Multitype Branching Research Articles

Unified model of short- and long-term HIV viral rebound for clinical trial planning.

Antiretroviral therapy (ART) effectively controls HIV infection, suppressing HIV viral loads. Typically suspension of therapy is rapidly followed by rebound of viral loads to high, pre-therapy levels. Indeed, a recent study showed that approximately 90% of treatment interruption study participants show viral rebound within at most a few months of therapy suspension, but the remaining 10%, showed viral rebound some months, or years, after ART suspension. Some may even never rebound. We investigate and compare branching process models aimed at gaining insight into these viral dynamics. Specifically, we provide a theory that explains both short- and long-term viral rebounds, and post-treatment control, via a multitype branching process with time-inhomogeneous rates, validated with data from Li et al. (Li et al. 2016 AIDS 30, 343–353. (doi:10.1097/QAD.0000000000000953)). We discuss the associated biological interpretation and implications of our best-fit model. To test the effectiveness of an experimental intervention in delaying or preventing rebound, the standard practice is to suspend therapy and monitor the study participants for rebound. We close with a discussion of an important application of our modelling in the design of such clinical trials.

Quenched Local Convergence of Boltzmann Planar Maps

Stephenson (2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit.

Multitype branching and graph product theory of infectious disease outbreaks.

The heterogeneity of human populations is a challenge to mathematical descriptions of epidemic outbreaks. Numerical simulations are deployed to account for the many factors influencing the spreading dynamics. Yet, the results from numerical simulations are often as complicated as the reality, leaving us with a sense of confusion about how the different factors account for the simulation results. Here, using a multitype branching together with a graph tensor product approach, I derive a single equation for the effective reproductive number of an infectious disease outbreak. Using this equation I deconvolute the impact of crowd management, targeted testing, contact heterogeneity, stratified vaccination, mask use, and smartphone tracing app use. This equation can be used to gain a basic understanding of infectious disease outbreaks and their simulations.

Backbone Decomposition of Multitype Superprocesses

In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

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.

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.

Not all fun and games: Potential incidence of SARS-CoV-2 infections during the Tokyo 2020 Olympic Games.

The Tokyo 2020 Olympic and Paralympic Games represent the most diverse international mass gathering event held since the start of the coronavirus disease 2019 (COVID-19) pandemic. Postponed to summer 2021, the rescheduled Games were set to be held amidst what would become the highest-ever levels of COVID-19 transmission in the host city of Tokyo. At the same time, the Delta variant of concern was gaining traction as the dominant viral strain and Japan had yet to exceed fifteen percent of its population fully vaccinated against COVID-19. To quantify the potential number of secondary cases that might arise during the Olympic Games, we performed a scenario analysis using a multitype branching process model. We considered the different contributions to transmission of Games accredited individuals, the general Tokyo population, and domestic spectators. In doing so, we demonstrate how transmission might evolve in these different groups over time, cautioning against any loosening of infection prevention protocols and supporting the decision to ban all spectators. If prevention measures were well observed, we estimated that the number of new cases among Games accredited individuals would approach zero by the end of the Games. However, if transmission was not controlled our model indicated hundreds of Games accredited individuals would become infected and daily incidence in Tokyo would reach upwards of 4,000 cases. Had domestic spectators been allowed (at 50% venue capacity), we estimated that over 250 spectators might have arrived infected to Tokyo venues, potentially generating more than 300 additional secondary infections while in Tokyo/at the Games. We also found the number of cases with infection directly attributable to hypothetical exposure during the Games was highly sensitive to the local epidemic dynamics. Therefore, reducing and maintaining transmission levels below epidemic levels using public health measures would be necessary to prevent cross-group transmission.

Multitype branching processes in random environment

В работе дан обзор результатов по теории многотипных ветвящихся процессов, эволюционирующих в случайной среде. Библиография: 104 названия.

Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China.

In this paper, deterministic and stochastic models are proposed to study the transmission dynamics of the Coronavirus Disease 2019 (COVID-19) in Wuhan, China. The deterministic model is formulated by a system of ordinary differential equations (ODEs) that is built upon the classical SEIR framework. The stochastic model is formulated by a continuous-time Markov chain (CTMC) that is derived based on the ODE model with constant parameters. The nonlinear CTMC model is approximated by a multitype branching process to obtain an analytical estimate for the probability of a disease outbreak. The local and global dynamics of the disease are analyzed by using the deterministic model with constant parameters, and the result indicates that the basic reproduction number $ \mathcal{R}_0 $ serves as a sharp disease threshold: the disease dies out if $ \mathcal{R}_0\le 1 $ and persists if $ \mathcal{R}_0 > 1 $. In contrast to the deterministic dynamics, the stochastic dynamics indicate that the disease may not persist when $ \mathcal{R}_0 > 1 $. Parameter estimation and validation are performed to fit our ODE model to the public reported data. Our result indicates that both the exposed and infected classes play an important role in shaping the epidemic dynamics of COVID-19 in Wuhan, China. In addition, numerical simulations indicate that a second wave of the ongoing pandemic is likely to occur if the prevention and control strategies are not implemented properly.

Stochastic investigation of HIV infection and the emergence of drug resistance.

Drug-resistant HIV-1 has caused a growing concern in clinic and public health. Although combination antiretroviral therapy can contribute massively to the suppression of viral loads in patients with HIV-1, it cannot lead to viral eradication. Continuing viral replication during sub-optimal therapy (due to poor adherence or other reasons) may lead to the accumulation of drug resistance mutations, resulting in an increased risk of disease progression. Many studies also suggest that events occurring during the early stage of HIV-1 infection (i.e., the first few hours to days following HIV exposure) may determine whether the infection can be successfully established. However, the numbers of infected cells and viruses during the early stage are extremely low and stochasticity may play a critical role in dictating the fate of infection. In this paper, we use stochastic models to investigate viral infection and the emergence of drug resistance of HIV-1. The stochastic model is formulated by a continuous-time Markov chain (CTMC), which is derived based on an ordinary differential equation model proposed by Kitayimbwa et al. that includes both forward and backward mutations. An analytic estimate of the probability of the clearance of HIV infection of the CTMC model near the infection-free equilibrium is obtained by a multitype branching process approximation. The analytical predictions are validated by numerical simulations. Unlike the deterministic dynamics where the basic reproduction number R0 serves as a sharp threshold parameter (i.e., the disease dies out if R0<1 and persists if R0>1), the stochastic models indicate that there is always a positive probability for HIV infection to be eradicated in patients. In the presence of antiretroviral therapy, our results show that the chance of clearance of the infection tends to increase although drug resistance is likely to emerge.

Controlling Fake News by Collective Tagging: A Branching Process Analysis

The spread of fake news on online social networks (OSNs) has become a matter of concern. These platforms are also used for propagating important authentic information. Thus, there is a need for mitigating fake news without significantly influencing the spread of real news. We leverage users’ inherent capabilities of identifying fake news and propose a warning-based control mechanism to curb this spread. Warnings are based on previous users’ responses that indicate the authenticity of the news. We use population-size dependent continuous-time multi-type branching processes to describe the spreading under the warning mechanism. We also have new results towards these branching processes. The (time) asymptotic proportions of the individual populations are derived using stochastic approximation tools. Using these, relevant type 1, type 2 performances are derived and an appropriate optimization problem is solved. The proposed mechanism effectively controls fake news, with negligible influence on the propagation of authentic news. We validate performance measures using Monte Carlo simulations on network connections provided by Twitter data.

Limit theorems for multi-type general branching processes with population dependence

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

The effect of demographic and environmental variability on disease outbreak for a dengue model with a seasonally varying vector population

Seasonal changes in temperature, humidity, and rainfall affect vector survival and emergence of mosquitoes and thus impact the dynamics of vector-borne disease outbreaks. Recent studies of deterministic and stochastic epidemic models with periodic environments have shown that the average basic reproduction number is not sufficient to predict an outbreak. We extend these studies to time-nonhomogeneous stochastic dengue models with demographic variability wherein the adult vectors emerge from the larval stage vary periodically. The combined effects of variability and periodicity provide a better understanding of the risk of dengue outbreaks. A multitype branching process approximation of the stochastic dengue model near the disease-free periodic solution is used to calculate the probability of a disease outbreak. The approximation follows from the solution of a system of differential equations derived from the backward Kolmogorov differential equation. This approximation shows that the risk of a disease outbreak is also periodic and depends on the particular time and the number of the initial infected individuals. Numerical examples are explored to demonstrate that the estimates of the probability of an outbreak from that of branching process approximations agree well with that of the continuous-time Markov chain. In addition, we propose a simple stochastic model to account for the effects of environmental variability on the emergence of adult vectors from the larval stage.

Disease Emergence in Multi-Patch Stochastic Epidemic Models with Demographic and Seasonal Variability.

Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.

Using multitype branching models to analyze bacterial pathogenicity

We apply multitype, continuous time Markov branching models to study pathogenicity in E. coli, a bacterium belonging to the genus Escherichia. First, we examine briefly the properties of multitype ...

Populations in environments with a soft carrying capacity are eventually extinct

Consider a population whose size changes stepwise by its members reproducing or dying (disappearing), but is otherwise quite general. Denote the initial (non-random) size by Z_0 and the size of the nth change by C_n, n= 1, 2, ldots . Population sizes hence develop successively as Z_1=Z_0+C_1, Z_2=Z_1+C_2 and so on, indefinitely or until there are no further size changes, due to extinction. Extinction is thus assumed final, so that Z_n=0 implies that Z_{n+1}=0, without there being any other finite absorbing class of population sizes. We make no assumptions about the time durations between the successive changes. In the real world, or more specific models, those may be of varying length, depending upon individual life span distributions and their interdependencies, the age-distribution at hand and intervening circumstances. We could consider toy models of Galton–Watson type generation counting or of the birth-and-death type, with one individual acting per change, until extinction, or the most general multitype CMJ branching processes with, say, population size dependence of reproduction. Changes may have quite varying distributions. The basic assumption is that there is a carrying capacity, i.e. a non-negative number K such that the conditional expectation of the change, given the complete past history, is non-positive whenever the population exceeds the carrying capacity. Further, to avoid unnecessary technicalities, we assume that the change C_n equals -1 (one individual dying) with a conditional (given the past) probability uniformly bounded away from 0. It is a simple and not very restrictive way to avoid parity phenomena, it is related to irreducibility in Markov settings. The straightforward, but in contents and implications far-reaching, consequence is that all such populations must die out. Mathematically, it follows by a supermartingale convergence property and positive probability of reaching the absorbing extinction state.

Lineage reconstruction from clonal correlations

A central task in developmental biology is to learn the sequence of fate decisions that leads to each mature cell type in a tissue or organism. Recently, clonal labeling of cells using DNA barcodes has emerged as a powerful approach for identifying cells that share a common ancestry of fate decisions. Here we explore the idea that stochasticity of cell fate choice during tissue development could be harnessed to read out lineage relationships after a single step of clonal barcoding. By considering a generalized multitype branching process, we determine the conditions under which the final distribution of barcodes over observed cell types encodes their bona fide lineage relationships. We then propose a method for inferring the order of fate decisions. Our theory predicts a set of symmetries of barcode covariance that serves as a consistency check for the validity of the method. We show that broken symmetries may be used to detect multiple paths of differentiation to the same cell types. We provide computational tools for general use. When applied to barcoding data in hematopoiesis, these tools reconstruct the classical hematopoietic hierarchy and detect couplings between monocytes and dendritic cells and between erythrocytes and basophils that suggest multiple pathways of differentiation for these lineages.

Evolutionary Rescue and Drug Resistance on Multicopy Plasmids.

Bacteria often carry "extra DNA" in the form of plasmids in addition to their chromosome. Many plasmids have a copy number greater than one such that the genes encoded on these plasmids are present in multiple copies per cell. This has evolutionary consequences by increasing the mutational target size, by prompting the (transitory) co-occurrence of mutant and wild-type alleles within the same cell, and by allowing for gene dosage effects. We develop and analyze a mathematical model for bacterial adaptation to harsh environmental change if adaptation is driven by beneficial alleles on multicopy plasmids. Successful adaptation depends on the availability of advantageous alleles and on their establishment probability. The establishment process involves the segregation of mutant and wild-type plasmids to the two daughter cells, allowing for the emergence of mutant homozygous cells over the course of several generations. To model this process, we use the theory of multitype branching processes, where a type is defined by the genetic composition of the cell. Both factors-the availability of advantageous alleles and their establishment probability-depend on the plasmid copy number, and they often do so antagonistically. We find that in the interplay of various effects, a lower or higher copy number may maximize the probability of evolutionary rescue. The decisive factor is the dominance relationship between mutant and wild-type plasmids and potential gene dosage effects. Results from a simple model of antibiotic degradation indicate that the optimal plasmid copy number may depend on the specific environment encountered by the population.

ESTIpop: a computational tool to simulate and estimate parameters for continuous-time Markov branching processes.

ESTIpop is an R package designed to simulate and estimate parameters for continuous-time Markov branching processes with constant or time-dependent rates, a common model for asexually reproducing cell populations. Analytical approaches to parameter estimation quickly become intractable in complex branching processes. In ESTIpop, parameter estimation is based on a likelihood function with respect to a time series of cell counts, approximated by the Central Limit Theorem for multitype branching processes. Additionally, simulation in ESTIpop via approximation can be performed many times faster than exact simulation methods with similar results. ESTIpop is available as an R package on Github (https://github.com/michorlab/estipop). Supplementary data are available at Bioinformatics online.

Stochastic Model of Bovine Babesiosis with Juvenile and Adult Cattle

A stochastic model for Bovine Babesiosis (BB) including ticks, and both juvenile and adult cattle is developed. This model is formulated by a system of continuous-time Markov chains (CTMCs) that is derived based on an extension of the deterministic ordinary differential equation model developed by Saad-Roy et al. (Bull Math Biol 77:514-547, 2015). The nonlinear CTMC model is approximated by a multitype branching process, giving a theoretical estimate of the probability of an outbreak of BB. Unlike the deterministic dynamics where the basic reproduction number is a sharp threshold parameter, the stochastic model indicates that there is always a positive probability of disease extinction within the cattle population. For parameter values from Colombia data, conditional probability distributions are numerically obtained for the time to disease extinction or outbreak, and are found to depend on the host type at the initiation of infection. The models with and without the inclusion of juvenile cattle are compared, and our result highlights that neglecting juvenile bovine in the models may lead to faulty predictions of critical disease statistics: particularly, it may underestimate the risk of infection. Endemic disease prevalence in adult cattle is examined for certain parameter values in the corresponding deterministic model. Notably, with long-lasting immunity, increased tick to juvenile infectivity decreases the proportion of infectious adults.