Theory Of Branching Processes Research Articles

The impact of effective participation in stopping misinformation: an approach based on branching processes

The emergence of research that focuses on understanding the spreading and impact of disinformation is increasing year after year. Most of the time, the purpose of those who start the spreading of intentionally false information that is designed to cause harm is to catalyze its fast transformation into misinformation, which is the false content shared by people who do not realize it is false or misleading. Our interest is in discussing the role of people who decide to adopt an active role in stopping the propagation of information when they realize that it is false. For this, we formulate two simple probabilistic models to compare misinformation spreading in possible scenarios for which there is a passive or an active environment of aware individuals. With aware individuals, we mean those individuals who realize that a piece of given information is false or misleading. In the passive environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it. In the active environment, we assume that if one of an aware individual is exposed to the misinformation then he/she will not spread it, but also he/she will stop the propagation to other individuals from the individual who contacted him/her. We appeal to the theory of branching processes to analyze propagation in both scenarios, and we discuss the role and the impact of effective participation in stopping misinformation. We show that the propagation reduces drastically, provided we assume an active environment. We also obtain theoretical and computational results to measure such a reduction, which in turn depends on the proportion of aware individuals and the number of potential contacts of each individual, which is assumed to be random.

Behaviours of honeybees can reduce the probability of deformed wing virus outbreaks in Varroa destructor-infested colonies

AbstractHoneybees are important plant pollinators. Unfortunately, there is a growing increase in the loss of honeybee colonies, and this is having a serious economic impact on crop farmers. A major cause of these losses is the parasitic mite Varroadestructor, which is a vector of deformed wing virus (DWV). Some bee species have resistant mechanisms, such as grooming and hygienic behaviours, against Varroa mites. A clear understanding of the effects of these control behaviours on the mites and the viruses they transmit can be important in reducing colony losses. Here, a stochastic model is formulated and analysed to consider the extent to which these control behaviours reduce the probability of an outbreak of DWV in honeybee colonies. Vector and bee-to-bee transmission routes are considered. Using branching process theory, it is shown that without any hygienic or grooming behaviour, a large probability of a DWV outbreak is possible. Also, if bees apply grooming or hygienic behaviour, this can reduce the probability of a virus outbreak, especially in the case of vector transmission, where it can be reduced to zero. Hygienic behaviour is the most significant factor in reducing a DWV outbreak. Thus, bee selection for hygienic behaviour may be important to reduce honeybee colony losses caused by DWV.

Loewner Theory for Bernstein Functions I: Evolution Families and Differential Equations

AbstractOne-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous branching processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein’s representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish a sufficient condition for families of Bernstein functions, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner–Kufarev differential equation. Most of these results are then applied in the sequel paper [35] to study continuous-state branching processes.

Branching Models in Practice: Tackling Real-Life Challenges Beyond Theoretical Constructs Using Python Programming

The mathematical modeling of branching processes finds its origin in attempts to model reproduction patterns in species. For systems with replicating and dying components (like cells or particles), branching process theory provides the mathematical tools for understanding their probabilistic progression.. They are used to describe random systems such as chain reactions, survival of family names, pest eradication, population development and gene propagation. This study showcases clear illustrations of the likelihood of extinction, the duration until extinction, and the probability related to the entire offspring. Observations of past elite families often highlight their eventual decrease, leading to myriad speculations. Through Python-based modeling, it was found that if every person, on average, yields slightly over one offspring, there is no absolute guarantee of family extinction. However, if the mean number of offspring per individual  is one or less, the process is guaranteed to become extinct.

A stochastic multi-host model for West Nile virus transmission

When initially introduced into a susceptible population, a disease may die out or result in a major outbreak. We present a Continuous-Time Markov Chain model for enzootic WNV transmission between two avian host species and a single vector, and use multitype branching process theory to determine the probability of disease extinction based upon the type of infected individual initially introducing the disease into the population – an exposed vector, infectious vector, or infectious host of either species. We explore how the likelihood of disease extinction depends on the ability of each host species to transmit WNV, vector biting rates on host species, and the relative abundance of host species, as well as vector abundance. Theoretical predictions are compared to the outcome of stochastic simulations. We find the community composition of hosts and vectors, as well as the means of disease introduction, can greatly affect the probability of disease extinction.

A continuous-time N-interaction random graph model

In this paper a continuous-time evolving random graph model is defined and examined. The main units of the model are complete graphs on N vertices, where is a fixed integer. At each birth event a new vertex and random number of edges are added to the graph. The asymptotic behaviour of the number of vertices and the asymptotic behaviour of the number of m-cliques () are studied. The proofs are based on general results of the theory of branching processes.

BRANCHING PROCESS MODELLING: A TOOL FOR DECIPHERING COMPLEX REAL-WORLD PHENOMENA USING R PROGRAMMING

The reproduction of organisms such as human beings, cells or neutrons can be modeled via branching processes.The theory of branching processes offers suitable mathematical models to depict the chance-based progression of systems, where elements like cells, particles, or general individuals replicate and perish over a certain lifespan. These models are employed to illustrate stochastic systems like chain reactions, lineage continuation of surnames, elimination of pests, population growth, and gene spread. This study clearly showcases the likelihood of extinction, its timing, and the odds of complete offspring through vivid examples. The decline of prominent families from history has often been observed and has sparked numerous speculations. The branching process is a very useful tool in many real life problems (non-deterministic problems) and random phenomenon. Research indicates that when the average number of descendants for each entity exceeds 1 (meaning individuals reproduce at a rate slightly higher than self-replacement), the branching process doesn't necessarily cease. On the other hand, if this average number is 1 or lower, the process will inevitably face extinction.

Tau Pathology induces Whole‐Brain Subcritical Brain Dynamics across the spectrum of Alzheimer’s Disease

AbstractBackgroundCognitive dysfunction in Alzheimer’s Disease (AD) is caused by dysfunctional brain region interactions. Although functional connectivity alterations in AD have been described in several studies, their associations to tau pathology have not been completely understood. Using whole‐brain network modeling we can mechanistically associate the functional impact of tau on brain function in silico.MethodUsing tau PET and fMRI data from the ADNI3 study, we linked regional tau burdens to fMRI activity using a Hopf dynamical model of brain activity using linear regressions. The Hopf model faithfully recreates oscillatory brain activity and adjust the bifurcation of the regional oscillations to either to a noisy (subcritical) or a more regular (supracritical) regime (Deco et al., 2017). The similarity of simulated and empirical time series was assessed by comparing the resulting (dynamical) functional connectivity differences using root mean squared error and Kolmogorov‐Smirnov statistics (as described in Kobeleva et al., 2021). We compared the resulting bifurcation parameters of the Hopf model between healthy controls with no SUVR above 2, healthy controls with at least one region with a SUVR above 2 (considered as at‐risk subjects with a high tau burden) and patients with mild cognitive impairment (MCI).ResultWe achieved a good fit of the tau‐dependent model of brain activity, ensuring a realistic representation of brain activity in the model. Mean bifurcation parameters differed across the three groups (controls with low and high tau burden, and MCI patients). Specifically, we found increasingly more subcritical bifurcation parameters in controls with high tau burden and MCI patients, quantitatively associating a higher tau burden associated with noisier brain oscillations.ConclusionOur results show that biology‐inspired models of AD contain parameters with clinically relevant information. We observed a shift to a more subcritical regime in subjects with high tau and even more so in MCI, showing that tau is associated with nosier brain oscillations, possibly contributing to impaired information transmission and observed cognitive dysfunction. These results constitute a missing link to the branching process theory of cortical network function in AD and contribute to understanding impaired cortical excitability in AD.

New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, and states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the “R0 alternative” of Van den Driessche and Watmough, which states that the local stability condition of the disease-free equilibrium may be expressed as R0<1, where R0 is the famous basic reproduction number, which also plays a major role in the theory of branching processes. The literature suggests that it is impossible to find general laws concerning the endemic points. However, it is quite common that 1. When R0>1, there exists a unique fixed endemic point, and 2. the endemic point is locally stable when R0>1. One would like to establish these properties for a large class of realistic epidemic models (and we do not include here epidemics without casualties). We have introduced recently a “simple” but broad class of “SIR-PH models” with varying populations, with the express purpose of establishing for these processes the two properties above. Since that seemed still hard, we have introduced a further class of “SIR-PH-FA” models, which may be interpreted as approximations for the SIR-PH models, and which include simpler models typically studied in the literature (with constant population, without loss of immunity, etc.). For this class, the first “endemic law” above is “almost established”, as explicit formulas for a unique endemic point are available, independently of the number of infectious compartments, and it only remains to check its belonging to the invariant domain. This may yet turn out to be always verified, but we have not been able to establish that. However, the second property, the sufficiency of R0>1 for the local stability of an endemic point, remains open even for SIR-PH-FA models, despite the numerous particular cases in which it was checked to hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH and SIR-PH-FA, and also for a second, more refined “intermediate approximation” SIR-PH-IA. We illustrate the current status-quo by presenting new results on a generalization of the SAIRS epidemic model.

Hybrid fitness effects modify fixation probabilities of introgressed alleles.

Hybridization is a common occurrence in natural populations, and introgression is a major source of genetic variation. Despite the evolutionary importance of adaptive introgression, classical population genetics theory does not take into account hybrid fitness effects. Specifically, heterosis (i.e. hybrid vigor) and Dobzhansky–Muller incompatibilities influence the fates of introgressed alleles. Here, we explicitly account for polygenic, unlinked hybrid fitness effects when tracking a rare introgressed marker allele. These hybrid fitness effects quickly decay over time due to repeated backcrossing, enabling a separation-of-timescales approach. Using diffusion and branching process theory in combination with computer simulations, we formalize the intuition behind how hybrid fitness effects affect introgressed alleles. We find that hybrid fitness effects can significantly hinder or boost the fixation probability of introgressed alleles, depending on the relative strength of heterosis and Dobzhansky–Muller incompatibilities effects. We show that the inclusion of a correction factor (α, representing the compounded effects of hybrid fitness effects over time) into classic population genetics theory yields accurate fixation probabilities. Despite having a strong impact on the probability of fixation, hybrid fitness effects only subtly change the distribution of fitness effects of introgressed alleles that reach fixation. Although strong Dobzhansky–Muller incompatibility effects may expedite the loss of introgressed alleles, fixation times are largely unchanged by hybrid fitness effects.

A stochastic epidemic model for the dynamics and control of maize streak disease

ABSTRACT Maize is an essential staple food crop and is vulnerable to diseases. Maize streak disease is one of the serious illnesses that affect maize, especially in sub-Saharan Africa where the disease is endemic. We developed a stochastic epidemic model with three control measure types (mechanical, chemical and preventative) that mitigate the disease. A dynamical system analysis of the deterministic version of the model is provided. The probability of maize streak disease extinction or persistence is determined using the theory of a multi-type branching process. Using these results, the degrees to which these control measures are effective in reducing maize streak disease are considered. It is shown that chemical and mechanical control measures, preferably together, are better than preventive controls in reducing disease prevalence. However, considering the possible negative effects of a chemical control, it is shown that sufficient mechanical control combined with a small degree of each of preventative and chemical control could be the most viable strategy to limit maize streak disease.

Parameter estimation in branching processes with almost sure extinction

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is z≥1. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation n and let n→∞; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property (C-consistency). This leads us to define the concept of Q-consistency, and we prove that our estimators are Q-consistent and asymptotically normal. To investigate the circumstances in which a C-consistent estimator is preferable to a Q-consistent estimator, we then provide two C-consistent estimators for subcritical Galton–Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.

Multitype branching processes in random environment

A survey of results in the theory of multitype branching processes evolving in a random environment is presented. Bibliography: 104 titles.

Disease clearance of tuberculosis infection: An in-host continuous-time Markov chain model

The clearance of tuberculosis infection shows an elimination of infectious Mycobacterium tuberculosis (Mtb) pathogens and infected macrophage cells. The evidence shows the existence of individuals, who are still tested negative in tuberculin skin test after living with people with active tuberculosis for up to six months. Since the Mtb pathogen is spread from person to person through airborne particles, we build a continuous-time Markov chain (CTMC) model to describe the initial infection with small amount of inhaled bacteria. The CTMC model successfully simulates sample paths presenting disease clearance. We apply the theory of multitype branching processes to analytically approximate the probability of disease clearance. We also estimate the disease clearance time, which is as less than a month for R0∈[1,1.5]. Our results demonstrate that the host immune factors affect both the probability and the time of the disease clearance. These relationships are linked by the basic reproduction number R0. Our findings provide new mechanisms for disease prevention and therapy developments.

On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results.

A Gamma Ornstein\u2013Uhlenbeck model driven by a Hawkes process

We propose an extension of the Gamma -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the Gamma -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the Gamma -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the Gamma -OU Barndorff-Nielsen and Shephard model.

Transition to multitype mixing in d-dimensional spreading dynamics.

The spreading dynamics of infectious diseases is determined by the interplay between geography and population mixing. There is homogeneous mixing at the local level and human mobility between distant populations. Here I model spatial location as a type and the population mixing by intra- and intertype mixing patterns. Using the theory of multitype branching process, I calculate the expected number of new infections as a function of time. In one dimension the analysis is reduced to the eigenvalue problem of a tridiagonal Teoplitz matrix. In d dimensions I take advantage of the graph cartesian product to construct the eigenvalues and eigenvectors from the eigenvalue problem in 1 one dimension. Using numerical simulations I uncover a transition from linear to multitype mixing exponential growth with increasing the population size. Given that most countries are characterized by a network of cities with more than 100 000 habitants, I conclude that the multitype mixing approximation should be the prevailing scenario.

Extinction Moment for a Branching Tree Evolution with Birth Rate and Death Rate Both Depending on Age

In this paper, a branching tree evolution is established, in which the birth rate and the death rate are both dependent on node’s age. The extinction probability and the t-pre-extinction (extinct before time t ) probability are studied, by which the distribution of the extinction moment can be given. The analytical formula and the approximation algorithm for the distribution of extinction moment are given; furthermore, the analytical formula and the approximation algorithm of extinction probability are given, and a necessary and sufficient condition of extinction with probability 1 is given. It is the first time to study the distribution of extinction time for the branching process with birth rate and the death rate both depending on node’s age, and the results will do great help in the theory of branching process. It is expected to be applied in the fields of biology, genetics, medicine, epidemiology, demography, nuclear physics, actuarial mathematics, algorithm, and data structures, etc.

Demographic Dynamics in Multitype Populations with Migrations

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

ОБ АСИМПТОТИЧЕСКИХ СООТНОШЕНИЯХ ДЛЯ КРИТИЧЕСКОГО ПРОЦЕССА ГАЛЬТОНА – ВАТСОНА

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.