Branching Process Model Research Articles

Assessing the transmission dynamics of measles in Japan, 2016

ObjectivesDespite the verification of measles elimination, Japan experienced multiple generations of measles transmission following importation events in 2016. The purpose of the present study was to analyze the transmission dynamics of measles in Japan, 2016, estimating the transmission potential in the partially vaccinated population. MethodsAll diagnosed measles cases were notified to the government, and the present study analyzed two pieces of datasets independently, i.e., the transmission tree of the largest outbreak in Osaka (n=49) and the final size distribution of all importation events (n=23 events). Branching process model was employed to estimate the effective reproduction number Rv, and the analysis of transmission tree in Osaka enabled us to account for the timing of introducing contact tracing and case isolation. ResultsEmploying a negative binomial distribution for the offspring distribution, the model with time-dependent decline in Rv due to interventions appeared to best fit to the transmission tree data with Rv of 9.20 (95% CI (confidence interval): 2.08, 150.68) and the dispersion parameter 0.32 (95% CI: 0.07, 1.17) before interventions were introduced. The relative transmissibility in the presence of interventions from week 34 was estimated at 0.005. Analyzing the final size distribution, models for subcritical and supercritical processes fitted equally well to the observed data, and the estimated reproduction number from both models did not exclude the possibility that Rv>1. ConclusionsOur results likely reflect the highly contagious nature of measles, indicating that Japan is at risk of observing multiple generations of measles transmission given imported cases. Considering that importation events may continue in the future, supplementary vaccination of adults needs to be considered.

Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics.

We consider the cell population dynamics with n different phenotypes. Both the Markovian branching process model (stochastic model) and the ordinary differential equation (ODE) system model (deterministic model) are presented, and exploited to investigate the dynamics of the phenotypic proportions. We will prove that in both models, these proportions will tend to constants regardless of initial population states (“phenotypic equilibrium”) under weak conditions, which explains the experimental phenomenon in Gupta et al.’s paper. We also prove that Gupta et al.’s explanation is the ODE model under a special assumption. As an application, we will give sufficient and necessary conditions under which the proportion of one phenotype tends to 0 (die out) or 1 (dominate). We also extend our results to non-Markovian cases.

Branching processes in continuous time as models of mutations: Computational approaches and algorithms

The appearance of mutations in cancer development plays a crucial role in the disease control and its medical treatment. Motivated by the practical significance, it is of interest to model the event of occurrence of a mutant cell that will possibly lead to a path of indefinite survival. A two-type branching process model in continuous time is proposed for describing the relationship between the waiting time till the first escaping extinction mutant cell is born and the lifespan distribution of cells, which due to the applied treatment have small reproductive ratio. A numerical method and related algorithm for solving the integral equations are developed, in order to estimate the distribution of the waiting time to the escaping extinction mutant cell is born. Numerical studies demonstrate that the proposed approximation algorithm reveals the substantial difference of the results in discrete-time setting. In addition, to study the time needed for the mutant cell population to reach high levels a simulation algorithm for continuous two-type decomposable branching process is proposed. Two different computational approaches together with the theoretical studies might be applied to different kinds of cancer and their proper treatment.

The critical domain size of stochastic population models

Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population’s ability to persist. Here we explore ways of including this variability. We study populations with distinct dispersal and sedentary stages, which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework. Individual-based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up to the population level using the Central Limit Theorem, and the other a variation on both Galton–Watson branching processes and branching processes in random environments. These branching process models closely approximate the IBM and yield insight into the factors determining the critical domain size for a given population subject to stochasticity.

Dendritic growth model of multilevel marketing

Biologically inspired dendritic network growth is utilized to model the evolving connections of a multilevel marketing (MLM) enterprise. Starting from agents at random spatial locations, a network is formed by minimizing a distance cost function controlled by a parameter, termed the balancing factor bf, that weighs the wiring and the path length costs of connection. The paradigm is compared to an actual MLM membership data and is shown to be successful in statistically capturing the membership distribution, better than the previously reported agent based preferential attachment or analytic branching process models. Moreover, it recovers the known empirical statistics of previously studied MLM, specifically: (i) a membership distribution characterized by the existence of peak levels indicating limited growth, and (ii) an income distribution obeying the 80−20 Pareto principle. Extensive types of income distributions from uniform to Pareto to a “winner-take-all” kind are also modeled by varying bf. Finally, the robustness of our dendritic growth paradigm to random agent removals is explored and its implications to MLM income distributions are discussed.

Reply trees in Twitter: data analysis and branching process models

Structure of networks constructed from mentioning relationships between posts in online media may be valuable for understanding how information and opinions spread in these media. We crawled Twitter to collect tweets and replies to construct a large number of so-called reply trees, each of which was rooted at a tweet and joined by replies. Consistent with the previous literature, we found that the empirical trees were characterized by some long path-like reply trees, large star-like trees, and long irregular trees, although their frequencies were not high. We tested several branching process models to explain the empirical frequency of these types of reply trees as well as more basic quantities such as the distributions of the size and depth of the reply tree. Based on our modeling results, we suggest that the in-degree of the tweet that initiates a reply tree (i.e., the number of times that the tweet is directly mentioned by other reply posts) may play an important role in forming the global shape of the reply tree.

Assessing local population vulnerability with branching process models: an application to wind energy development

Quantifying the impact of anthropogenic development on local populations is important for conservation biology and wildlife management. However, these local populations are often subject to demographic stochasticity because of their small population size. Traditional modeling efforts such as population projection matrices do not consider this source of variation whereas individual‐based models, which include demographic stochasticity, are computationally intense and lack analytical tractability. One compromise between approaches is branching process models because they accommodate demographic stochasticity and are easily calculated. These models are known within some sub‐fields of probability and mathematical ecology but are not often applied in conservation biology and applied ecology. We applied branching process models to quantitatively compare and prioritize species locally vulnerable to the development of wind energy facilities. Specifically, we examined species vulnerability using branching process models for four representative species: A cave bat (a long‐lived, low fecundity species), a tree bat (short‐lived, moderate fecundity species), a grassland songbird (a short‐lived, high fecundity species), and an eagle (a long‐lived, slow maturation species). Wind turbine‐induced mortality has been observed for all of these species types, raising conservation concerns. We simulated different mortality rates from wind farms while calculating local extinction probabilities. The longer‐lived species types (e.g., cave bats and eagles) had much more pronounced transitions from low extinction risk to high extinction risk than short‐lived species types (e.g., tree bats and grassland songbirds). High‐offspring‐producing species types had a much greater variability in baseline risk of extinction than the lower‐offspring‐producing species types. Long‐lived species types may appear stable until a critical level of incidental mortality occurs. After this threshold, the risk of extirpation for a local population may rapidly increase with only minimal increases in wind mortality. Conservation biologists and wildlife managers may need to consider this mortality pattern when issuing take permits and developing monitoring protocols for wind facilities. We also describe how our branching process models may be generalized across a wider range of species for a larger assessment project and then describe how our methods may be applied to other stressors in addition to wind.

TESS: an R package for efficiently simulating phylogenetic trees and performing Bayesian inference of lineage diversification rates.

Many fundamental questions in evolutionary biology entail estimating rates of lineage diversification (speciation-extinction) that are modeled using birth-death branching processes. We leverage recent advances in branching-process theory to develop a flexible Bayesian framework for specifying diversification models-where rates are constant, vary continuously, or change episodically through time-and implement numerical methods to estimate parameters of these models from molecular phylogenies, even when species sampling is incomplete. We enable both statistical inference and efficient simulation under these models. We also provide robust methods for comparing the relative and absolute fit of competing branching-process models to a given tree, thereby providing rigorous tests of biological hypotheses regarding patterns and processes of lineage diversification. The source code for TESS is freely available at http://cran.r-project.org/web/packages/TESS/ CONTACT: Sebastian.Hoehna@gmail.com.

Probability and time to fixation of an evolving sequence

The propagation of a sequence in a population is traced using a branching process model of Poisson distributions. The sequence, initially inserted in a location of the genome of a single individual, is under selective pressure and can undergo detrimental, beneficial or even neutral mutations that allow for adaptive possibilities in the future. The exact solution for the ultimate probability of fixation (u) of the sequence in this model is a Lambert W function of mutation rates and selective values:u=1+W(−γ⋅e−(γ+β))γ, where γ=1+s−μd−μb, β=2αμb, s is the intrinsic coefficient of selection of the sequence (selective advantage is set by s>0), μd and μb are the deleterious and beneficial rates of mutations that occur in the sequence and α is the effect of beneficial mutations. Predictions are adaptable to a wide range of situations, e.g., fixation of favourable mutations, local mutators or gene duplicates.The equation for the probability of fixation can be recursively nested, which leads to a non-obvious conclusion: the ultimate probability of fixation of the favourable combination of a number of rare events occurring in the sequence is not the product of the individual probabilities of the events but tends to be approximately proportional to the product of their square roots, which results in a remarkably high chance of fixation. This conclusion has implications in several topics of evolutionary genetics. The analysis also gives the distribution of the times of occurrence of the beneficial mutations that will lead to sequence fixation.

Modelling circulating tumour cells for personalised survival prediction in metastatic breast cancer.

Ductal carcinoma is one of the most common cancers among women, and the main cause of death is the formation of metastases. The development of metastases is caused by cancer cells that migrate from the primary tumour site (the mammary duct) through the blood vessels and extravasating they initiate metastasis. Here, we propose a multi-compartment model which mimics the dynamics of tumoural cells in the mammary duct, in the circulatory system and in the bone. Through a branching process model, we describe the relation between the survival times and the four markers mainly involved in metastatic breast cancer (EPCAM, CD47, CD44 and MET). In particular, the model takes into account the gene expression profile of circulating tumour cells to predict personalised survival probability. We also include the administration of drugs as bisphosphonates, which reduce the formation of circulating tumour cells and their survival in the blood vessels, in order to analyse the dynamic changes induced by the therapy.We analyse the effects of circulating tumour cells on the progression of the disease providing a quantitative measure of the cell driver mutations needed for invading the bone tissue. Our model allows to design intervention scenarios that alter the patient-specific survival probability by modifying the populations of circulating tumour cells and it could be extended to other cancer metastasis dynamics.

Estimation in second order branching processes with application to swine flu data

This paper discusses inferential issues related to estimation of offspring mean and variance in a second order branching process, when both the offspring distributions are assumed to have identical mean and variance. Estimating equation approach is used to find the estimator of the offspring mean and the fact that a second order branching process model can be modeled as an autoregressive process is utilized to obtain the estimator of the offspring variance. Both the estimators are shown to be consistent and asymptotically normal. The second order branching process model is applied to H1N1 data for Pune, India, and Mexico and is found to be a suitable model. The estimates obtained from this model are used to compute the proportion of vaccination required for elimination of the disease.

The effective founder effect in a spatially expanding population.

The gradual loss of diversity and the establishment of clines in allele frequencies associated with range expansions are patterns observed in many species, including humans. These patterns can result from a series of founder events occurring as populations colonize previously unoccupied areas. We develop a model of an expanding population and, using a branching process approximation, show that spatial gradients reflect different amounts of genetic drift experienced by different subpopulations. We then use this model to measure the net average strength of the founder effect, and we demonstrate that the predictions from the branching process model fit simulation results well. We further show that estimates of the effective founder size are robust to potential confounding factors such as migration between subpopulations. We apply our method to data from Arabidopsis thaliana. We find that the average founder effect is approximately three times larger in the Americas than in Europe, possibly indicating that a more recent, rapid expansion occurred.

Predicting localised measles outbreak potential in Australia

IntroductionAustralia has achieved measles elimination as announced in March 2014 by the WHO Western Pacific Regional Committee, based on several lines of evidence. However, despite strong national evidence for elimination, there remains substantial regional variation in vaccine coverage, has resulted in recent outbreaks and potential for increased frequency in the future. MethodsIn this study, we apply a multiple cohort model of measles immunity, stratified by age and local geographic area to predict trends in the measles reproduction number R. In addition, we use branching process models of outbreak risks to predict state-level probabilities of the occurrence of measles outbreaks over the next 20 years. ResultsOur results suggest increasing risks of large measles outbreaks over this period, in particular in the states of Queensland and New South Wales. In addition, there is wide variation in predicted R values by smaller geographic areas, although uncertainty in age-specific immunity limits the precision of our results. DiscussionOur predictions align with observed outbreaks in Australian states and suggest our approach to determining future outbreak risks could be applied more widely in elimination or near-elimination settings.

Ebola cases and health system demand in Liberia.

Author SummaryThere is considerable uncertainty regarding the steps needed to contain the ongoing Ebola crisis in West Africa, the timeline required to achieve control, and the projected burden of mortality. To address these issues, we develop a branching process model for Ebola transmission that focuses on offspring distributions (i.e., the numbers of new infections caused by each case). We use the model to assess the likely progression of Ebola in Liberia. The model assesses the feedback between new cases and hospital demand under a range of plausible intervention scenarios, particularly ramping-up of treatment facilities over time and increasing the number of individuals seeking hospital treatment through outreach and education. Transmission scenarios—to health care workers in hospitals, to caregivers in the community, to hospital visitors, and to individuals preparing bodies for funerals—are described by distinct offspring distributions based on available data. Results suggest that the outcome of the epidemic depends on both hospital capacity and individual behavior. Additionally, the model highlights the conditions under which transmission might have outpaced hospital capacity, and projects possible epidemic trajectories into 2015.

Assessing Measles Transmission in the United States Following a Large Outbreak in California.

The recent increase in measles cases in California may raise questions regarding the continuing success of measles control. To determine whether the dynamics of measles is qualitatively different in comparison to previous years, we assess whether the 2014-2015 measles outbreak associated with an Anaheim theme park is consistent with subcriticality by calculating maximum-likelihood estimates for the effective reproduction numbe given this year’s outbreak, using the Galton-Watson branching process model. We find that the dynamics after the initial transmission event are consistent with prior transmission, but does not exclude the possibilty that the effective reproduction number has increased.

Detecting differential transmissibilities that affect the size of self-limited outbreaks.

Our ability to respond appropriately to infectious diseases is enhanced by identifying differences in the potential for transmitting infection between individuals. Here, we identify epidemiological traits of self-limited infections (i.e. infections with an effective reproduction number satisfying ) that correlate with transmissibility. Our analysis is based on a branching process model that permits statistical comparison of both the strength and heterogeneity of transmission for two distinct types of cases. Our approach provides insight into a variety of scenarios, including the transmission of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) in the Arabian peninsula, measles in North America, pre-eradication smallpox in Europe, and human monkeypox in the Democratic Republic of the Congo. When applied to chain size data for MERS-CoV transmission before 2014, our method indicates that despite an apparent trend towards improved control, there is not enough statistical evidence to indicate that has declined with time. Meanwhile, chain size data for measles in the United States and Canada reveal statistically significant geographic variation in , suggesting that the timing and coverage of national vaccination programs, as well as contact tracing procedures, may shape the size distribution of observed infection clusters. Infection source data for smallpox suggests that primary cases transmitted more than secondary cases, and provides a quantitative assessment of the effectiveness of control interventions. Human monkeypox, on the other hand, does not show evidence of differential transmission between animals in contact with humans, primary cases, or secondary cases, which assuages the concern that social mixing can amplify transmission by secondary cases. Lastly, we evaluate surveillance requirements for detecting a change in the human-to-human transmission of monkeypox since the cessation of cross-protective smallpox vaccination. Our studies lay the foundation for future investigations regarding how infection source, vaccination status or other putative transmissibility traits may affect self-limited transmission.

Asymptotics of Symmetric Compound Poisson Population Models

Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radiusrof convergence. It is shown that the asymptotic behaviour of symmetric compound Poisson models is mainly determined by the characteristic value θrφ′(r−). If θrφ′(r−)≥1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ′(r−) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Dirac coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.

Explanation of Temporal Clustering of Tsunami Sources Using the Epidemic-Type Aftershock Sequence Model

Temporal clustering of tsunami sources is examined in terms of a branching process model. It previously was observed that there are more short interevent times between consecutive tsunami sources than expected from a stationary Poisson process. The epidemic‐type aftershock sequence (ETAS) branching process model is fitted to tsunami catalog events, using the earthquake magnitude of the causative event from the Centennial and Global Centroid Moment Tensor (CMT) catalogs and tsunami sizes above a completeness level as a mark to indicate that a tsunami was generated. The ETAS parameters are estimated using the maximum‐likelihood method. The interevent distribution associated with the ETAS model provides a better fit to the data than the Poisson model or other temporal clustering models. When tsunamigenic conditions (magnitude threshold, submarine location, dip‐slip mechanism) are applied to the Global CMT catalog, ETAS parameters are obtained that are consistent with those estimated from the tsunami catalog. In particular, the dip‐slip condition appears to result in a near zero magnitude effect for triggered tsunami sources. The overall consistency between results from the tsunami catalog and that from the earthquake catalog under tsunamigenic conditions indicates that ETAS models based on seismicity can provide the structure for understanding patterns of tsunami source occurrence. The fractional rate of triggered tsunami sources on a global basis is approximately 14%. Online Material: Figures of residual analysis for tsunami and earthquake catalogs.

A Crump-Mode-Jagers Branching Process Model of Prion Loss in Yeast

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.

A Crump-Mode-Jagers Branching Process Model of Prion Loss in Yeast

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.