A branching stochastic evolutionary model of the B-cell repertoire.

We propose a stochastic framework to describe the evolution of the B-cell repertoire during germinal center (GC) reactions. Our model is formulated as a multitype age-dependent branching process with time-varying immigration. The immigration process captures the mechanism by which founder B cells initiate clones by gradually seeding GC over time, while the branching process describes the temporal evolution of the composition of these clones. The model assigns a type to each cell to represent attributes of interest. Examples of attributes include the binding affinity class of the B cells, their clonal family, or the nucleotide sequence of the heavy and light chains of their receptors. The process is generally non-Markovian. We present its properties, including as t → ∞ when the process is supercritical, the most relevant case to study expansion of GC B cells. We introduce temporal alpha and beta diversity indices for multitype branching processes. We focus on the dynamics of clonal dominance, highlighting its non-stationarity, and the accumulation of somatic hypermutations in the context of sequential immunization. We evaluate the impact of the ongoing seeding of GC by founder B cells on the dynamics of the B-cell repertoire, and quantify the effect of precursor frequency and antigen availability on the timing of GC entry. An application of the model illustrates how it may help with interpretation of BCR sequencing data.

Statistical modelling of COVID-19 pandemic development applying branching processes.

In this paper, a statistical model for COVID-19 infection dynamics is described, using only the observed daily statistics of infected individuals. For this purpose, two special classes of branching processes without or with an immigration component are considered. These models are intended to estimate the main parameter of the infection and to give a prediction of the mean value of the non-observed population of the infected individuals. This is a serious advantage in comparison with other more complicated models where the officially reported data are not sufficient for estimation of the model parameters. The model is applied for different regions in the world and the corresponding parameters of the infection dynamics are estimated.

Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration.

We consider a class of Sevastyanov branching processes with non-homogeneous Poisson immigration. These processes relax the assumption required by the Bellman-Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper, we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include novel LLN and CLT which emerge from the non-homogeneity of the immigration process.

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.

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.

Two-Type Reducible Age-Dependent Branching Processes With Non-Homogeneous Poisson Immigration.

Two type reducible age-dependent branching stochastic processes with non-homogeneous Poisson immigration are considered as models of renewal cell population dynamics. The asymptotic behaviour of the first moments of the process with or without immigration is investigated. Several classes of asymptotic behavior are identified for the population dynamics. Our results are also useful for developing associated methods of statistical inference.

Two-Type Age-Dependent Branching Processes with Inhomogeneous Immigration as Models of Renewing Cell Populations.

Two-type reducible age-dependent branching processes with inhomogeneous immigration are considered to describe the kinetics of renewing cell populations. This class of processes can be used to model the generation of oligodendrocytes in the central nervous system in vivo or the kinetics of leukemia cells. The asymptotic behavior of the first and second moments, including the correlation, of the process is investigated.

Modeling Cell Kinetics using Branching Processes with Non-Homogeneous Poisson Immigration.

Age-dependent branching processes with non-homogeneous Poisson immigration are proposed as models of cell proliferation kinetics. The asymptotic behaviour of the first and second-order moments is investigated and the obtained results are use to develop a relevant statistical inference.

Limiting Distributions for Multitype Branching Processes.

In this paper the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The paper considers also the relative frequencies of distinct types of individuals which is motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.