Branching Process Research Articles

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.

Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

The basic reproduction number R0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R0 is not a function of the original ODE model, unless we also include in it a certain (F,V) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (F,V) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (F,V) gradient decomposition (and hence of the resulting R0). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the (F,V) decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (F,V) decompositions are available as well.

A stochastic target problem for branching diffusion processes

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain-based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. Next, we show that the value function can be simplified into a novel function with the use of a finite-dimensional argument through a concept known as the branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

Incorporating testing volume into estimation of effective reproduction number dynamics.

Branching process inspired models are widely used to estimate the effective reproduction number-a useful summary statistic describing an infectious disease outbreak-using counts of new cases. Case data is a real-time indicator of changes in the reproduction number, but is challenging to work with because cases fluctuate due to factors unrelated to the number of new infections. We develop a new model that incorporates the number of diagnostic tests as a surveillance model covariate. Using simulated data and data from the SARS-CoV-2 pandemic in California, we demonstrate that incorporating tests leads to improved performance over the state of the art.

The impacts of spatial–temporal heterogeneity of human-to-human contacts on the extinction probability of infectious disease from branching process model

In this study, we focus on the impacts of spatial–temporal heterogeneity of human-to-human contacts on the spread of infectious diseases and develop a multi-type branching process model by introducing random human-to-human contact mode into a structured population. We provide the general formulas of the generation size, extinction probability, and basic reproduction number of the proposed branching process model. The result shows that the natural temporal heterogeneity (i.e. random contacts over time) can lead to a higher extinction probability while remains the same basic reproduction number and generation size. This is also numerically verified by choosing the real contact distributions from different circumstances of four countries. In addition, we observe a non-monotonic pattern of the differences, against the transmission probability and the mean contact rate, between the extinction probabilities under the constant and random contact patterns. Given the spatial heterogeneity, we show that it can contribute to the increase of basic reproduction number, but also increase the extinction probability of the infectious disease. This study adds novel insights to the course of the impact of heterogeneity on the transmission dynamics and also provides additional evidence for the limited role of reproduction numbers.

Exact convergence rate in central limit theorem for a supercritical branching process with immigration in a random environment

Let {Z n } be a supercritical branching process with immigration in an independent, identically distributed(i.i.d.) environment ξ. The Berry-Esseen bound for log⁡Z n has been established by Wang et al. (2021). To refine that, under the less restrictive moment conditions, we calculate the exact convergence rate in the central limit theorem for log⁡Z n under the annealed law.

Speed of extinction for continuous-state branching processes in a weakly subcritical Lévy environment

AbstractWe continue with the systematic study of the speed of extinction of continuous-state branching processes in Lévy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu (2018) and Palau et al. (2016), where it is assumed that the branching mechanism is stable, and complement the recent articles of Bansaye et al. (2021) and Cardona-Tobón and Pardo (2021), where the critical and the strongly and intermediate subcritical cases were treated, respectively. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes, and the asymptotic behaviour of exponential functionals of Lévy processes. Our approach is inspired by the last two previously cited papers, and by Afanasyev et al. (2012), where the analogue was obtained.

Exact calculation of end-of-outbreak probabilities using contact tracing data.

A key challenge for public health policymakers is determining when an infectious disease outbreak has finished. Following a period without cases, an estimate of the probability that no further cases will occur in future (the end-of-outbreak probability) can be used to inform whether or not to declare an outbreak over. An existing quantitative approach (the Nishiura method), based on a branching process transmission model, allows the end-of-outbreak probability to be approximated from disease incidence time series, the offspring distribution and the serial interval distribution. Here, we show how the end-of-outbreak probability under the same transmission model can be calculated exactly if data describing who-infected-whom (the transmission tree) are also available (e.g. from contact tracing studies). In that scenario, our novel approach (the traced transmission method) is straightforward to use. We demonstrate this by applying the method to data from previous outbreaks of Ebola virus disease and Nipah virus infection. For both outbreaks, the traced transmission method would have determined that the outbreak was over earlier than the Nishiura method. This highlights that collection of contact tracing data and application of the traced transmission method may allow stringent control interventions to be relaxed quickly at the end of an outbreak, with only a limited risk of outbreak resurgence.

The coalescent structure of uniform and Poisson samples from multitype branching processes

The coalescent structure of uniform and Poisson samples from multitype branching processes

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 N ≥ 3 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 ( 2 ≤ m ≤ N ) are studied. The proofs are based on general results of the theory of branching processes.

A mathematical model of Marburg virus disease outbreaks and the potential role of vaccination in control

BackgroundMarburg virus disease is an acute haemorrhagic fever caused by Marburg virus. Marburg virus is zoonotic, maintained in nature in Egyptian fruit bats, with occasional spillover infections into humans and nonhuman primates. Although rare, sporadic cases and outbreaks occur in Africa, usually associated with exposure to bats in mines or caves, and sometimes with secondary human-to-human transmission. Outbreaks outside of Africa have also occurred due to importation of infected monkeys. Although all previous Marburg virus disease outbreaks have been brought under control without vaccination, there is nevertheless the potential for large outbreaks when implementation of public health measures is not possible or breaks down. Vaccines could thus be an important additional tool, and development of several candidate vaccines is under way.MethodsWe developed a branching process model of Marburg virus transmission and investigated the potential effects of several prophylactic and reactive vaccination strategies in settings driven primarily by multiple spillover events as well as human-to-human transmission. Linelist data from the 15 outbreaks up until 2022, as well as an Approximate Bayesian Computational framework, were used to inform the model parameters.ResultsOur results show a low basic reproduction number which varied across outbreaks, from 0.5 [95% CI 0.05–1.8] to 1.2 [95% CI 1.0–1.9] but a high case fatality ratio. Of six vaccination strategies explored, the two prophylactic strategies (mass and targeted vaccination of high-risk groups), as well as a combination of ring and targeted vaccination, were generally most effective, with a probability of potential outbreaks being terminated within 1 year of 0.90 (95% CI 0.90–0.91), 0.89 (95% CI 0.88–0.90), and 0.88 (95% CI 0.87–0.89) compared with 0.68 (0.67–0.69) for no vaccination, especially if the outbreak is driven by zoonotic spillovers and the vaccination campaign initiated as soon as possible after onset of the first case.ConclusionsOur study shows that various vaccination strategies can be effective in helping to control outbreaks of MVD, with the best approach varying with the particular epidemiologic circumstances of each outbreak.

A generating-function approach to modelling complex contagion on clustered networks with multi-type branching processes

Abstract Understanding cascading processes on complex network topologies is paramount for modelling how diseases, information, fake news and other media spread. In this article, we extend the multi-type branching process method developed in Keating et al., (2022), which relies on networks having homogenous node properties, to a more general class of clustered networks. Using a model of socially inspired complex contagion we obtain results, not just for the average behaviour of the cascades but for full distributions of the cascade properties. We introduce a new method for the inversion of probability generating functions to recover their underlying probability distributions; this derivation naturally extends to higher dimensions. This inversion technique is used along with the multi-type branching process to obtain univariate and bivariate distributions of cascade properties. Finally, using clique-cover methods, we apply the methodology to synthetic and real-world networks and compare the theoretical distribution of cascade sizes with the results of extensive numerical simulations.

Role of O2 on nitrous oxide fuel blend ethylene auto-ignition sensitivity

An experimental and kinetic modeling study of the influence of O2 addition on the ignition behaviors of Ar diluted nitrous oxide/ethylene blend was performed. Ignition delay time (IDT) measurements were taken by detecting pressure evolution profiles in a heated rapid compression machine (RCM) at temperatures and pressures ranging from 910–1500 K to 35–45 bar. Results show that the addition of O2 significantly promotes the ignition of N2O/C2H4/Ar blend. With the O2 blending ratio increase from 10 % to 100 %, the IDT first decreases and then increases. Subsequently, the IDT data were employed to validate several kinetic models in literature. A reaction model developed in our previous work [Combustion and Flam, 2020, 221: 64-73] can reproduce the measured data and predict the promotion effect of O2 on N2O/C2H4/Ar blend well. Reaction pathway and rate of production (ROP) analyses show that high decomposition temperature of N2O inhibits the ignition of N2O/C2H4/Ar blend, whose ignition is mainly driven by reactions of N2O (+M) = N2 + O (+M) and N2O + H = N2 + OH. The dominant pathway is chain branching process: C2H4 + H/OH→C2H3 (+M)→C2H2 + O→HCCO + H2→CH2CO + H→CH3CO (+M)→CO + CH3. However, the presence of O2 in mixture induces reaction flux changes and decreases the global activation energy for chemical reaction. C2H3 radical is the main intermediate product during the ignition initial stage of C2H4 blends, whose ROP can reflect ignition process of C2H4 blends. For N2O/C2H4/O2/Ar and C2H4/O2/Ar ignition, C2H3 radical pool can be rapidly established via H-abstraction reaction of C2H4. However, the reaction of N2O + H = N2 + OH enhances the ROP of OH radical in N2O/C2H4/O2/Ar blends, resulting in C2H3 radical pool rapid establishment and a faster ignition.

ABSORBING MARKOV CHAINS TO CHARACTERIZE AND PREDICT METASTASIS PATHWAYS IN CHILDHOOD CANCER

Cancer and its metastasis in children have a high degree of lethality and side effects, so its characteristics need to be studied independently of adult cancer. The aim of this work is to model the metastasis pathways of the main childhood cancers worldwide by Absorbing Markov Chains, an important mathematical tool used for different applications in science. Statistical information was collected to detect the main affected organs (primary sites) and those where cancer cells generally spread and metastasize (secondary sites). Taking into account that it is a branching process, a directed graph was developed, and the associated transition matrices for the first and second metastases were constructed. Organs whose cancers generally remain encapsulated and do not spread their cancer cells are considered absorbing states in terms of Markov processes. For the selected organs, the probability of ending up in each of the absorption states according to the primary site was calculated, as well as the number of possible previous metastases until reaching one of these states. Although the lung in childhood cancer is not a characteristic primary site, it is one of the main sites of metastasis. Therefore, this work dedicates a section to including this organ as a site of possible metastasis.

Frequent asymmetric migrations suppress natural selection in spatially structured populations.

Natural microbial populations often have complex spatial structures. This can impact their evolution, in particular the ability of mutants to take over. While mutant fixation probabilities are known to be unaffected by sufficiently symmetric structures, evolutionary graph theory has shown that some graphs can amplify or suppress natural selection, in a way that depends on microscopic update rules. We propose a model of spatially structured populations on graphs directly inspired by batch culture experiments, alternating within-deme growth on nodes and migration-dilution steps, and yielding successive bottlenecks. This setting bridges models from evolutionary graph theory with Wright-Fisher models. Using a branching process approach, we show that spatial structure with frequent migrations can only yield suppression of natural selection. More precisely, in this regime, circulation graphs, where the total incoming migration flow equals the total outgoing one in each deme, do not impact fixation probability, while all other graphs strictly suppress selection. Suppression becomes stronger as the asymmetry between incoming and outgoing migrations grows. Amplification of natural selection can nevertheless exist in a restricted regime of rare migrations and very small fitness advantages, where we recover the predictions of evolutionary graph theory for the star graph.

Formulation of Radiality Constraints for Optimal Microgrid Formation

Forming microgrids (MG) in distribution systems (DSs) is an operational measure proposed to enhance power system flexibility. The computational performance of the applied optimization techniques varies greatly according to the constraints ensuring network radiality. Formulations of radiality constraints differ from each other by model complexity and the search space of the relaxed linear programming (LP) problem. In this paper, widely used formulations of radiality constraints are first revisited and flexibly enhanced to address the complex operating environment of modern DSs. To incorporate various radiality constraints and provide an optimization basis for comparison, a flexible MG formation problem is proposed with the ability to automatically allocate distributed generators (DGs) into multiple MGs. Combined formulations with parent-child (PC) constraints are then developed to shrink the LP feasible region. Moreover, to accelerate the branching process by applying the branch-and-cut (B&C) algorithm, an efficient formulation of radiality constraints is proposed by adding extra constraints that exclude fundamental loops into the single-commodity flow formulation. To further accelerate the branching process, a dynamic constraint generation algorithm is developed without substantially increasing the complexity of the LP problem. DSs of various complexities and sizes are employed to verify the computational efficiency of the proposed formulations. The paper concludes by suggesting the most efficient formulations of radiality constraints according to the scale of the targeted DS.

Population Size of a Critical Branching Process Evolving in an Unfavorable Environment

Population Size of a Critical Branching Process Evolving in an Unfavorable Environment

Asymptotics of the distribution and harmonic moments for a supercritical branching process in a random environment

Asymptotics of the distribution and harmonic moments for a supercritical branching process in a random environment

Small Molecules Temporarily Induce Neuronal Features in Adult Canine Dermal Fibroblasts.

Several methods have been developed to generate neurons from other cell types for performing regeneration therapy and in vitro studies of central nerve disease. Small molecules (SMs) can efficiently induce neuronal features in human and rodent fibroblasts without transgenes. Although canines have been used as a spontaneous disease model of human central nerve, efficient neuronal reprogramming method of canine cells have not been well established. We aimed to induce neuronal features in adult canine dermal fibroblasts (ACDFs) by SMs and assess the permanency of these changes. ACDFs treated with eight SMs developed a round-shaped cell body with branching processes and expressed neuronal proteins, including βIII-tubulin, microtubule-associated protein 2 (MAP2), and neurofilament-medium. Transcriptome profiling revealed the upregulation of neuron-related genes, such as SNAP25 and GRIA4, and downregulation of fibroblast-related genes, such as COL12A1 and CCN5. Calcium fluorescent imaging demonstrated an increase in intracellular Ca2+ concentration upon stimulation with glutamate and KCl. Although neuronal features were induced similarly in basement membrane extract droplet culture, they diminished after culturing without SMs or in vivo transplantation into an injured spinal cord. In conclusion, SMs temporarily induce neuronal features in ACDFs. However, the analysis of bottlenecks in the neuronal induction is crucial for optimizing the process.

The Erlang weighted tree, a new branching process

AbstractIn this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if .