Search

AbstractWe consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the globalextinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partialextinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Markovian binary trees are a special class of branching processes in which the lifetime of an individual is controlled by a transient Markovian arrival process. A Markovian binary tree is characterized by the 4-tuple (?,D0,B, d), where ? is the vector of initial phase distribution of the first individual, D0 is the matrix of phase transition rates between birth and death events, B is the matrix of birth rates and d is the vector of death rates. In order to use the Markovian binary tree to model the evolution of a real population, we need to determine the parameters (?,D0,B, d) from observations of that population. In the absence of migration, the only observable changes in a population are those associated with a birth or a death event; no phase transition in the underlying process can actually been seen. We are thus dealing with a problem of parameter estimation from incomplete data, and one way to solve this statistical problem is to make use of the EM algorithm. Our purpose here is thus to specify this algorithm to the Markovian binary tree setting. In the first part of this paper, we introduce a discrete time terminating marked Markov arrival process (MMAP), based on which a class of discrete multivariate phase-type (MPH) distributions is defined. The discrete MPH-distributions hold many of the properties possessed by continuous MPH-distributions (Assaf, et al. (1983), Kulkarni (1988), and O'Cinneide (1990)). It is known that the joint distribution functions of continuous MPH are fairly complicated and difficult to calculate. In contrast, for the discrete MPH introduced here, we provide recursive formulas the joint probabilities and explicit expressions for means, variances, and co-variances.

Recent measurement studies [8] have shown that peer-to-peer (P2P) file sharing applications are the major traffic source in the Internet. P2P applications, such as eDonkey, Kazaa, or BitTorrent, form overlay networks on the application layer and offer its peers to download and share their files with other peers in a highly distributed way. As a consequence, peers act simultaneously as both clients and servers. For a comprehensive survey of P2P technology, we refer to [7].

We perform an analytical sensitivity analysis for a model of a continuous-time branching process evolving on a fixed network. This allows us to determine the relative importance of the model parameters to the growth of the population on the network. We then apply our results to the early stages of an influenza-like epidemic spreading among a set of cities connected by air routes in the USA. We also consider vaccination and analyse the sensitivity of the total size of the epidemic with respect to the fraction of vaccinated people. Our analysis shows that the epidemic growth is more sensitive with respect to transmission rates within cities than travel rates between cities. More generally, we highlight the fact that branching processes offer a powerful stochastic modelling tool with analytical formulas for sensitivity which are easy to use in practice.

We consider multi-type Markovian branching processes subject to catastrophes which kill random numbers of living individuals at random epochs. It is well known that the criteria for extinction of such a process is related to the conditional growth rate of the population, given the history of the process of catastrophes, and that it is usually hard to evaluate. We give a simple characterization in the case when all individuals have the same probability of surviving a catastrophe, and we determine upper and lower bounds in the case where survival depends on the type of the individual. The upper bound appears to be often much tighter than the lower bound.