Extinction in lower Hessenberg branching processes with countably many types

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$

A low-rank technique for computing the quasi-stationary distribution of subcritical Galton-Watson processes

We present a new algorithm for computing the quasi-stationary distribution of subcritical Galton--Watson branching processes. This algorithm is based on a particular discretization of a well-known functional equation that characterizes the quasi-stationary distribution of these processes. We provide a theoretical analysis of the approximate low-rank structure that stems from this discretization, and we extend the procedure to multitype branching processes. We use numerical examples to demonstrate that our algorithm is both more accurate and more efficient than other approaches

Extinction probabilities of branching processes with countably infinitely many types

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods

The probabilities of extinction in a branching random walk on a strip

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative method to compute the vector $\boldsymbol{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\boldsymbol{q}(A)$ in the set of fixed points of the progeny generating vector

Linking Population-Size-Dependent and Controlled Branching Processes

Population-size dependent branching processes (PSDBP) and controlled branching processes (CBP) are two classes of branching processes widely used to model biological populations that exhibit logistic growth. In this paper we develop connections between the two, with the ultimate goal of determining when a population is more appropriately modelled with a PSDBP or a CBP. In particular, we state conditions for the existence of equivalent PSDBPs and CBPs, we then consider the subclass of CBPs with deterministic control functions (DCBPs), stating a necessary and sufficient condition for DCBP-PSDBP equivalence. Finally, we derive an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size, and show that under certain conditions this bound tends to zero as the initial population size becomes large

Sensitivity analysis of a branching process evolving on a network with application in epidemiology

We perform an analytical sensitivity analysis for a model of a continuous-time branching process evolving on a fixed network. This allows us to determine the relative importance of the model parameters to the growth of the population on the network. We then apply our results to the early stages of an influenza-like epidemic spreading among a set of cities connected by air routes in the United States. We also consider vaccination and analyze the sensitivity of the total size of the epidemic with respect to the fraction of vaccinated people. Our analysis shows that the epidemic growth is more sensitive with respect to transmission rates within cities than travel rates between cities. More generally, we highlight the fact that branching processes offer a powerful stochastic modeling tool with analytical formulas for sensitivity which are easy to use in practice.Comment: 17 pages (30 with SI), Journal of Complex Networks, Feb 201

Consistent least squares estimation in population-size-dependent branching processes

We consider discrete-time parametric population-size-dependent branching processes (PSDBPs) with almost sure extinction and propose a new class of weighted least-squares estimators based on a single trajectory of population size counts. We prove that, conditional on non-extinction up to a finite time $n$, our estimators are consistent and asymptotic normal as $n\to\infty$. We pay particular attention to estimating the carrying capacity of a population. Our estimators are the first conditionally consistent estimators for PSDBPs, and more generally, for Markov models for populations with a carrying capacity. Through simulated examples, we demonstrate that our estimators outperform other least squares estimators for PSDBPs in a variety of settings. Finally, we apply our methods to estimate the carrying capacity of the endangered Chatham Island black robin population