Bellman-Harris Branching Process Research Articles

Reduced critical Bellman–Harris branching processes for small populations

Abstract A critical Bellman-Harris branching process {Z(t),t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z(t) ≤ φ(t), where either φ(t) = o(t) as t → ∞ or φ(t) = at,a>0, we study the structure of the process where Z(s,t) is the number of particles in the initial process at moment s which either survive up to moment t or have a positive number of descendants at this moment.

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

Coalescence on Supercritical Bellman-Harris Branching Processes

We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$ random chosen individuals. In this paper, we study the distributions of $D_k(t)$ and $X_k(t)$ and their limit distributions as $t \to \infty$.

Critical Bellman-Harris branching processes with long-living particles

A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter β ∈ (0, 1]. It is shown that, contrary to the critical indecomposable Bellman-Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment t is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment t given that the population contains particles of the first type at this moment.

Identifiability of Age-Dependent Branching Processes from Extinction Probabilities and Number Distributions

Consider an age-dependent, single-species branching process defined by a progeny number distribution, and a lifetime distribution associated with each independent particle. In this paper, we focus on the associated inverse problem where one wishes to for- mally solve for the progeny number distribution or the lifetime distribution that defines the Bellman-Harris branching process. We derive results for the existence and uniqueness (the identifiability) of these two distributions given one of two types of information: the extinc- tion time probability of the entire process (extinction time distribution), or the distribution of the total number of particles at one fixed time. We demonstrate that perfect knowledge of the distribution of extinction times allows us to formally determine either the progeny number distribution or the lifetime distribution. Furthermore, we show that these constructions are unique. We then consider "data" consisting of a perfectly known total number distribution given at one specific time. For a process with known progeny number distribution and ex- ponentially distributed lifetimes, we show that the rate parameter is identifiable. For general lifetime distributions, we also show that the progeny distribution is globally unique. Our results are presented through four theorems, each describing the constructions in the four distinct cases.

Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={a t,i : 1≤ i≤ Z(t)}, where a t,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let D k(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of D k(t) and its limit distribution as t→∞.

Local Particles Numbers in Critical Branching Random Walk

Critical catalytic branching random walk on d-dimensional integer lattice is investigated for all d. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of particles presence at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman-Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on an integer lattice.

Statistics for the Luria-Delbrück distribution

The Luria-Delbr\"uck distribution is a classical model of mutations in cell kinetics. It is obtained as a limit when the probability of mutation tends to zero and the number of divisions to infinity. It can be interpreted as a compound Poisson distribution (for the number of mutations) of exponential mixtures (for the developing time of mutant clones) of geometric distributions (for the number of cells produced by a mutant clone in a given time). The probabilistic interpretation, and a rigourous proof of convergence in the general case, are deduced from classical results on Bellman-Harris branching processes. The two parameters of the Luria-Delbr\"uck distribution are the expected number of mutations, which is the parameter of interest, and the relative fitness of normal cells compared to mutants, which is the heavy tail exponent. Both can be simultaneously estimated by the maximum likehood method. However, the computation becomes numerically unstable as soon as the maximal value of the sample is large, which occurs frequently due to the heavy tail property. Based on the empirical generating function, robust estimators are proposed and their asymptotic variance is given. They are comparable in precision to maximum likelihood estimators, with a much broader range of calculability, a better numerical stability, and a negligible computing time.

Branching dynamics of viral information spreading

Despite its importance for rumors or innovations propagation, peer-to-peer collaboration, social networking, or marketing, the dynamics of information spreading is not well understood. Since the diffusion depends on the heterogeneous patterns of human behavior and is driven by the participants' decisions, its propagation dynamics shows surprising properties not explained by traditional epidemic or contagion models. Here we present a detailed analysis of our study of real viral marketing campaigns where tracking the propagation of a controlled message allowed us to analyze the structure and dynamics of a diffusion graph involving over 31,000 individuals. We found that information spreading displays a non-Markovian branching dynamics that can be modeled by a two-step Bellman-Harris branching process that generalizes the static models known in the literature and incorporates the high variability of human behavior. It explains accurately all the features of information propagation under the "tipping point" and can be used for prediction and management of viral information spreading processes.

Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling

The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination policies.

Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling

The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination policies.

Determining the expected variability of immune responses using the cyton model

During an adaptive immune response, lymphocytes proliferate for 5-20 cell divisions, then stop and die over a period of weeks. The cyton model for regulation of lymphocyte proliferation and survival was introduced by Hawkins et al. (Proc. Natl. Acad. Sci. USA 104, 5032-5037, 2007) to provide a framework for understanding this response and its regulation. The model assumes stochastic values for division and survival times for each cell in a responding population. Experimental evidence indicates that the choice of times is drawn from a skewed distribution such as the lognormal, with the fate of individual cells being potentially highly variable. For this reason we calculate the higher moments of the model so that the expected variability can be determined. To do this we formulate a new analytic framework for the cyton model by introducing a generalization to the Bellman-Harris branching process. We use this framework to introduce two distinct approaches to predicting variability in the immune response to a mitogenic signal. The first method enables explicit calculations for certain distributions and qualitatively exhibits the full range of observed immune responses. The second approach does not facilitate analytic solutions, but allows simple numerical schemes for distributions for which there is little prospect of analytic formulae. We compare the predictions derived from the second method to experimentally observed lymphocyte population sizes from in vivo and in vitro experiments. The model predictions for both data sets are remarkably accurate. The important biological conclusion is that there is limited variation around the expected value of the population size irrespective of whether the response is mediated by small numbers of cells undergoing many divisions or for many cells pursuing a small number of divisions. Therefore, we conclude the immune response is robust and predictable despite the potential for great variability in the experience of each individual cell.

Pseudo-likelihood estimation for discretely observed multitype Bellman–Harris branching processes

This paper presents a method for parametric estimation in the class of multitype Bellman–Harris branching processes when the data consist of cell counts collected on several colonies observed at discrete time points. This sampling scheme arises frequently in biology to analyze cell proliferation in tissue culture. We investigate the use of a pseudo-likelihood approach in this context. It is defined from the mean vectors and variance–covariance matrices of the numbers of cells. The proposed estimator is strongly consistent and asymptotically normal as the number of observed colonies goes to infinity. In situations where these two moments have no closed-form expressions their values can be replaced by simulation-based approximations. The resulting simulated pseudo-maximum likelihood estimator is asymptotically equivalent to the pseudo-maximum likelihood estimator as the number of simulation increases. We illustrate the proposed method via two examples, and evaluate its finite sample properties through simulations.

Estimating the life-span of oligodendrocytes from clonal data on their development in cell culture

This paper presents a new method to analyze clonal data on oligodendrocyte development in cell culture. The process of oligodendrocyte generation from precursor cells is modelled as a multi-type Bellman–Harris branching process as suggested in an earlier paper [K. Boucher, A. Zorin, A.Y. Yakovlev, M. Mayer-Pröschel, M. Noble, An alternative stochastic model of generation of oligodendrocytes in cell culture, J. Math. Biol. 43 (2001) 22]. This model has been extended to allow for death of oligodendrocytes as well as a dissimilar distribution of the first mitotic cycle duration as compared to the subsequent cycles of precursor cells, which lengths are assumed to be independent and identically distributed random variables. Since the time-span of oligodendrocytes is not directly observable in clonal data, plausible parametric assumptions are invoked to make estimation problems tractable. In particular, the time to cell death follows a two-parameter gamma distribution, while the lapse of time between the event of cell death and the event of cell disintegration is assumed to be exponentially distributed. A simulated pseudo maximum likelihood method for estimation of model parameters has been developed using simulation-based approximations of the expected numbers and variance-covariance matrices for different types of cells. Finite sample properties of the estimation procedure are studied by computer simulations. The proposed method is illustrated with an analysis of the clonal development of O-2A progenitor cells isolated from the rat optic nerve and the corpus callosum.

Asymptotics for sums of random variables with local subexponential behaviour

We study distributions F on [0, infinity) such that for some T less than or equal to infinity F*(2)(x, x + T] similar to 2F(x, x + T]. The case T = infinity corresponds to F being subexponential, and our analysis shows that the properties for T < I are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman-Harris branching processes.

Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration

The branching processes with state-dependent immigration are considered as alternating regenerative processes. The main purpose is to demonstrate some new “regenerative” methods. Critical Bellman–Harris branching processes with state-dependent immigration are investigated and new limit theorems are obtained in the case of an infinite offspring variance and possibly infinite mean of the immigrants.

A branching process model of telomere shortening

A branching process model of a certain cell population is considered. The population is divided into types according to the number of remaining chromosome end units. This leads to a reducible multi-type Bellman-Harris branching process which turns out to exhibit polynomial growth dynamics: let be the expected number of j-type cells of age less than a at time t starting from a k-type ancestor cell. Then, as where the constant C depends on a,j and k. and can be given explicitely. The proof is fairly short and simple, using elementary results from renewal theory and a Tauberian theorem for Laplace-Stieltjes transforms

Branching Processes with Final Types of Particles and Random Trees

This paper considers a Bellman–Harris branching process whose probability generating function $f(s)$ of the number of direct descendants of particles satisfies the relation $f(s) = s + (1 - s)^{1 + \alpha } L(1 - s),0 < \alpha \leqq 1$. Let $\tau $ be the moment of extinction of the process and let $\nu_\Delta $ be the total number of particles the number of direct descendants of each of which belongs to the set $\Delta ,\Delta \subset \{ 0,1, \ldots ,n, \ldots \} $. The paper gives conditions under which, for any $x \in ( - \infty , + \infty )$ and some scaling constants $b(N)$, a nondegenerate limit, $\lim _{N \to \infty } {\bf P}\{ \tau b(N) \leqq x|\nu_\Delta = N\} $, exists.

The Total Number of Particles in a Reduced Bellman-Harris Branching Process

Let $z(t)$ be the number of particles at time t in a Bellman-Harris branching process with generating function $f(s)$ of the number of direct descendants and distribution $G(t)$ of particle lifelength satisfying the conditions \[f'(1) = 1,\qquad f(s) = s + (1 - s)^{1 + \alpha } L(1 - s),\] where $\alpha \in ( {0,1} ]$, the function $L(x)$ varies slowly as $x \to 0 + $, and \[\mathop {\lim }\limits_{n \to \infty } \frac{{n\left( {1 - G\left( n \right)} \right)}} {{1 - f_n \left( 0 \right)}} = 0,\] where $f_n ( s )$ is the nth iteration of $f(s)$. Denote by $\{ z(\tau ,t), 0 \leq \tau \leq t\}$ the corresponding reduced Bellman-Harris branching process, where $z(\tau ,t)$ is the number of particles in the initial process at time $\tau $ whose descendants or they themselves are alive at time t. Let $\nu (t)$ be the number of dead particles of the reduced process to time t. The paper finds the limiting distribution of $\nu(t)$ under the conditions $z(t) > 0$ and $t \to \infty $.