Markov Branching Process Research Articles

Nonlinear Markov branching processes with immigration and resurrection

The nonlinear Markov branching process with immigration and resurrection (NMBPIR) is considered in this paper. We give the criteria for regularity and uniqueness of NMBPIR, and then investigate the extinction probability and the mean extinction time. Furthermore, we give a condition for the Feller minimal NMBPIR to be recurrence, and provide an example to illustrate our results.

Total-current population-dependent branching processes: analysis via stochastic approximation

We consider continuous-time two-type population size-dependent Markov Branching Processes. The offspring distribution can depend on the current (alive) and total (dead and alive) populations. We assume finite second-moment conditions and use the stochastic approximation technique for the analysis. In particular, we identify an appropriate ordinary differential equation (ODE) and show that either of the two events occurs with a certain probability: (a) the time-asymptotic proportion of the populations converges to the attractors or saddle points of the ODE, (b) it enters every neighbourhood and exits some neighbourhood of a saddle point infinitely often. We also prove a finite time approximation result for the stochastic trajectory. Further, we analyse a branching process with attack and acquisition, which captures the competition in online viral markets; for this case, the probability of approximation equals one.

The multiple birth properties of multi-type Markov branching processes

The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can reveal the multiple birth characteristics. Then the joint probability distribution of multiple birth of multi-type Markov branching process until any time t is obtained by using the new process. Furthermore, the probability distribution of multiple birth until the extinction of the process is also given.

Critical Markov branching process with infinite variance allowing Poisson immigration with increasing intensity

The article studies a single-type critical Markov branching process with infinite variance of the offspring distribution. The process admits also an immigration component at the time points of a non-homogeneous Poisson process. If additionally the mean number of immigrants is infinite, then proper limit distributions are obtained, under suitable normalization of the sample paths, depending on the rate of change of the intensity of the Poisson process.

On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Abstract The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Fluctuation Theory of Continuous-Time, Skip-Free Downward Markov Chains with Applications to Branching Processes with Immigration

We develop a comprehensive methodology for the fluctuation theory of continuous-time, skip-free Markov chains, extending and improving the recent work of Choi and Patie for discrete-time, skip-free Markov chains. As a significant application, we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classic family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes.

A Countable-Type Branching Process Model for the Tug-of-War Cancer Cell Dynamics.

We consider a time-continuous Markov branching process of proliferating cells with a countable collection of types. Among-type transitions are inspired by the Tug-of-War process introduced by McFarland et al. (Proc Natl Acad Sci 111(42):15138-15143, 2014) as a mathematical model for competition of advantageous driver mutations and deleterious passenger mutations in cancer cells. We introduce a version of the model in which a driver mutation pushes the type of the cell L-units up, while a passenger mutation pulls it 1-unit down. The distribution of time to divisions depends on the type (fitness) of cell, which is an integer. The extinction probability given any initial cell type is strictly less than 1, which allows us to investigate the transition between types (type transition) in an infinitely long cell lineage of cells. The analysis leads to the result that under driver dominance, the type transition process escapes to infinity, while under passenger dominance, it leads to a limit distribution. Implications in cancer cell dynamics and population genetics are discussed.

Many-to-few for non-local branching Markov process

Many-to-few for non-local branching Markov process

Spread of parasites affecting death and division rates in a cell population

We introduce a general class of branching Markov processes for the modelling of a parasite infection in a cell population. Each cell contains a quantity of parasites which evolves as a diffusion with positive jumps. The drift, diffusive function and positive jump rate of this quantity of parasites depend on its current value. The division rate of the cells also depends on the quantity of parasites they contain. At division, a cell gives birth to two daughter cells and shares its parasites between them. Cells may also die, at a rate which may depend on the quantity of parasites they contain. We study the long-time behaviour of the parasite infection.

Convergence of genealogies through spinal decomposition with an application to population genetics

Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when N rightarrow infty . Informally, the moment of order k of the population is obtained by observing the genealogy of k individuals chosen uniformly at random after size-biasing the population at time N by its kth factorial moment. We show that the sampled genealogy can be expressed in terms of a k-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying k-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright–Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

Harmonic Moments and Large Deviations for the Markov Branching Process with Immigration

Harmonic Moments and Large Deviations for the Markov Branching Process with Immigration

Subcritical multitype Markov branching processes with immigration generated by Poisson random measures

We investigate multitype subcritical Markov branching processes with immigration driven by Poisson random measures. Limiting distributions are established for various rates of the Poisson measures when they are asymptotically equivalent to exponential or regularly varying functions. Results analogous to a strong LLN are proved, and limiting normal distributions are obtained when the local intensity of the Poisson measure increases with time. When it decreases, conditional limiting distributions are established. When the intensity converges to a constant, a stationary limiting distribution is obtained. The asymptotic behavior of the first and second moments of the processes is also investigated.

Extended Sibuya Distribution in Subcritical Markov Branching Processes

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t > 0 follows the shifted extended Sibuya distribution, with parameters depending on the time t > 0. The conditional limit probability is the logarithmic series distribution supported by the positive integers.

Wright function in the solution to the Kolmogorov equation of the Markov branching process with geometric reproduction of particles*

Wright function in the solution to the Kolmogorov equation of the Markov branching process with geometric reproduction of particles*

Estimating mutation rates in a Markov branching process using approximate Bayesian computation

Estimating microbial mutation rates is an essential task in evolutionary biology, with wide range applications in related fields such as virology, epidemiology, clinic and public health, and antibiotic research. Significant progress has been made on this research since 1943 when Luria–Delbrück fluctuation analysis was first introduced. However, existing estimators of mutation rates are heavily reliant on model assumptions in fluctuation analysis, and become less applicable to real microbial experiments which deviate from the model assumptions. To overcome this difficulty, we propose to model fluctuation experimental data by a two-type Markov branching process (MBP) and use approximate Bayesian computation (ABC) to estimate the mutation probability parameters. Such an ABC-based mutation rate estimator is based on intensive simulations from the mutation process, thereby taking advantage of modern computing power. Most importantly, its likelihood-free feature allows more complex and realistic setups of the mutation process, especially when the distribution of the number of mutants cannot be easily derived. To further improve computation efficiency, we use a Gaussian process surrogate to substitute the simulator in the ABC algorithm, and call the resulting estimator GPS-ABC. Simulation studies show that, when used to estimate constant mutation rate in MBP, ABC-based estimators generally outperform traditional moment or likelihood-based estimators. When mutations occur in two stages, i.e., in MBP with a piece-wise constant mutation rate function, traditional mutation rate estimators become not applicable, yet GPS-ABC still achieves reasonable estimates. Finally, the proposed GPS-ABC estimator is used to analyze real fluctuation experimental datasets for studying drug resistance.

Sturm–Liouville theory and decay parameter for quadratic markov branching processes

AbstractFor a quadratic Markov branching process (QMBP), we show that the decay parameter is equal to the first eigenvalue of a Sturm–Liouville operator associated with the partial differential equation that the generating function of the transition probability satisfies. The proof is based on the spectral properties of the Sturm–Liouville operator. Both the upper and lower bounds of the decay parameter are given explicitly by means of a version of Hardy’s inequality. Two examples are provided to illustrate our results. The important quantity, the Hardy index, which is closely linked to the decay parameter of the QMBP, is deeply investigated and estimated.

Asymptotic behavior for supercritical branching processes

Let {Z(t);t⩾0} be a Markov branching process (MBP). There exists a well-known sequence {C(t);t⩾0} such that W(t)≔Z(t)/C(t) a.s. converges to a non-degenerate random variable W as t→∞. This paper attempts to study the asymptotic behavior of P(Z(t)=kt) and P(0⩽Z(t)⩽kt) with kte−λt→0 as t→∞ for MBPs, which helps to study large deviations of Z(t+s)/Z(t). Moreover, we obtain the local limit theorem of this process as an additional finding. During the argumentation, the Cramér method is applied to analyze the large deviation of the sum of random variables.

Monte Carlo solution of heat conductivity problems with quadratic nonlinearity on the boundary of domain

In this work, we study the approach connected with branching Markov processes. Branching Markov random processes are created to solve boundary-value problems (BVR) with polynomial nonlinearities. The realization of these processes creates so called “trees”. Unbiased estimators of the solution of the nonlinear problem are constructed on these random trees. We also calculated in parallel way the variance (statistical error) of the constructed unbiased estimators. We offer computational algorithms for solving some nonlinear diffusion problems which frequently appears in engineering problems, particularly in heat conductivity. The case of a quadratic nonlinearity for the boundary BVP is considered in detail. Algorithms, for diffusion BVPs with nonlinear boundary conditions, differ from proposed for linear diffusion BVPs algorithms early and we compare their efficiencies. The described algorithms applied to the computational problem of thermal engineering in the presence of a nonlinear boundary condition. Also, this problem is solved as a boundary value problem of conductive heat transfer.

Yaglom limit for critical nonlocal branching Markov processes

We consider the classical Yaglom limit theorem for a branching Markov process X=(Xt,t≥0), with nonlocal branching mechanism in the setting that the mean semigroup is critical, that is, its leading eigenvalue is zero. In particular, we show that there exists a constant c(f) such that Law(⟨f,Xt⟩ t|⟨1,Xt⟩>0)→ec(f),t→∞, where ec(f) is an exponential random variable with rate c(f) and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, (Probab. Theory Related Fields 173 (2019) 999–1062; Electron. Commun. Probab. 23 (2018) 42), these results do not allow for nonlocal branching which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the kth martingale moments of X (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport for which the nonlocality is essential, complementing recent developments in this domain (Ann. Appl. Probab. 30 (2020) 2573–2612; Ann. Appl. Probab. 30 (2020) 2815–2845; SIAM J. Appl. Math. 81 (2021) 982–1001; Cox et al. (2021); J. Stat. Phys. 176 (2019) 425–455).

Doubly stochastic Yule cascades (Part I): The explosion problem in the time-reversible case

Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations (NSE), we identify a new class of stochastic cascade models, referred to as doubly stochastic Yule cascades. We establish non-explosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov process. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell ageing models introduced by Aldous and Shields (1988) and independently by Athreya (1985).