Multitype Galton-Watson Branching Process Research Articles

Analysis of Discrete-Time Queues with Branching Arrivals

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply π-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes.

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

The probabilities of extinction in a branching random walk on a strip

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 global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction 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.

Determining the effects of transplacental and direct transmission on the probability of persistence in a bluetongue virus model in temperate and tropical regions

Bluetongue virus (BTV) is an important concern for the ruminant livestock industry owing to its high morbidity rate. It is mainly transmitted by a midge to a host (vectorial transmission) but even though its replication in the midge ceases during winter in temperate countries, outbreaks re-emerge in the next season. Transplacental and direct transmission have been proposed as mechanisms that facilitate its overwintering (persistence), but to date, their effects on the probability of BTV persistence are not fully understood. In addition, the effects of these mechanisms on the probability of BTV persistence in tropical regions where the virus is endemic are not clear. A deterministic model that accounts for BTV transmission to cattle is formulated and analysed. Using the next generation approach the basic reproduction number (R0) is determined. When R0<1, the model exhibits a backward bifurcation indicating that the virus persists. When R0>1, a continuous-time Markov chain (CTMC) model derived from the deterministic model is used to estimate the probability of BTV persistence. By approximating the CTMC model with a multitype Galton–Watson branching process, it is shown that both mechanisms can have a high and a low positive effect on the probability of BTV persistence in temperate and tropical regions, respectively. These results indicate the importance of transplacental and direct transmission for BTV persistence in temperate regions, but not in tropical regions where vectorial transmission is the dominant mechanism as expected.

On aggregation of multitype Galton–Watson branching processes with immigration

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton–Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton–Watson processes with immigration as well.

Recurrence and transience of contractive autoregressive processes and related Markov chains

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence.

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton-Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.

Contagions in random networks with overlapping communities

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

Contagions in random networks with overlapping communities

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Variational characterization of multitype Galton-Watson branching processes

In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.

Vaginal Immunization to Elicit Primary T-Cell Activation and Dissemination

Primary T-cell activation at mucosal sites is of utmost importance for the development of vaccination strategies. T-cell priming after vaginal immunization, with ovalbumin and CpG oligodeoxynucleotide adjuvant as model vaccine formulation, was studied in vivo in hormone-synchronized mice and compared to the one induced by the nasal route. Twenty-four hours after both vaginal or nasal immunization, antigen-loaded dendritic cells were detected within the respective draining lymph nodes. Vaginal immunization elicited a strong recruitment of antigen-specific CD4+ T cells into draining lymph nodes that was more rapid than the one observed following nasal immunization. T-cell clonal expansion was first detected in iliac lymph nodes, draining the genital tract, and proliferated T cells disseminated towards distal lymph nodes and spleen similarly to what observed following nasal immunization. T cells were indeed activated by the antigen encounter and acquired homing molecules essential to disseminate towards distal lymphoid organs as confirmed by the modulation of CD45RB, CD69, CD44 and CD62L marker expression. A multi-type Galton Watson branching process, previously used for in vitro analysis of T-cell proliferation, was applied to model in vivo CFSE proliferation data in draining lymph nodes 57 hours following immunization, in order to calculate the probabilistic decision of a cell to enter in division, rest in quiescence or migrate/die. The modelling analysis indicated that the probability of a cell to proliferate was higher following vaginal than nasal immunization. All together these data show that vaginal immunization, despite the absence of an organized mucosal associated inductive site in the genital tract, is very efficient in priming antigen-specific CD4+ T cells and inducing their dissemination from draining lymph nodes towards distal lymphoid organs.

Optimization and Phenotype Allocation

We study the phenotype allocation problem for the stochastic evolution of a multitype population in a random environment. Our underlying model is a multitype Galton–Watson branching process in a random environment. In the multitype branching model, different types denote different phenotypes of offspring, and offspring distributions denote the allocation strategies. Two possible optimization targets are considered: the long-term growth rate of the population conditioned on nonextinction, and the extinction probability of the lineage. In a simple and biologically motivated case, we derive an explicit formula for the long-term growth rate using the random Perron–Frobenius theorem, and we give an approximation to the extinction probability by a method similar to that developed by Wilkinson. Then we obtain the optimal strategies that maximize the long-term growth rate or minimize the approximate extinction probability, respectively, in a numerical example. It turns out that different optimality criteria can lead to different strategies.

The asymptotic relative generation distribution of the closest common ancestor of the multitype Galton–Watson tree conditioned on non-extinction

We extend Geiger’s size-biased tree method to the multitype case to give another proof of Zubkov’s result that the asymptotic relative generation distribution of the closest common ancestor of the multitype Galton–Watson branching process conditioned on non-extinction is uniform.

Non-parametric Bayesian estimation for multitype branching processes through simulation-based methods

The problem of statistical inference from a Bayesian outlook is studied for the multitype Galton–Watson branching process, considering a non-parametric framework. The only data assumed to be available are each generation's population size vectors. The Gibbs sampler is used in estimating the posterior distributions of the main parameters of the model, and the predictive distributions for as yet unobserved generations. The algorithm provided is independent of whether the process becomes extinct or not. The method is illustrated with simulated examples.

Dynamics of escape mutants

We use multi-type Galton–Watson branching processes to model the evolution of populations that, due to a small reproductive ratio of the individuals, are doomed to extinction. Yet, mutations occurring during the reproduction process, may lead to the appearance of new types of individuals that are able to escape extinction. We provide examples of such populations in medical, biological and environmental contexts and give results on (i) the probability of escape/extinction, (ii) the distribution of the waiting time to produce the first individual whose lineage does not get extinct and (iii) the distribution of the time it takes for the number of mutants to reach a high level. Special attention is dedicated to the case where the probability of mutation is very small and approximations for (i)–(iii) are derived.

APPROXIMATE REDUCTION OF MULTI-TYPE GALTON–WATSON PROCESSES WITH TWO TIME SCALES

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler "aggregated system" governed by a few "macrovariables". Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales. In this work we deal with the approximate aggregation of a model for a population subjected to demographic stochasticity and whose dynamics is controlled by two processes with different time scales. There are no restrictions on the slow process while the fast process is supposed to be conservative of the total number of individuals. The incorporation of the effects of demographic stochasticity in the dynamics of the population makes both the fast and the slow processes being modelled by two multi-type Galton–Watson branching processes. We present a multi-type global model that incorporates the combined effect of the fast and slow processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining an "aggregated" simpler system. We show that, given the separation of time scales between the two processes is high enough, we can obtain relevant information about the behavior of the multi-type global model through the study of this simple aggregated system.

Approximate reduction of multiregional birth-death models with fast migration

In this work, we deal with the reduction of a time discrete model for a population distributed among N spatial patches and whose dynamics is controlled both by reproduction and by migration. These processes take place at different time scales in the sense of the latter being much faster than the former. We incorporate the effect of demographic stochasticity into the population, which results in both dynamics being modelled by multitype Galton-Watson branching processes. We present a multitype global model that incorporates the effect of the two processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining a unitype “aggregated” process that approximates the evolution of the total size of the population. We show that, given the separation of time scales between the birth-death process and the migration process is sufficiently high, we can obtain both qualitative and quantitative information about the behavior of the multitype global model through the study of this simple aggregated model.

The probability distribution of the height of the genealogical tree of a multi-type Galton-Watson branching process

Article The probability distribution of the height of the genealogical tree of a multi-type Galton-Watson branching process was published on January 1, 1999 in the journal Discrete Mathematics and Applications (volume 9, issue 2).

A discrete-time, multi-type generational inheritance branching process model of cell proliferation.

Mammalian cell populations, such as tumors, may contain subpopulations differing in parameters such as cell lifetimes, even if the populations are derived from single cells. The mode of inheritance of cell lifetimes has previously been the subject of experimental and mathematical investigation. To obtain data on cell lifetimes over more cell generations then previously available, Axelrod et al. [Cell Prolif. 26:235-249(1988)] measured the number of cells in primary colonies and secondary colonies derived form the primary colonies. The experimental results indicated large variance of cells per colony and highly significant correlations between the numbers of cells in primary and secondary colonies. To mathematically model these results we derive, for previously uninvestigated multi-type Galton-Watson branching process models, the covariance of the cell counts in the primary and secondary colonies. As a result, we are able to successfully model the data with two subpopulations having differing proliferation rates, in which the proliferation rate of a daughter cell is primarily determined by the proliferation rate of its mother. Interestingly, simulations display a trade-off between high values of variances and correlation coefficients. The values obtained from experiment are located on the boundary of the region attainable by simulation.