Multitype Branching Processes Research Articles

Small value probabilities for supercritical multitype branching processes with immigration

We consider the small value probability of a supercritical multi-type branching process with immigration (Zn,n≥0). It is well known that with a suitable normalization, ρ−nZn converges to a finite and non-negative limit, which can be written as Wt where W is a random variable and t is a deterministic vector. Based on the large deviation results in Jones (2004) about the supercritical multitype branching processes, and techniques dealing with immigration used in Chu et al. (2014), we obtain the convergence rate of P(W≤ε) as ε→0+, which is known as the small value probability of W.

Cancer initiation with epistatic interactions between driver and passenger mutations

We investigate the dynamics of cancer initiation in a mathematical model with one driver mutation and several passenger mutations. Our analysis is based on a multi-type branching process: we model individual cells which can either divide or undergo apoptosis. In the case of a cell division, the two daughter cells can mutate, which potentially confers a change in fitness to the cell. In contrast to previous models, the change in fitness induced by the driver mutation depends on the genetic context of the cell, in our case on the number of passenger mutations. The passenger mutations themselves have no or only a very small impact on the cell׳s fitness. While our model is not designed as a specific model for a particular cancer, the underlying idea is motivated by clinical and experimental observations in Burkitt Lymphoma. In this tumor, the hallmark mutation leads to deregulation of the MYC oncogene which increases the rate of apoptosis, but also the proliferation rate of cells. This increase in the rate of apoptosis hence needs to be overcome by mutations affecting apoptotic pathways, naturally leading to an epistatic fitness landscape. This model shows a very interesting dynamical behavior which is distinct from the dynamics of cancer initiation in the absence of epistasis. Since the driver mutation is deleterious to a cell with only a few passenger mutations, there is a period of stasis in the number of cells until a clone of cells with enough passenger mutations emerges. Only when the driver mutation occurs in one of those cells, the cell population starts to grow rapidly.

Limit distributions for multitype branching processes of $m$-ary search trees

Soit $m\geq3$ un entier. Très populaire en informatique fondamentale, l’arbre $m$-aire de recherche est une chaîne de Markov à temps discret qui modélise de célèbres algorithmes de tri et de recherche de données. Ce processus aléatoire vérifie une transition de phase bien connue : lorsque $m\leq26$, le comportement asymptotique du processus est gaussien. En revanche, lorsque $m\geq27$, il n’est plus gaussien et fait apparaître la limite $W^{DT}$ d’une martingale à valeurs complexes. Dans cet article, on considère le processus de branchement multitype qui est le plongement en temps continu de l’arbre $m$-aire de recherche. Ce processus fait l’objet d’une transition de phase du même type. En particulier, lorsque $m\ge27$, son asymptotique s’exprime à l’aide de la limite $W$ d’une martingale complexe. Grâce à la propriété de branchement, la loi de $W$ est solution d’une équation en distribution du type $Z\stackrel{ \mathcal{L}}{=}\mathrm{e}^{-\lambda T}(Z^{(1)}+\cdots+Z^{(m)})$ où $\lambda$ est un nombre complexe particulier, les $Z^{(k)}$ sont des variables aléatoires complexes indépendantes dont la loi est celle de $Z$, $T$ est une variable aléatoire réelle positive indépendante des $Z^{(k)}$, et $\stackrel{ \mathcal{L}}{=}$ désigne l’égalité en distribution. On étudie cette équation en loi par des approches variées. L’existence et l’unicité de solutions sont prouvées par des méthodes de contraction. L’absolue continuité de $W$ et le fait que son support soit le plan complexe tout entier sont démontrés par analyse de Fourier. Enfin, on obtient l’existence de moments exponentiels en considérant $W$ comme la limite d’une cascade de Mandelbrot à valeurs complexes.

Multispecies Virial Expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange-Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs.

Using multitype branching processes to quantify statistics of disease outbreaks in zoonotic epidemics.

Branching processes have served as a model for chemical reactions, biological growth processes, and contagion (of disease, information, or fads). Through this connection, these seemingly different physical processes share some common universalities that can be elucidated by analyzing the underlying branching process. In this work we focus on coupled branching processes as a model of infectious diseases spreading from one population to another. An exceedingly important example of such coupled outbreaks are zoonotic infections that spill over from animal populations to humans. We derive several statistical quantities characterizing the first spillover event from animals to humans, including the probability of spillover, the first passage time distribution for human infection, and disease prevalence in the animal population at spillover. Large stochastic fluctuations in those quantities can make inference of the state of the system at the time of spillover difficult. Focusing on outbreaks in the human population, we then characterize the critical threshold for a large outbreak, the distribution of outbreak sizes, and associated scaling laws. These all show a strong dependence on the basic reproduction number in the animal population and indicate the existence of a novel multicritical point with altered scaling behavior. The coupling of animal and human infection dynamics has crucial implications, most importantly allowing for the possibility of large human outbreaks even when human-to-human transmission is subcritical.

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.

Random Fluid Limit of an Overloaded Polling Model

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

Random Fluid Limit of an Overloaded Polling Model

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.

Bottom-up model of self-organized criticality on networks

The Bak-Tang-Wiesenfeld (BTW) sandpile process is an archetypal, stylized model of complex systems with a critical point as an attractor of their dynamics. This phenomenon, called self-organized criticality, appears to occur ubiquitously in both nature and technology. Initially introduced on the two-dimensional lattice, the BTW process has been studied on network structures with great analytical successes in the estimation of macroscopic quantities, such as the exponents of asymptotically power-law distributions. In this article, we take a microscopic perspective and study the inner workings of the process through both numerical and rigorous analysis. Our simulations reveal fundamental flaws in the assumptions of past phenomenological models, the same models that allowed accurate macroscopic predictions; we mathematically justify why universality may explain these past successes. Next, starting from scratch, we obtain microscopic understanding that enables mechanistic models; such models can, for example, distinguish a cascade's area from its size. In the special case of a 3-regular network, we use self-consistency arguments to obtain a zero-parameter mechanistic (bottom-up) approximation that reproduces nontrivial correlations observed in simulations and that allows the study of the BTW process on networks in regimes otherwise prohibitively costly to investigate. We then generalize some of these results to configuration model networks and explain how one could continue the generalization. The numerous tools and methods presented herein are known to enable studying the effects of controlling the BTW process and other self-organizing systems. More broadly, our use of multitype branching processes to capture information bouncing back and forth in a network could inspire analogous models of systems in which consequences spread in a bidirectional fashion.

Estimation of the infection parameter of an epidemic modeled by a branching process

The aim of this paper is to build estimators of the infection parameter in the different phases of an epidemic (growth and decay phases). The epidemic is modeled by a Markovian process of order $d\geqslant1$, and can be written as a multitype branching process. We propose three estimators suitable for the different classes of criticality of the process, and consequently for different phases of the epidemic. We prove their consistency and asymptotic normality when the number of ancestors (resp. number of generations) tends to infinity.

Intrinsic Branching Structure within Random Walk on Z

We reveal the branching structure for a nonhomogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,\infty)$ by the walk starting from $0,$ could be expressed in terms of a nonhomogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk with bounded jumps in random environment and specify the explicit invariant density for the Markov chain of “the environment viewed from the particles.” The invariant density and the limit velocity could be expressed explicitly in terms of the environment.

Synchronization-induced persistence versus selection for habitats in spatially coupled ecosystems

Critical population phase transitions, in which a persistent population becomes extinction-prone owing to environmental changes, are fundamentally important in ecology, and their determination is a key factor in successful ecosystem management. To persist, a species requires a suitable environment in a sufficiently large spatial region. However, even if this condition is met, the species does not necessarily persist, owing to stochastic fluctuations. Here, we develop a model that allows simultaneous investigation of extinction due to either stochastic or deterministic reasons. We find that even classic birth-death processes in spatially extended ecosystems exhibit phase transitions between extinction-prone and persistent populations. Sometimes these are first-order transitions, which means that environmental changes may result in irreversible population collapse. Moreover, we find that higher migration rates not only lead to higher robustness to stochastic fluctuations, but also result in lower sustainability in heterogeneous environments by preventing efficient selection for suitable habitats. This demonstrates that intermediate migration rates are optimal for survival. At low migration rates, the dynamics are reduced to metapopulation dynamics, whereas at high migration rates, the dynamics are reduced to a multi-type branching process. We focus on species persistence, but our results suggest a unique method for finding phase transitions in spatially extended stochastic systems in general.

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 → ∞.

Conditional Distributions and Waiting Times in Multitype Branching Processes

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

Conditional Distributions and Waiting Times in Multitype Branching Processes

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X t :tÂ?0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism Â?(Â?)=Â?Â?Â?+AŸÂ? 2+Â?(0,+Â?)(e Â?Â?x Â?1+Â?x)n(dx), where Â?=Â?Â?Â?(0+)>0, AŸÂ?0, and n is a measure on (0,Â?) such that Â?(0,+Â?) x 2 n(dx)<+Â?. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W t =e Â?Â?t Â?X t Â? is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W Â? such that W t Â?W Â?, $\mathbb{P} _{\mu}$ -a.s. as tÂ?Â?. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer Â?(f) in term of the spectral decomposition of f. In the small branching rate case Â? 2Â?(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miloś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miloś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.

Markov-modulated stochastic recursive equations with applications to delay-tolerant networks

We investigate a class of Markov-modulated stochastic recursive equations. This class includes multi-type branching processes with immigration as well as linear stochastic equations. Conditions are established for the existence of a stationary solution and expressions for the first two moments of this solution are found. Furthermore, the transient characteristics of the stochastic recursion are investigated: we obtain the first two moments of the transient solution as well. Finally, to illustrate our approach, the results are applied to the performance evaluation of packet forwarding in delay-tolerant mobile ad-hoc networks.

Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models

Environmental heterogeneity, spatial connectivity, and movement of individuals play important roles in the spread of infectious diseases. To account for environmental differences that impact disease transmission, the spatial region is divided into patches according to risk of infection. A system of ordinary differential equations modeling spatial spread of disease among multiple patches is used to formulate two new stochastic models, a continuous-time Markov chain, and a system of stochastic differential equations. An estimate for the probability of disease extinction is computed by approximating the Markov chain model with a multitype branching process. Numerical examples illustrate some differences between the stochastic models and the deterministic model, important for prevention of disease outbreaks that depend on the location of infectious individuals, the risk of infection, and the movement of individuals.

Uncovering epidemiological dynamics in heterogeneous host populations using phylogenetic methods

Host population structure has a major influence on epidemiological dynamics. However, in particular for sexually transmitted diseases, quantitative data on population contact structure are hard to obtain. Here, we introduce a new method that quantifies host population structure based on phylogenetic trees, which are obtained from pathogen genetic sequence data. Our method is based on a maximum-likelihood framework and uses a multi-type branching process, under which each host is assigned to a type (subpopulation). In a simulation study, we show that our method produces accurate parameter estimates for phylogenetic trees in which each tip is assigned to a type, as well for phylogenetic trees in which the type of the tip is unknown. We apply the method to a Latvian HIV-1 dataset, quantifying the impact of the intravenous drug user epidemic on the heterosexual epidemic (known tip states), and identifying superspreader dynamics within the men-having-sex-with-men epidemic (unknown tip states).

Risk of Infectious Disease Outbreaks by Imported Cases with Application to the European Football Championship 2012

The European Centre for Disease Prevention and Control called the attention in March 2012 to the risk of measles in Ukraine among visitors to the 2012 UEFA European Football Championship. Large populations of supporters travelled to various locations in Poland and Ukraine, depending on the schedule of Euro 2012 and the outcome of the games, possibly carrying the disease from one location to another. In the present study, we propose a novel two-phase multitype branching process model with immigration to describe the risk of a major epidemic in connection with large-scale sports-related mass gathering events. By analytic means, we calculate the expected number and the variance of imported cases and the probability of a major epidemic caused by the imported cases in their home country. Applying our model to the case study of Euro 2012 we demonstrate that the results of the football games can be highly influential to the risk of measles outbreaks in the home countries of supporters. To prevent imported epidemics, it should be emphasized that vaccinating travellers would most efficiently reduce the risk of epidemic, while requiring the minimum doses of vaccines as compared to other vaccination strategies. Our theoretical framework can be applied to other future sport tournaments too.