Classical and Modern Branching Processes

Controlled Branching Processes

Branching processes are stochastic models describing the evolution of populations in which individuals reproduce and die independently over time. In the classical setting, an individual’s reproductive capacity is fixed throughout its lifetime. However, in real-world situations, fertility typically rises during a juvenile phase, peaks at maturity, and subsequently declines. In order to capture this feature, we introduce a branching random walk with ageing, as an extension of the classical branching random walk, by assigning each individual an age-dependent reproductive rate. Our model differs from classical age-dependent processes such as the Bellman–Harris model, where the remaining lifespan depends on age, while the rate of reproduction is fixed within that lifetime. As in the classical case, branching random walks with ageing are parametrised by λ>0, which tunes the reproductive speed and may be seen as a characteristic of the population. The thresholds of λ separating extinction and survival are the global and local critical parameters. We characterise the value of the local critical parameter and provide a lower bound for the global critical parameter. We identify a class of ageing branching random walks for which this lower bound coincides with the global critical parameter. We study how local modifications to the reproduction and ageing rates may change the critical parameters. This is of practical interest: in species preservation, one may want to lower the critical parameters, so that λ exceeds them, and there is a positive probability of survival. On the other hand, in epidemic control, the goal is to increase the critical parameters, since if λ is below them, then the epidemic is eventually going to disappear. We compute the expected number of individuals alive in a branching process with ageing and show that, contrary to the behaviour of classical branching processes, it may exhibit an initial growth even when the population is ultimately destined for extinction.

Let μ ( t ) be the number of particles at time t of a continuous-time critical branching process. It is known that the probability of non-extinction of the process at time t Q ( t ) = P { μ ( t ) > 0 | μ (0) = 1} → 0 as t → ∞. Hence it follows that Q m 0 = P { μ ( t ) > 0 | μ (0) = m } ∼ mQ ( t ) → 0 for any m = 2,3, . . . Let for any integer m > r ≥ 1 In this paper, we prove that Q mr ( t ) ∼ ( m − r ) Q ( t ) as t → ∞ for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.

Recent experimental work has shown that activity in living neural networks can propagate as a critical branching process that revisits many metastable states. Neural network theory suggests that attracting states could store information, but little is known about how a branching process could form such states. Here we use a branching process to model actual data and to explore metastable states in the network. When we tune the branching parameter to the critical point, we find that metastable states are most numerous and that network dynamics are not attracting, but neutral.

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit.Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

The capacity of pluripotent embryonic stem cells to differentiate into any cell type in the body makes them invaluable in the field of regenerative medicine. However, because of the complexity of both the core pluripotency network and the process of cell fate computation it is not yet possible to control the fate of stem cells. We present a theoretical model of stem cell fate computation that is based on Halley and Winkler’s Branching Process Theory (BPT) and on Greaves et al.’s agent-based computer simulation derived from that theoretical model. BPT abstracts the complex production and action of a Transcription Factor (TF) into a single critical branching process that may dissipate, maintain, or become supercritical. Here we take the single TF model and extend it to multiple interacting TFs, and build an agent-based simulation of multiple TFs to investigate the dynamics of such coupled systems. We have developed the simulation and the theoretical model together, in an iterative manner, with the aim of obtaining a deeper understanding of stem cell fate computation, in order to influence experimental efforts, which may in turn influence the outcome of cellular differentiation. The model used is an example of self-organization and could be more widely applicable to the modelling of other complex systems. The simulation based on this model, though currently limited in scope in terms of the biology it represents, supports the utility of the Halley and Winkler branching process model in describing the behaviour of stem cell gene regulatory networks. Our simulation demonstrates three key features: (i) the existence of a critical value of the branching process parameter, dependent on the details of the cistrome in question; (ii) the ability of an active cistrome to “ignite” an otherwise fully dissipated cistrome, and drive it to criticality; (iii) how coupling cistromes together can reduce their critical branching parameter values needed to drive them to criticality.

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

We consider a sequence of almost critical branching processes with immigration supposing that the immigration process is weakly stationary. The rate of growth and asymptotic properties of fluctuations of such branching processes are investigated.

Asymptotic properties and absolute continuity of laws stable by random weighted mean

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi (t), Gi (0+) = 0. At the end of its life, cell i is replaced by j 1 new cells of type 1, j 2 new cells of type 2, ···, jm new cells of type m with probability , and we define the generating functions for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timekn withkn/n→β. We attach to each vertexv of this tree a random variableX(v) and define\(S(v) = \Sigma _{w \varepsilon \pi (0,v)} X(w)\), where π(0,v) is the unique path in the family tree from its root tov. FinallyMnis the maximal displacement of the branching random walkS(·), that isMn=max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn−1/2Mn converges in distribution. In particular {n−1/2Mn}n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Part 1 Random Walk of a random Field: Brownian Motion of a Poisson Field Extreme Value Problems Changing the Initial Process and the Motion. Part 2 Branching Random Walk: Branching Random Walk Starting with One Particle Branching Random Walks of a Random Field Branching Wiener Process Starting with One Particle Critical Branching Random Walk Starting with One Particle Critical Branching Random Walks of a Random Field Multitype Branching Random Walk. Part 3 Strassen Type Theorems: Infinitely Many Independent Particles Branching Random Walk.