44 research outputs found
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 , in which
individuals of type may give birth to offspring of type only.
For this class of processes, we study the set 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 and whose maximum is the partial extinction probability
vector . In the case where
, we derive a global extinction
criterion which holds under second moment conditions, and when
we develop necessary and sufficient
conditions for
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 whose mean progeny matrices have a
block lower Hessenberg form. For these processes, the probability
of extinction in subsets of types may differ from the global extinction probability
and the partial extinction probability
. After deriving partial and global extinction
criteria, we develop conditions for
. We then present an
iterative method to compute the vector for any set .
Finally, we investigate the location of the vectors 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
, our estimators are consistent and asymptotic normal as . 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