the different biological parameters affect the system's functionality. We identify the existence of an optimal value of the selection rate, able to maximize the number of selected B-cells for a given generation.

Stability and busy periods in a multiclass queue with state-dependent arrival rates

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton–Watson processes, the Laplace–Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.

The survival probability of a critical multi-type branching process in i.i.d. random environment

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with respect to an empirical offspring measure, $\rho,$ is the Legendre dual of $U_{\skrik}(\,\cdot,\,\pi).$ From the LLDP we deduce a conditional large deviation principle and a weak variant of the classical McMillian Theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure $\varpi,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined entropy.

Distribution of the quasispecies for a Galton–Watson process on the sharp peak landscape

Abstract We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a - 1)(ak/k!)∑i≥ 1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutation probability. The probability distribution Q(σ, a) on the nonnegative integers given by the above equation is the quasispecies distribution with parameters σ and a.

Beyond the Q-process: various ways of conditioning the multitype Galton-Watson process

Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated Q-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the Q-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained Q-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a Q-process. It does not however coincide with the original associated Q-process, except in the critical regime.

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper, we form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the first passage time of corresponding random walks. Especially, for recurrent random walk which could stay at its position with probability 0 < r < 1, we show that the tail probability of the first passage time decays as \(\frac{2} {{\sqrt {\pi (1 - r)} }}\frac{1} {{\sqrt n }} \) when n → ∞.

Multitype subcritical branching processes in a random environment

We investigate a multitype Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Assuming that the mean of the increment X of the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices is negative and the random variable XeX has zero mean, we find the asymptotics of the survival probability at time n as n → ∞.

A Branching Process Approximation of the Final Size of a Multitype Collective Reed-Frost Model

We consider the asymptotic behavior of the final size of a multitype collective Reed-Frost process. This type of models was introduced by (9) and include most known epidemic models of the type SIR (Susceptible, Infected, Removed) as special cases. Under certain conditions, we show that, when the initial number of susceptible is very large and the initial number of infected individuals is finite, the infection process behaves as a multitype Galton-Watson process. This fact is proved using a simple argument based on Bernstein polynomials. We use this result to study the final size of the epidemic.

On subcritical multi-type branching process in random environment

We investigate a multi-type Galton-Watson process in a random environment generated by a sequence of independent identically distributed random variables. Suppose that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices has negative mean and assuming some additional conditions, we find the asymptotics of the survival probability at time $n$ as $n \to \infty$.

Limit laws of estimators for critical multi-type Galton–Watson processes

We consider the asymptotics of various estimators based on a large sample of branching trees from a critical multi-type Galton–Watson process, as the sample size increases to infinity. The asymptotics of additive functions of trees, such as sizes of trees and frequencies of types within trees, a higher-order asymptotic of the “relative frequency” estimator of the left eigenvector of the mean matrix, a higher-order joint asymptotic of the maximum likelihood estimators of the offspring probabilities and the consistency of an estimator of the right eigenvector of the mean matrix, are established.

A discrete branching process model for the spread of HIV via steady sexual partnerships.

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.

Transient behaviour of a Galton–Watson process with a large number of types

Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different modelling assumptions, for instance different values of the number of types.

Transient behaviour of a Galton–Watson process with a large number of types

Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different modelling assumptions, for instance different values of the number of types.