Extinction times for a birth–death process with weak competition

Sagitov, Serik; Shaimerdenova, Altynay

doi:10.1007/s10986-013-9204-x

Cited by 8 publications

(5 citation statements)

References 14 publications

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“…For both regimes, Nåsell (2011) poses as an open problem the determination of the mean extinction time E T N . Our results have some similarity with Theorem 2(ii) of Sagitov and Shaimerdenova (2013), who study the distribution of the extinction time for a different version of the logistic model in a completely different limit. Barbour, Hamza, Kaspi and Klebaner (2015) study a very general class of population models, which includes this one.…”

supporting

confidence: 84%

Extinction times in the subcritical stochastic SIS logistic epidemic

2018

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Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size N. We study the behaviour of the process as the population size N tends to infinity. Our results cover the entire subcritical regime, including the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to 0 as [Formula: see text] but more slowly than [Formula: see text]. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory.

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confidence: 84%

Extinction times in the subcritical stochastic SIS logistic epidemic

2018

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“…We are aware of only a few mathematical results related to our work. In [9], the authors do not study the quasi-stationary distribution but only the mean time to expectation starting from a state of order K for which they obtain the asymptotic behavior in K (see also [21]). Here we are able to control this quantity for all initial states and also for the quasi-stationary distribution as a starting distribution.…”

mentioning

confidence: 99%

Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

Chazottes

Collet

Méléard

2015

Probab. Theory Relat. Fields

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We study a general class of birth-and-death processes with state space N that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a 'carrying capacity' K. When K is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of K, for large K. We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when K is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process.

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“…The selection of these was based on the mathematical results of Sagitov and Shaimerdenova (2013) on the non‐spatial stochastic logistic model, thus corresponding to our case of

ε = 0

. Denoting by MTE the mean time to extinction, Sagitov and Shaimerdenova (2013) showed that the

\log (MTE)

increases linearly with

\log (U)

for the critical value of

m = m_{c}

, whereas

\log (MTE)

increases sublinearly with

\log (U)

for

m > m_{c}

and superlinearly with

\log (U)

for

m < m_{c}

. This behaviour is illustrated in Figure 2, where we have simulated the model with different domain sizes, for values of the parameter m that are both just below and just above the critical value of

m_{c} = A^{+}

.…”

Section: Methodsmentioning

confidence: 99%

Mathematical and simulation methods for deriving extinction thresholds in spatial and stochastic models of interacting agents

2020

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1. In ecology, one of the most fundamental questions relates to the persistence of populations, or conversely to the probability of their extinction. Deriving extinctionThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

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Extinction times for a birth–death process with weak competition

Cited by 8 publications

References 14 publications

Extinction times in the subcritical stochastic SIS logistic epidemic

Extinction times in the subcritical stochastic SIS logistic epidemic

Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

Mathematical and simulation methods for deriving extinction thresholds in spatial and stochastic models of interacting agents

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