Age-dependent Branching Process Research Articles

Modeling escape from extinction with decomposable multi-type Sevastyanov branching processes

Biological populations under stress often face certain extinction unless they adapt toward unfavorable circumstances. In some scenarios such adaptation can assume the form of mutations within the genome of the population (e.g., cancer cells resisting chemotherapy, viruses developing resistance toward a vaccine), while in other scenarios adaptation can be in the form of movement toward some physical location (e.g., spreading of cancer cells to parts of the organism unaffected by treatment, animal populations fleeing polluted areas or areas struck by disaster). Regardless of the particular situation, it is often the case that cells/individuals with different levels of adaptation (which we may group into types) emerge among the cells/individuals of a stressed population. We propose a decomposable multi-type Sevastyanov branching process (possibly with multiple supercritical types) for modeling relevant aspects of the dynamics of such populations. The branching process developed within this paper is a generalization of the decomposable multi-type age-dependent branching process with a single supercritical type considered in Slavtchova-Bojkova and Vitanov. With respect to Slavtchova-Bojkova and Vitanov, we introduce additional, possibly supercritical, types into the interaction scheme between types, further, we incorporate possible dependence of the reproductive capabilities of cells/individuals from their age. We obtain a system of integral equations for the probability generating function of the new process and accordingly expand previous results from Slavtchova-Bojkova and Vitanov concerning probabilities of extinction, number of occurred mutations, waiting time to escape mutant, and immediate risk of escaping extinction. We also provide a general numerical scheme for calculating obtained systems of integral equations.

Age-dependent branching processes and applications to the Luria-Delbruck experiment

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbrück that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution have spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by Lea, Coulson, and Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/56/abstr.html

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs, geometric inhomogeneous random graphs and scale-free percolation. In hyperbolic random graphs, n=Θ(eR∕2) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In geometric inhomogeneous random graphs, and in scale-free percolation, each vertex is given an independent weight and location from an underlying measured metric space and Zd, respectively, and two vertices are connected independently with a probability that is a function of their distance and their weights. We assign independent and identically distributed (i.i.d.) weights to the edges of the obtained random graphs, and investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical weighted distance. In scale-free percolation, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity.In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ∈(2,3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution, that is, the branching process produces infinitely many individuals in finite time. We show that in all three models, typical distances within the giant/infinite component converge in distribution, that is, no re-scaling is necessary. This solves an open question in Hofstad and Komjáthy (2017).The main tools of our proof are to develop elaborate couplings of the models to infinite versions, to follow the shortest paths to infinity and then to connect these paths by using weight-dependent percolation on the graphs, that is, we delete edges attached to vertices with higher weight with higher probability. We realise the percolation using the edge-weights: only very short edges connected to high weight vertices are allowed to stay, hence establishing arbitrarily short upper bounds for connections.

Multi-type age-dependent branching processes as models of metastasis evolution

Metastasis, the spread of cancer cells from a primary tumor to secondary location(s) in the human organism, is the ultimate cause of death for the majority of cancer patients. Although studied for more than 180 years, increasing efforts in recent years have significantly contributed to a better understanding of this aspect of tumor development. Adding to this understanding, our current paper proposes a multi-type branching process model, through which various topics, such as extinction of cancer cell clones, occurrence of successful mutants, and immediate risk of escaping extinction, are investigated. More specifically, the relationship between the lifespan distribution of cancer cells with subcritical reproduction and the waiting time until the occurrence of a mutant (having supercritical reproduction) cancer cell, escaping extinction, is highlighted. The theoretical studies formulated in this paper may allow our model to be tailored to real data available for particular kinds of cancer and chemotherapy.

Tight Fluctuations of Weight-Distances in Random Graphs with Infinite-Variance Degrees

We prove results for first-passage percolation on the configuration model with degrees having asymptotic finite mean, infinite variance and i.i.d. edge-weights with strictly positive support of the form Y=a+X, where a is a positive constant and the excess edge-weight X is a non-negative random variable with zero as the infimum of its support. We prove that the weight of the optimal path between two uniformly chosen vertices has tight fluctuations around the asymptotic mean of the graph-distance if and only if the following condition holds: the random variable X is such that the age-dependent branching process describing first-passage percolation exploration in the same graph with edge-weights from distribution X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost-shortest paths in the graph-distance proliferate, in the sense that there even exist paths having tight total excess edge-weight for appropriate edge-weight distributions. Our proof makes use of a degree-dependent percolation model that we believe is interesting in its own right, as well as tightness results for distances in scale-free configuration models that we prove to hold under rather weak conditions on the degrees.

Weighted Distances in Scale-Free Configuration Models

In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent tau in (2,3). We assign independent and identically distributed (i.i.d.) weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time—called explosive branching process—Baroni, Hofstad and the second author showed in Baroni et al. (J Appl Probab 54(1):146–164, 2017) that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the tau in (2,3) case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is O(log log n), where n is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices.

Nonparametric estimation of the division rate of an age dependent branching process

We study the nonparametric estimation of the branching rate B(x) of a supercritical Bellman–Harris population: a particle with age x has a random lifetime governed by B(x); at its death time, it gives rise to k≥2 children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over [0,T]. Asymptotics are taken as T→∞; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity properties, we construct a kernel-based estimator of B(x) that achieves the rate of convergence exp(−λBβ2β+1T), where λB is the Malthus parameter and β>0 is the smoothness of the function B(x) in a vicinity of x. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time T only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.

A test of homogeneity for age-dependent branching processes with immigration.

We propose a novel procedure to test whether the immigration process of a discretely observed age-dependent branching process with immigration is time-homogeneous. The construction of the test is motivated by the behavior of the coefficient of variation of the population size. When immigration is time-homogeneous, we find that this coefficient converges to a constant, whereas when immigration is time-inhomogeneous we find that it is time-dependent, at least transiently. Thus, we test the assumption that the immigration process is time-homogeneous by verifying that the sample coefficient of variation does not vary significantly over time. The test is simple to implement and does not require specification or fitting any branching process to the data. Simulations and an application to real data on the progression of leukemia are presented to illustrate the approach.

ON CLASSES OF EQUIVALENCE AND IDENTIFIABILITY OF AGE-DEPENDENT BRANCHING PROCESSES.

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper, we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. This result is applicable to study identifiability of the offspring and lifespan distributions for parametric families of branching processes. For example, we identify classes of Markov processes that are not identifiable. We show that age-dependent processes with (non-exponential) gamma distributed lifespan are identifiable and that Smith-Martin processes are not always identifiable.

On Classes of Equivalence and Identifiability of Age-Dependent Branching Processes

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. This result can be used to study identifiability of the offspring and lifespan distributions for parametric families of branching processes. For example, we identify classes of Markov processes that are not identifiable. We show that age-dependent processes with (nonexponential) gamma-distributed lifespans are identifiable and that Smith-Martin processes are not always identifiable.

Stochastic modeling of stress erythropoiesis using a two-type age-dependent branching process with immigration

The erythroid lineage is a particularly sensitive target of radiation injury. We model the dynamics of immature (BFU-E) and mature (CFU-E) erythroid progenitors, which have markedly different kinetics of recovery, following sublethal total body irradiation using a two-type reducible age-dependent branching process with immigration. Properties of the expectation and variance of the frequencies of both types of progenitors are presented. Their explicit expressions are derived when the process is Markovian, and their asymptotic behavior is identified in the age-dependent (non-Markovian) case. Analysis of experimental data on the kinetics of BFU-E and CFU-E reveals that the probability of self-renewal increases transiently for both cell types following sublethal irradiation. In addition, the probability of self-renewal increased more for CFU-E than for BFU-E. The strategy adopted by the erythroid lineage ensures replenishment of the BFU-E compartment while optimizing the rate of CFU-E recovery. Finally, our analysis also indicates that radiation exposure causes a delay in BFU-E recovery consistent with injury to the hematopoietic stem/progenitor cell compartment that give rise to BFU-E. Erythroid progenitor self-renewal is thus an integral component of the recovery of the erythron in response to stress.

The closure property of 2RV under random sum

Under certain conditions, the survival function of random sum of a sequence of independent and identically distributed nonnegative random variables with a common survival function F¯, preserves the second-order regular variation (2RV) whenever F¯ is of 2RV. Applications in the Cramér–Lundberg model and the age-dependent branching process are presented.

Sevastyanov branching processes with non-homogeneous Poisson immigration

Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity r(t). The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of r(t). Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity.

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.

Asymptotic Behavior of Cell Populations Described by Two-Type Reducible Age-Dependent Branching Processes With Non-Homogeneous Immigration

Stem and precursor cells play a critical role in tissue development, maintenance, and repair throughout the life. Often, experimental limitations prevent direct observation of the stem cell compartment, thereby posing substantial challenges to the analysis of such cellular systems. Two-type age-dependent branching processes with immigration are proposed to model populations of progenitor cells and their differentiated progenies. Immigration of cells into the pool of progenitor cells is formulated as a non-homogeneous Poisson process. The asymptotic behavior of the process is governed by the largest of two Malthusian parameters associated with embedded Bellman-Harris processes. Asymptotic approximations to the expectations of the total cell counts are improved by Markov compensators.

Generalization of the Beck-Cohen superstatistics

Generalized superstatistics, i.e., a "statistics of superstatistics," is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random control parameter that determines both the density of energy states and the distribution of the intensive parameter for each superstatistical subsystem, thereby forming the third, upper level of dynamics. Generalized superstatistics can be used for nonstationary nonequilibrium systems. The system in which a supercritical multitype age-dependent branching process takes place is an example of a nonstationary generalized superstatistical system. The theory is applied to pair production in a neutron star magnetosphere.

Limit theorems for supercritical age-dependent branching processes with neutral immigration

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

Limit theorems for supercritical age-dependent branching processes with neutral immigration

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

Mathematical and Experimental Approaches to Identify and Predict the Effects of Chemotherapy on Neuroglial Precursors

The adverse effects of chemotherapy on normal cells of the body create substantial clinical problems for many cancer patients. However, relatively little is known about the effects, other than promotion of cell death, of such agents on the function of normal precursor cells critical in tissue homeostasis and repair. We have combined mathematical and experimental analyses to identify the effects of sublethal doses of chemotherapy on glial precursor cells of the central nervous system. We modeled the temporal development of a population of precursor and terminally differentiated cells exposed to sublethal doses of carmustine (BCNU), a classic alkylating chemotherapeutic agent used in treatment of gliomas and non-Hodgkin's lymphomas, as a multitype age-dependent branching process. We fitted our model to data from in vitro clonal experiments using the method of pseudo-likelihood. This approach identifies several novel drug effects, including modification of the cell cycle length, the time between division and differentiation, and alteration in the probability of undergoing self-renewal division in precursor cells. These changes of precursor cell function in the chemotherapy-exposed brain may have profound clinic implications. We applied our computational approach to analyze the effects of BCNU on clonal cultures of oligodendrocyte progenitor cells-one of the best-characterized neural progenitor cells in the mammalian brain. Our analysis reveals that transient exposures to BCNU increased the cell cycle length of progenitor cells and decreased their time to differentiation, while also decreasing the likelihood that they will undergo self-renewing divisions. By investigating the behavior of our mathematical model, we demonstrate that precursor cell populations should recover spontaneously from transient modifications of the timing of division and of differentiation, but such recovery will not happen after alteration of cell fate. These studies identify means by which precursor cell function can be critically compromised by transient exposure to chemotherapy with long-term consequences on the progenitor cell pool even in the absence of drug-induced apoptosis. These analyses also provide novel tools that apply broadly to identify effects of chemotherapeutic agents and other physiological stressors.

A composite likelihood approach to the analysis of longitudinal clonal data on multitype cellular systems under an age-dependent branching process.

A recurrent statistical problem in cell biology is to draw inference about cell kinetics from observations collected at discrete time points. We investigate this problem when multiple cell clones are observed longitudinally over time. The theory of age-dependent branching processes provides an appealing framework for the quantitative analysis of such data. Likelihood inference being difficult in this context, we propose an alternative composite likelihood approach, where the estimation function is defined from the marginal or conditional distributions of the number of cells of each observable cell type. These distributions have generally no closed-form expressions but they can be approximated using simulations. We construct a bias-corrected version of the estimating function, which also offers computational advantages. Two algorithms are discussed to compute parameter estimates. Large sample properties of the estimator are presented. The performance of the proposed method in finite samples is investigated in simulation studies. An application to the analysis of the generation of oligodendrocytes from oligodendrocyte type-2 astrocyte progenitor cells cultured in vitro reveals the effect of neurothrophin-3 on these cells. Our work demonstrates also that the proposed approach outperforms the existing ones.