Multitype Branching Research Articles

Reduced multitype critical branching processes in random environment

Abstract We consider a multitype critical branching process Z n , n = 0, 1,…, in an i.i.d. random environment. Let Z m,n be the number of particles in this process at time m having descendants at time n. A limit theorem is proved for the logarithm of Z nt,n at moments nt,0 ≤ t ≤ 1, conditioned on the survival of the process Z n up to moment n when n → ∞.

Abstract P5-08-01: Reconstructing the evolutionary paths of BIG 1-98

Abstract We are reconstructing the dynamics of tumour growth, treatment response, and origins of resistance of the Breast International Group (BIG) 1-98 study to determine the evolutionary paths of non-responders. We established the somatic alteration landscape of primary tumours collected at surgery from BIG 1-98 patients using targeted DNA sequencing of 287 genes of FFPE samples from 538 patients. Combined analysis of genetic and clinical data indicates increased genomic instability; TP53 mutations and copy number variations of 11q13 and 8p11 are associated with poor prognosis. We developed an in silico simulation framework to investigate if the non-responder outcome is due to (over)-treatment or intrinsic to evolutionary selection prior to diagnosis. To reconstruct the evolutionary history of tumours, we created a computational multitype branching process that tracks the expansion of diverse clonal lineages as they acquire driver and passenger mutations that alter their proliferation and mutation rates. To account for the heterogeneity between patients, we created a fitting procedure based on the Cramer-von Misses statistic to find the likelihood of parameters of our computational model that recreate the mutational landscape and clinicopathological factors observed in each patient. Once the fitting procedure is done, we simulate the course of treatment following the study arm scheme accounting for the cell-cycle action mechanism of Tamoxifen and Letrozole. Using our tool, we are characterising the range of tumour development scenarios covering different degrees of aggressiveness and genomic instability. Our computational model allows simulation of diverse adjuvant-schemes to predict optimised treatment regimes for validation in patient-derived tumour xenograft mouse models. Citation Format: Lara-Gonzalez LL, Loi S, Ferrari D, Papenfuss A, Goode DL. Reconstructing the evolutionary paths of BIG 1-98 [abstract]. In: Proceedings of the 2017 San Antonio Breast Cancer Symposium; 2017 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2018;78(4 Suppl):Abstract nr P5-08-01.

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.

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type L and type S) in which only particles of type L have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least $1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)! e^{k-2}} + o(1)$. These bounds are sharp.

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.

Almost sure growth of supercritical multi-type continuous-state branching process

In Li (2011), Example 2.2, the notion of a multi-type continuous-state branching process (MCSBP) was introduced with a finite number of types, with the countably infinite case being proposed in Kyprianou and Palau (2017). One may consider such processes as a super-Markov chain on a countable state-space of types, which undertakes both local and non-local branching. In Kyprianou and Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists a lead eigenvalue which characterises the spectral radius of the linear semigroup associated to the process. Moreover, in a qualitative sense, the sign of this eigenvalue distinguishes between the cases where there is local extinction and exponential growth. In this paper, we continue in this vein and show that, when the number of types is finite, the lead eigenvalue gives the precise almost sure rate of growth of each type. This result matches perfectly classical analogues for multi-type Galton--Watson processes.

Multitype Branching Processes in Random Environment: Probability of Survival for the Critical Case

We investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in an environment generated by a sequence of independent identically distribute...

Duration of a minor epidemic

Disease outbreaks in stochastic SIR epidemic models are characterized as either minor or major. When , all epidemics are minor, whereas if , they can be minor or major. In 1955, Whittle derived formulas for the probability of a minor or a major epidemic. A minor epidemic is distinguished from a major one in that a minor epidemic is generally of shorter duration and has substantially fewer cases than a major epidemic. In this investigation, analytical formulas are derived that approximate the probability density, the mean, and the higher-order moments for the duration of a minor epidemic. These analytical results are applicable to minor epidemics in stochastic SIR, SIS, and SIRS models with a single infected class. The probability density for minor epidemics in more complex epidemic models can be computed numerically applying multitype branching processes and the backward Kolmogorov differential equations. When is close to one, minor epidemics are more common than major epidemics and their duration is significantly longer than when or .

Mutation in populations governed by a Galton–Watson branching process

A population genetics model based on a multitype branching process, or equivalently a Galton–Watson branching process for multiple alleles, is presented. The diffusion limit forward Kolmogorov equation is derived for the case of neutral mutations. The asymptotic stationary solution is obtained and has the property that the extant population partitions into subpopulations whose relative sizes are determined by mutation rates. An approximate time-dependent solution is obtained in the limit of low mutation rates. This solution has the property that the system undergoes a rapid transition from a drift-dominated phase to a mutation-dominated phase in which the distribution collapses onto the asymptotic stationary distribution. The changeover point of the transition is determined by the per-generation growth factor and mutation rate. The approximate solution is confirmed using numerical simulations.

Bivariate Markov chains converging to Lamperti transform Markov additive processes

Motivated by various applications, we describe the scaling limits of bivariate Markov chains (X,J) on Z+×{1,…,κ} where X can be viewed as a position marginal and {1,…,κ} is a set of κ types. The chain starts from an initial value (n,i)∈Z+×{1,…,κ}, with i fixed and n→∞, and typically we will assume that the macroscopic jumps of the marginal X are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that X is non-increasing. We then observe different asymptotic regimes according to whether the rate of type change is proportional to, faster than, or slower than the macroscopic jump rate. In these different situations, we obtain in the scaling limit Lamperti transforms of Markov additive processes, that sometimes reduce to standard positive self-similar Markov processes. As first examples of applications, we study the number of collisions in coalescents in varying environment and the scaling limits of Markov random walks with a barrier. This completes previous results obtained by Haas and Miermont (2011) and Bertoin and Kortchemski (2016) in the monotype setting. In a companion paper, we will use these results as a building block to study the scaling limits of multi-type Markov branching trees, with applications to growing models of random trees and multi-type Galton–Watson trees.

Extinction properties of multi-type continuous-state branching processes

Recently in Barczy et al. (2015), the notion of a multi-type continuous-state branching process (with immigration) having d-types was introduced as a solution to an d-dimensional vector-valued SDE. Preceding that, work on affine processes, originally motivated by mathematical finance, in Duffie et al. (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero and Pérez Garmendia (2017). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Engländer and Kypriano (2004) we explore their extinction properties and pose a number of open problems.

Multi-Type Branching Processes Modeling of Nosocomial Epidemics

AbstractNosocomial epidemics are infectious diseases which spread among different types of susceptible individuals in a health-care facility. To model this type of epidemics, we use a multi-type branching process with a multivariate negative binomial offspring distribution. In particular, we estimate the basic reproduction number

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches.

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth-death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle's study of a susceptible-infected-recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.

Queueing for an infinite bus line and aging branching process

We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right, and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is less than $$d_i$$ stations before, where $$d_i$$ is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually coalesce almost surely by coupling arguments. Three regimes appear, two of which leading to a.s. coalescing of the buses. The approach relies on a connection with age-structured branching processes with immigration and varying environment. We need to prove a Kesten–Stigum-type theorem, i.e., the a.s. convergence of the successive size of the branching process normalized by its mean. The techniques developed combine a spine approach for multitype branching process in varying environment and geometric ergodicity along the spine to control the increments of the normalized process.

A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures

This paper deals with the dynamic modeling and simulation of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy session during in vitro assays. Each cell is described by a finite number of phenotypic states with possible transitions between them. The population dynamics is then given by an age-dependent multi-type branching process. From this representation, we obtain formulas for the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. The proposed model has been implemented into Matlab© and the numerical results corroborate the ability of the model to reproduce four major types of cell responses: delayed growth, anti-proliferative, cytostatic and cytotoxic.

Predicting correlation coefficients for Monte Carlo eigenvalue simulations with multitype branching process

This paper provides a prediction method of the generation-to-generation correlations as observed when solving large scale eigenvalue problems such as full core nuclear reactor simulations. Knowing the correlations enables correction of the variance underestimation that occurs when assuming that the active generations are independent. The Monte Carlo power iteration is cast in the Multitype Branching Process (MBP) framework by discretizing the neutron phase space which allows calculation of spatial and temporal moments. These moments can then provide auto-correlation coefficients between the generations of MBP and are shown to accurately predict the auto-correlation coefficients of the original Monte Carlo simulation. This prediction capability was demonstrated on the full core 2D PWR BEAVRS benchmark and compared successfully with variance estimates from independent simulations.

Expectation maximization estimates of the offspring probabilities in a class of multitype branching processes with binary family trees

ABSTRACTWhen proliferating cells are counted in several independent colonies at some time points, the maximum likelihood estimates of the parameters of the multitype branching process are obtained trough an expectation maximization algorithm. In the case of an offspring distribution governed by a Markov branching process with binary family trees, this method, relying then on a partial knowledge of the tree, yields the same estimates as those computed with the complete knowledge of the tree.

Multitype Bellman-Harris branching model provides biological predictors of early stages of adult hippocampal neurogenesis

BackgroundAdult hippocampal neurogenesis, the process of formation of new neurons, occurs throughout life in the hippocampus. New neurons have been associated with learning and memory as well as mood control, and impaired neurogenesis has been linked to depression, schizophrenia, autism and cognitive decline during aging. Thus, understanding the biological properties of adult neurogenesis has important implications for human health. Computational models of neurogenesis have attempted to derive biologically relevant knowledge, hard to achieve using experimentation. However, the majority of the computational studies have predominantly focused on the late stages of neurogenesis, when newborn neurons integrate into hippocampal circuitry. Little is known about the early stages that regulate proliferation, differentiation, and survival of neural stem cells and their immediate progeny.ResultsHere, based on the branching process theory and biological evidence, we developed a computational model that represents the early stage hippocampal neurogenic cascade and allows prediction of the overall efficiency of neurogenesis in both normal and diseased conditions. Using this stochastic model with a simulation program, we derived the equilibrium distribution of cell population and simulated the progression of the neurogenic cascade. Using BrdU pulse-and-chase experiment to label proliferating cells and their progeny in vivo, we quantified labeled newborn cells and fit the model on the experimental data. Our simulation results reveal unknown but meaningful biological parameters, among which the most critical ones are apoptotic rates at different stages of the neurogenic cascade: apoptotic rates reach maximum at the stage of neuroblasts; the probability of neuroprogenitor cell renewal is low; the neuroblast stage has the highest temporal variance within the cell types of the neurogenic cascade, while the apoptotic stage is short.ConclusionAt a practical level, the stochastic model and simulation framework we developed will enable us to predict overall efficiency of hippocampal neurogenesis in both normal and diseased conditions. It can also generate predictions of the behavior of the neurogenic system under perturbations such as increase or decrease of apoptosis due to disease or treatment.

Fixation probability of a nonmutator in a large population of asexual mutators

In an adapted population of mutators in which most mutations are deleterious, a nonmutator that lowers the mutation rate is under indirect selection and can sweep to fixation. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large population of asexual mutators. We show that when beneficial mutations are absent, the fixation probability is a nonmonotonic function of the mutation rate of the mutator: it first increases sublinearly and then decreases exponentially. We also find that beneficial mutations can enhance the fixation probability of a nonmutator. Our analysis is relevant to an understanding of recent experiments in which a reduction in the mutation rates has been observed.