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Journal articles on the topic 'Extinction population branching processes'

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Published: 24 April 2022

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1

Wang, Han-xing. "Extinction of population-size-dependent branching processes in random environments." Journal of Applied Probability 36, no. 1 (1999): 146–54. http://dx.doi.org/10.1239/jap/1032374237.

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We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.
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2

Wang, Han-xing. "Extinction of population-size-dependent branching processes in random environments." Journal of Applied Probability 36, no. 01 (1999): 146–54. http://dx.doi.org/10.1017/s0021900200016922.

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We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Z n } n≥0 is associated with the stationary environment ξ− = {ξ n } n≥0, let B = {ω : Z n (ω) = for some n}, and q(ξ−) = P(B | ξ−, Z 0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.
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3

Hueter, Irene. "Branching processes in generalized autoregressive conditional environments." Advances in Applied Probability 48, no. 4 (2016): 1211–34. http://dx.doi.org/10.1017/apr.2016.71.

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AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environ
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4

Jagers, Peter, Fima C. Klebaner, and Serik Sagitov. "Markovian paths to extinction." Advances in Applied Probability 39, no. 2 (2007): 569–87. http://dx.doi.org/10.1239/aap/1183667624.

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Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a
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5

Wang, Han-Xing, and Dafan Fang. "Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments." Journal of Applied Probability 36, no. 2 (1999): 611–19. http://dx.doi.org/10.1239/jap/1032374477.

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A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particular
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6

Klebaner, Fima C. "Asymptotic behavior of near-critical multitype branching processes." Journal of Applied Probability 28, no. 3 (1991): 512–19. http://dx.doi.org/10.2307/3214487.

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Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.
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7

Klebaner, Fima C. "Asymptotic behavior of near-critical multitype branching processes." Journal of Applied Probability 28, no. 03 (1991): 512–19. http://dx.doi.org/10.1017/s0021900200042376.

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Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.
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8

Klebaner, F. C., and R. Liptser. "Likely path to extinction in simple branching models with large initial population." Journal of Applied Mathematics and Stochastic Analysis 2006 (May 14, 2006): 1–23. http://dx.doi.org/10.1155/jamsa/2006/60376.

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We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time, we study the Galton-Watson process, and in continuous time, the branching diffusion. The most likely paths are found with the help of the large deviation principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the nonnegativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent inte
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9

Hautphenne, S., G. Latouche, and G. Nguyen. "Extinction Probabilities of Branching Processes with Countably Infinitely Many Types." Advances in Applied Probability 45, no. 04 (2013): 1068–82. http://dx.doi.org/10.1017/s0001867800006777.

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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
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10

Hautphenne, S., G. Latouche, and G. Nguyen. "Extinction Probabilities of Branching Processes with Countably Infinitely Many Types." Advances in Applied Probability 45, no. 4 (2013): 1068–82. http://dx.doi.org/10.1239/aap/1386857858.

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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
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11

Wang, Han-Xing, and Dafan Fang. "Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments." Journal of Applied Probability 36, no. 02 (1999): 611–19. http://dx.doi.org/10.1017/s002190020001737x.

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A population-size-dependent branching process {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ > 1. When the environmental sequence {ξ n } is a irreducible positive recurrent Mar
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12

Jagers, Peter, Fima C. Klebaner, and Serik Sagitov. "Markovian paths to extinction." Advances in Applied Probability 39, no. 02 (2007): 569–87. http://dx.doi.org/10.1017/s0001867800001890.

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Subcritical Markov branching processes {Z t } die out sooner or later, say at time T < ∞. We give results for the path to extinction {Z uT , 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of x u−1 Z uT , as Z 0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by Z T-u , u > 0, and show that as Z 0= x → ∞ this process converges to a Markov process {Y u }, which we describe completely. Emphasis is on continuous time processes, those in disc
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13

González, Miguel, Cristina Gutiérrez, and Rodrigo Martínez. "Modeling Y-Linked Pedigrees through Branching Processes." Mathematics 8, no. 2 (2020): 256. http://dx.doi.org/10.3390/math8020256.

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A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-
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14

Li, Yuqiang. "On a Continuous-State Population-Size-Dependent Branching Process and Its Extinction." Journal of Applied Probability 43, no. 1 (2006): 195–207. http://dx.doi.org/10.1239/jap/1143936253.

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A continuous-state population-size-dependent branching process {Xt} is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt / t obtained when Xt tends to infinity.
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15

Li, Yuqiang. "On a Continuous-State Population-Size-Dependent Branching Process and Its Extinction." Journal of Applied Probability 43, no. 01 (2006): 195–207. http://dx.doi.org/10.1017/s0021900200001467.

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A continuous-state population-size-dependent branching process {X t } is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population X t is discussed, and the limit distribution of X t / t obtained when X t tends to infinity.
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16

Pakes, Anthony G. "Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements." Advances in Applied Probability 21, no. 02 (1989): 243–69. http://dx.doi.org/10.1017/s000186780001853x.

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The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems a
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17

Pakes, Anthony G. "Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements." Advances in Applied Probability 21, no. 2 (1989): 243–69. http://dx.doi.org/10.2307/1427159.

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The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems an
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18

Xing, Yongsheng, and Yongjin Wang. "On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes." Journal of Applied Probability 42, no. 01 (2005): 175–84. http://dx.doi.org/10.1017/s0021900200000140.

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In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.
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19

Xing, Yongsheng, and Yongjin Wang. "On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes." Journal of Applied Probability 42, no. 1 (2005): 175–84. http://dx.doi.org/10.1239/jap/1110381379.

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In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.
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20

Mitov, K. V., and N. M. Yanev. "Bellman-Harris branching processes with a special type of state-dependent immigration." Advances in Applied Probability 21, no. 02 (1989): 270–83. http://dx.doi.org/10.1017/s0001867800018541.

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We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.
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21

Mitov, K. V., and N. M. Yanev. "Bellman-Harris branching processes with a special type of state-dependent immigration." Advances in Applied Probability 21, no. 2 (1989): 270–83. http://dx.doi.org/10.2307/1427160.

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We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.
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22

Braunsteins, Peter, and Sophie Hautphenne. "The probabilities of extinction in a branching random walk on a strip." Journal of Applied Probability 57, no. 3 (2020): 811–31. http://dx.doi.org/10.1017/jpr.2020.35.

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AbstractWe 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, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}
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23

Huillet, Thierry, and Servet Martínez. "REGENERATIVE MUTATION PROCESSES RELATED TO THE SELFDECOMPOSABILITY OF SIBUYA DISTRIBUTIONS." Probability in the Engineering and Informational Sciences 33, no. 2 (2018): 291–325. http://dx.doi.org/10.1017/s0269964818000189.

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The Sibuya distribution is a discrete probability distribution on the positive integers which, while Poisson-compounding it, gives rise to the discrete-stable distribution of Steutel and van Harn. We first address the question of the discrete self-decomposability of Sibuya and Sibuya-related distributions. Discrete self-decomposable distributions arise as limit laws of pure-death branching processes with immigration, translating a balance between immigration events and systematic ageing and ultimate death of the immigrants at constant rate. Exploiting this fact, we design a new Luria–Delbrück-
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24

Molina, M., M. Mota, and A. Ramos. "Some Contributions to the Theory of Near-Critical Bisexual Branching Processes." Journal of Applied Probability 44, no. 02 (2007): 492–505. http://dx.doi.org/10.1017/s0021900200003119.

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We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to ot
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25

Molina, M., M. Mota, and A. Ramos. "Some Contributions to the Theory of Near-Critical Bisexual Branching Processes." Journal of Applied Probability 44, no. 02 (2007): 492–505. http://dx.doi.org/10.1017/s0021900200117978.

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We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to ot
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26

Molina, M., M. Mota, and A. Ramos. "Some Contributions to the Theory of Near-Critical Bisexual Branching Processes." Journal of Applied Probability 44, no. 2 (2007): 492–505. http://dx.doi.org/10.1239/jap/1183667416.

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We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to ot
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27

Kersting, Götz. "A unifying approach to branching processes in a varying environment." Journal of Applied Probability 57, no. 1 (2020): 196–220. http://dx.doi.org/10.1017/jpr.2019.84.

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AbstractBranching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead
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28

González, M., R. Martínez, and M. Slavtchova-Bojkova. "Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling." Journal of Applied Probability 47, no. 01 (2010): 58–71. http://dx.doi.org/10.1017/s0021900200006392.

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The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination poli
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29

González, M., R. Martínez, and M. Slavtchova-Bojkova. "Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling." Journal of Applied Probability 47, no. 1 (2010): 58–71. http://dx.doi.org/10.1239/jap/1269610816.

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The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination poli
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30

Ball, Frank, Tom Britton, and Peter Neal. "On expected durations of birth–death processes, with applications to branching processes and SIS epidemics." Journal of Applied Probability 53, no. 1 (2016): 203–15. http://dx.doi.org/10.1017/jpr.2015.19.

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Abstract We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, A
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31

Klebaner, Fima C. "Linear growth in near-critical population-size-dependent multitype Galton–Watson processes." Journal of Applied Probability 26, no. 3 (1989): 431–45. http://dx.doi.org/10.2307/3214402.

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We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combina
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32

Klebaner, Fima C. "Linear growth in near-critical population-size-dependent multitype Galton–Watson processes." Journal of Applied Probability 26, no. 03 (1989): 431–45. http://dx.doi.org/10.1017/s0021900200038043.

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We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combina
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33

Kabluchko, Zakhar. "Persistence and Equilibria of Branching Populations with Exponential Intensity." Journal of Applied Probability 49, no. 1 (2012): 226–44. http://dx.doi.org/10.1239/jap/1331216844.

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We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form eλ(du) = e-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλχ with intensity eλ. If λφ'(λ) < 0 then the branching population suffer
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34

Kabluchko, Zakhar. "Persistence and Equilibria of Branching Populations with Exponential Intensity." Journal of Applied Probability 49, no. 01 (2012): 226–44. http://dx.doi.org/10.1017/s0021900200008962.

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We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form e λ(du) = e-λu du, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλ χ with intensity e λ. If λφ'(λ) < 0 then the branching popul
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35

ZU, JIAN, JINLIANG WANG, and YASUHIRO TAKEUCHI. "COEVOLUTIONARY DYNAMICS OF PREDATOR-PREY INTERACTIONS." International Journal of Biomathematics 05, no. 03 (2012): 1260015. http://dx.doi.org/10.1142/s1793524512600157.

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In this paper, with the method of adaptive dynamics, we investigate the coevolution of phenotypic traits of predator and prey species. The evolutionary model is constructed from a deterministic approximation of the underlying stochastic ecological processes. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that evolutionary branching in the prey phenotype will occur when the frequency dependence in the prey carrying capacity is not strong. Furthermore, it is found that if the two prey branches mov
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36

Zhunwei, Lu, and Peter Jagers. "A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments." Journal of Applied Probability 41, no. 1 (2004): 176–86. http://dx.doi.org/10.1239/jap/1077134676.

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In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α1 ≤α0 < + ∞ and 0 < C1, C2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C1k-α0 and above by C2k-α1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.
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37

Lampert, Adam, and Alan Hastings. "Synchronization-induced persistence versus selection for habitats in spatially coupled ecosystems." Journal of The Royal Society Interface 10, no. 87 (2013): 20130559. http://dx.doi.org/10.1098/rsif.2013.0559.

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Critical population phase transitions, in which a persistent population becomes extinction-prone owing to environmental changes, are fundamentally important in ecology, and their determination is a key factor in successful ecosystem management. To persist, a species requires a suitable environment in a sufficiently large spatial region. However, even if this condition is met, the species does not necessarily persist, owing to stochastic fluctuations. Here, we develop a model that allows simultaneous investigation of extinction due to either stochastic or deterministic reasons. We find that eve
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38

Zhunwei, Lu, and Peter Jagers. "A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments." Journal of Applied Probability 41, no. 01 (2004): 176–86. http://dx.doi.org/10.1017/s0021900200014121.

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In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α 1 ≤α 0 < + ∞ and 0 < C 1, C 2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.
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39

Farrington, C. P., and A. D. Grant. "The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease." Journal of Applied Probability 36, no. 03 (1999): 771–79. http://dx.doi.org/10.1017/s0021900200017563.

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We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.
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40

Farrington, C. P., and A. D. Grant. "The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease." Journal of Applied Probability 36, no. 3 (1999): 771–79. http://dx.doi.org/10.1239/jap/1032374633.

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Abstract:
We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.
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41

Pénisson, Sophie, and Christine Jacob. "Stochastic Methodology for the Study of an Epidemic Decay Phase, Based on a Branching Model." International Journal of Stochastic Analysis 2012 (December 4, 2012): 1–32. http://dx.doi.org/10.1155/2012/598701.

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We present a stochastic methodology to study the decay phase of an epidemic. It is based on a general stochastic epidemic process with memory, suitable to model the spread in a large open population with births of any rare transmissible disease with a random incubation period and a Reed-Frost type infection. This model, which belongs to the class of multitype branching processes in discrete time, enables us to predict the incidences of cases and to derive the probability distributions of the extinction time and of the future epidemic size. We also study the epidemic evolution in the worst-case
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42

Jagers, Peter, and Sergei Zuyev. "Populations in environments with a soft carrying capacity are eventually extinct." Journal of Mathematical Biology 81, no. 3 (2020): 845–51. http://dx.doi.org/10.1007/s00285-020-01527-5.

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Abstract 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 $$Z_0$$ Z 0 and the size of the nth change by $$C_n$$ C n , $$n= 1, 2, \ldots $$ n = 1 , 2 , … . Population sizes hence develop successively as $$Z_1=Z_0+C_1,\ Z_2=Z_1+C_2$$ 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 $$Z_n=0$$ Z n = 0 implies that $$Z_{n+1}=0$$ Z n + 1 = 0 , without there being a
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43

Lambert, Amaury. "Branching Processes: Variation, Growth and Extinction of Populations P. Haccou, J. Jagers, V. A. Vatutin . 2005. Branching Processes: Variation, Growth and Extinction of Populations. Cambridge University Press.xii+. 316 15.5 × 23.5 cm, hardcover, $US95.00. ISBN: 0-521-83220-9." Ecoscience 13, no. 4 (2006): 560–61. http://dx.doi.org/10.2980/1195-6860(2006)13[560:bpvgae]2.0.co;2.

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44

Asmussen, Søren. "Branching Processes: Variation, Growth and Extinction of Populations Edited by P. Haccou, P. Jagers, and V. A. Vatutin." Biometrics 62, no. 4 (2006): 1269–70. http://dx.doi.org/10.1111/j.1541-0420.2006.00596_1.x.

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45

Jagers, Peter, and Andreas Lagerås. "General branching processes conditioned on extinction are still branching processes." Electronic Communications in Probability 13 (2008): 540–47. http://dx.doi.org/10.1214/ecp.v13-1419.

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46

Shiozawa, Yuichi. "Extinction of branching symmetric α-stable processes". Journal of Applied Probability 43, № 04 (2006): 1077–90. http://dx.doi.org/10.1017/s0021900200002448.

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We give a criterion for extinction or local extinction of branching symmetric α-stable processes in terms of the principal eigenvalue for time-changed processes of symmetric α-stable processes. Here the branching rate and the branching mechanism are spatially dependent. In particular, the branching rate is allowed to be singular with respect to the Lebesgue measure. We apply this criterion to some branching processes.
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47

Shiozawa, Yuichi. "Extinction of branching symmetric α-stable processes". Journal of Applied Probability 43, № 4 (2006): 1077–90. http://dx.doi.org/10.1239/jap/1165505209.

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We give a criterion for extinction or local extinction of branching symmetric α-stable processes in terms of the principal eigenvalue for time-changed processes of symmetric α-stable processes. Here the branching rate and the branching mechanism are spatially dependent. In particular, the branching rate is allowed to be singular with respect to the Lebesgue measure. We apply this criterion to some branching processes.
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48

Mayster, Penka. "Alternating branching processes." Journal of Applied Probability 42, no. 04 (2005): 1095–108. http://dx.doi.org/10.1017/s0021900200001133.

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We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.
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49

Mayster, Penka. "Alternating branching processes." Journal of Applied Probability 42, no. 4 (2005): 1095–108. http://dx.doi.org/10.1239/jap/1134587819.

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We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.
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50

Hautphenne, Sophie. "Extinction Probabilities of Supercritical Decomposable Branching Processes." Journal of Applied Probability 49, no. 03 (2012): 639–51. http://dx.doi.org/10.1017/s0021900200009438.

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We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific cla
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