Multitype Branching Processes Research Articles

Multitype branching processes with inhomogeneous Poisson immigration

Abstract In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

Regular variation in a fixed-point problem for single- and multi-class branching processes and queues

Abstract Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

Stochastic two-group models with transmission dependent on host infectivity or susceptibility

ABSTRACTStochastic epidemic models with two groups are formulated and applied to emerging and re-emerging infectious diseases. In recent emerging diseases, disease spread has been attributed to superspreaders, highly infectious individuals that infect a large number of susceptible individuals. In some re-emerging infectious diseases, disease spread is attributed to waning immunity in susceptible hosts. We apply a continuous-time Markov chain (CTMC) model to study disease emergence or re-emergence from different groups, where the transmission rates depend on either the infectious host or the susceptible host. Multitype branching processes approximate the dynamics of the CTMC model near the disease-free equilibrium and are used to estimate the probability of a minor or a major epidemic. It is shown that the probability of a major epidemic is greater if initiated by an individual from the superspreader group or by an individual from the highly susceptible group. The models are applied to Severe Acute Respiratory Syndrome and measles.

A Multi-type Preferential Attachment Tree

A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $\mathbb{N}^p$ to $\mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates. In the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)=\gamma_{ij}(n_1+n_2+\dots +n_p)+\beta_{ij}$, where $\gamma_{ij}$ and $\beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters $\{\gamma_{ij}\}, \{\beta_{ij}\}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.

Effects of Transmission Bottlenecks on the Diversity of Influenza A Virus.

We investigate the fate of de novo mutations that occur during the in-host replication of a pathogenic virus, predicting the probability that such mutations are passed on during disease transmission to a new host. Using influenza A virus as a model organism, we develop a life-history model of the within-host dynamics of the infection, deriving a multitype branching process with a coupled deterministic model to capture the population of available target cells. We quantify the fate of neutral mutations and mutations affecting five life-history traits: clearance, attachment, budding, cell death, and eclipse phase timing. Despite the severity of disease transmission bottlenecks, our results suggest that in a single transmission event, several mutations that appeared de novo in the donor are likely to be transmitted to the recipient. Even in the absence of a selective advantage for these mutations, the sustained growth phase inherent in each disease transmission cycle generates genetic diversity that is not eliminated during the transmission bottleneck.

Multi-type branching processes with time-dependent branching rates

Abstract Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Abstract B70: Mathematical model quantifies the effect of novel combination therapies in high-grade serous ovarian cancer

Abstract Purpose: We introduce here the most comprehensive mathematical model to date for high-grade serous ovarian cancer patient treatment response and use it to predict the effect of novel combination therapies. Introduction: Platinum-based chemotherapy constitutes the backbone for clinical care of advanced solid cancers, such as high-grade serous ovarian cancer (HGSOC). The vast majority of patients, however, become resistant to platinum, which eventually leads to generally fatal refractory disease. As HGSOC tumors are heterogeneous, cancer cells often have activated many resistant mechanisms, and targeting a single mechanism at a time is unlikely to result in significant improvement in survival rates. A systems biology approach, where resistance mechanisms are comprehensively considered, is required in efforts to overcome platinum resistance. Methods: Platinum resistance is a multifactorial process and the most important resistance mechanisms are: 1) reduced intake or increased efflux leading to reduced platinum bioavailability in a cell, 2) DNA repair mechanisms that overcome platinum-induced DNA adducts, and 3) dysfunctional apoptosis machinery. The herein established mathematical model is based on three major resistance mechanisms and it models tumor growth and evolution as a deterministic continuous time, multitype branching process. Importantly, we have used comprehensive pre- and postoperative clinical data, including high-resolution 18F-fluorodeoxyglucose positron emission tomography/computed tomography images, from a HGSOC patient cohort. This allows accurate calibration of the model parameters, which ensures that the model is able to capture clinical course of HGSOC patients. In addition to quantifying the effect of a platinum-taxane to tumor burden of an HGSOC patient, we have modeled the impact of debulking surgery. This enables us to use the mathematical model in simulating “virtual HGSOC patients” who undergo standard-of-care (SOC) therapy, i.e., debulking surgery and platinum-taxane therapy. Results: Our model-generated virtual patients show almost identical treatment responses with two validation HGSOC cohorts that together contain data from 105 HGSOC patients. The average virtual HGSOC patient has platinum-free interval (PFI) of six months, which is identical to the mean PFI in the training and independent validation cohorts. The ability to create in silico HGSOC patients whose responses correspond to real-life HGSOC cohorts was supported statistically by the Kolmogorov-Smirnov test. As our model can capture the heterogeneous responses of individual HGSOC patients, we utilized it to evaluate the added value of combining novel targeted therapies to the current first-line SOC therapy. The results from the virtual clinical trials with selected targeted agents are in line with the results from early-phase clinical trials and pinpoint the most effective combination strategies. Further, our results show that targeting the three most important platinum-resistant mechanisms simultaneously with clinically available agents provides a significant survival advance to HGSOC patients. Conclusions: High failure rate in translating preclinical results to successful clinical trials is a major issue in oncology. The herein introduced mathematical framework enables creating virtual patient cohorts that capture the response variation in real-life patient cohorts. This allows for rapid and reliable estimate of the added value of the novel therapy in comparison to SOC therapy to optimize clinical trial design. Here we focused on overcoming platinum resistance in HGSOC and our results show substantial added value of combining three early-phase clinical trial agents, which target the three major resistant mechanisms and have been individually combined with SOC. Citation Format: Sampsa Hautaniemi, Emilia Kozlowska, Anniina Färkkilä, Tuulia Vallius, Olli Carpen, Seija Grenman, Rainer Lehtonen, Johanna Hynninen, Sakari Hietanen. Mathematical model quantifies the effect of novel combination therapies in high-grade serous ovarian cancer. [abstract]. In: Proceedings of the AACR Conference: Addressing Critical Questions in Ovarian Cancer Research and Treatment; Oct 1-4, 2017; Pittsburgh, PA. Philadelphia (PA): AACR; Clin Cancer Res 2018;24(15_Suppl):Abstract nr B70.

Probability of a major infection in a stochastic within-host model with multiple stages

Establishment or spread of a viral infection within healthy individuals depends on exposure to a viral source, either through virus particles or through cells that have been infected. We assume that a potential infection has reached the site of the healthy target cells and we apply stochastic within-host models and multitype branching processes to investigate whether a major infection becomes established. The model includes multiple latent and actively infected stages. It is shown that the probability of a major infection is generally more likely after the virus has entered the target cell and the cell is actively infected. In some cases, the probability of a major infection is less likely if the burst size of actively infected cells is small.

Self-normalized large deviation for supercritical branching processes

Abstract We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

Statistical inference for partially observed branching processes with application to cell lineage tracking of in vivo hematopoiesis

Single-cell lineage tracking strategies enabled by recent experimental technologies have produced significant insights into cell fate decisions, but lack the quantitative framework necessary for rigorous statistical analysis of mechanistic models describing cell division and differentiation. In this paper, we develop such a framework with corresponding moment-based parameter estimation techniques for continuous-time, multi-type branching processes. Such processes provide a probabilistic model of how cells divide and differentiate, and we apply our method to study hematopoiesis, the mechanism of blood cell production. We derive closed-form expressions for higher moments in a general class of such models. These analytical results allow us to efficiently estimate parameters of much richer statistical models of hematopoiesis than those used in previous statistical studies. To our knowledge, the method provides the first rate inference procedure for fitting such models to time series data generated from cellular barcoding experiments. After validating the methodology in simulation studies, we apply our estimator to hematopoietic lineage tracking data from rhesus macaques. Our analysis provides a more complete understanding of cell fate decisions during hematopoiesis in nonhuman primates, which may be more relevant to human biology and clinical strategies than previous findings from murine studies. For example, in addition to previously estimated hematopoietic stem cell self-renewal rate, we are able to estimate fate decision probabilities and to compare structurally distinct models of hematopoiesis using cross validation. These estimates of fate decision probabilities and our model selection results should help biologists compare competing hypotheses about how progenitor cells differentiate. The methodology is transferrable to a large class of stochastic compartmental and multi-type branching models, commonly used in studies of cancer progression, epidemiology and many other fields.

Establishment of Locally Adapted Mutations Under Divergent Selection.

We study the establishment probabilities of locally adapted mutations using a multi-type branching process framework. We find a surprisingly simple and intuitive analytical approximation for the establishment probabilities in a symmetric two-deme model under the assumption of weak (positive) selection. This is the first analytical closed-form approximation for arbitrary migration rate to appear in the literature. We find that the establishment probability lies between the weak and the strong migration limits if we condition the origin of the mutation to the deme where it is advantageous. This is not the case when we condition the mutation to first occur in a deme where it is disadvantageous. In this case we find that an intermediate migration rate maximizes the probability of establishment. We extend our results to the cases of multiple demes, two demes with asymmetric rates of gene flow, and asymmetric carrying capacities. The latter case allows us to illustrate how density regulation can affect establishment probabilities. Finally, we use our results to investigate the role of gene flow on the rate of local adaptation and identify cases in which intermediate amounts of gene flow facilitate the rate of local adaptation as compared to two populations without gene flow.

Demographic variance in heterogeneous populations: matrix models and sensitivity analysis

The demographic consequences of stochasticity in processes such as survival and reproduction are modulated by the heterogeneity within the population. Therefore, to study effects of stochasticity on population growth and extinction risk, it is critical to use structured population models in which the most important sources of heterogeneity (e.g. age, size, developmental stage) are incorporated as i‐state variables.Demographic stochasticity in heterogeneous populations has often been studied using one of two approaches: multitype branching processes and diffusion approximations. Here, we link these approaches, through the demographic stochasticity in age‐ or stage‐structured matrix population models. We derive the demographic variance, σ2d, which measures the per capita contribution to the variance in population growth increment, and we show how it can be decomposed into contributions from transition probabilities and fertility across ages or stages. Furthermore, using matrix calculus we derive the sensitivity of σ2d to age‐ or stage‐specific mortality and fertility. We apply the methods to an extensive set of data from age‐classified human populations (long‐term time‐series for Sweden, Japan and the Netherlands; two hunter–gatherer populations, and the high‐fertility Hutterites), and to a size‐classified population of the herbaceous plant Calathea ovandensis. For the human populations our analysis reveals substantial temporal changes in the demographic variance as well as its main components across age.These new methods provide a powerful approach for calculating the demographic variance for any structured model, and for analyzing its main components and sensitivities. This will make possible new analyses of demographic variance across different kinds of heterogeneity in different life cycles, which will in turn improve our understanding of mechanisms underpinning extinction risk and other important biological outcomes.

Sur l’extinction des populations avec plusieurs types dans un environnement aléatoire

This study focuses on the extinction rate of a population that follows a continuous-time multi-type branching process in a random environment. Numerical computations in a particular example inspired by an epidemic model suggest an explicit formula for this extinction rate, but only for certain parameter values.

Reduced multitype critical branching processes in random environment

Abstract We consider a multitype critical branching process Z n , n = 0, 1,…, in an i.i.d. random environment. Let Z m,n be the number of particles in this process at time m having descendants at time n. A limit theorem is proved for the logarithm of Z nt,n at moments nt,0 ≤ t ≤ 1, conditioned on the survival of the process Z n up to moment n when n → ∞.

Abstract P5-08-01: Reconstructing the evolutionary paths of BIG 1-98

Abstract We are reconstructing the dynamics of tumour growth, treatment response, and origins of resistance of the Breast International Group (BIG) 1-98 study to determine the evolutionary paths of non-responders. We established the somatic alteration landscape of primary tumours collected at surgery from BIG 1-98 patients using targeted DNA sequencing of 287 genes of FFPE samples from 538 patients. Combined analysis of genetic and clinical data indicates increased genomic instability; TP53 mutations and copy number variations of 11q13 and 8p11 are associated with poor prognosis. We developed an in silico simulation framework to investigate if the non-responder outcome is due to (over)-treatment or intrinsic to evolutionary selection prior to diagnosis. To reconstruct the evolutionary history of tumours, we created a computational multitype branching process that tracks the expansion of diverse clonal lineages as they acquire driver and passenger mutations that alter their proliferation and mutation rates. To account for the heterogeneity between patients, we created a fitting procedure based on the Cramer-von Misses statistic to find the likelihood of parameters of our computational model that recreate the mutational landscape and clinicopathological factors observed in each patient. Once the fitting procedure is done, we simulate the course of treatment following the study arm scheme accounting for the cell-cycle action mechanism of Tamoxifen and Letrozole. Using our tool, we are characterising the range of tumour development scenarios covering different degrees of aggressiveness and genomic instability. Our computational model allows simulation of diverse adjuvant-schemes to predict optimised treatment regimes for validation in patient-derived tumour xenograft mouse models. Citation Format: Lara-Gonzalez LL, Loi S, Ferrari D, Papenfuss A, Goode DL. Reconstructing the evolutionary paths of BIG 1-98 [abstract]. In: Proceedings of the 2017 San Antonio Breast Cancer Symposium; 2017 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2018;78(4 Suppl):Abstract nr P5-08-01.

The survival probability of a critical multi-type branching process in i.i.d. random environment

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type L and type S) in which only particles of type L have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least $1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)! e^{k-2}} + o(1)$. These bounds are sharp.

Multitype Branching Processes in Random Environment: Probability of Survival for the Critical Case

We investigate the asymptotic behavior of the survival probability of a critical multitype branching process evolving in an environment generated by a sequence of independent identically distribute...

Recurrence and transience of contractive autoregressive processes and related Markov chains

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.