On the probability of extinction in a periodic environment

Bacaër, Nicolas; Dads, El Hadi Ait

doi:10.1007/s00285-012-0623-9

Cited by 25 publications

(34 citation statements)

References 31 publications

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“…Nevertheless, no analytical progress can be made along the same lines for the seasonally forced model we use here. We are aware of only one study [62] that addressed extinction probabilities in the periodic context using theoretical methods, but the method of Bacar & Ait Dads [62] has a limitation because it applies to the large population limit only. The development of the approaches to compute the time to extinction in seasonally forced models will be therefore a subject of further research.…”

Section: Discussionmentioning

confidence: 99%

Characterizing the dynamics of rubella relative to measles: the role of stochasticity

Rozhnova

Metcalf

Grenfell

2013

J. R. Soc. Interface.

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Rubella is a completely immunizing and mild infection in children. Understanding its behaviour is of considerable public health importance because of congenital rubella syndrome, which results from infection with rubella during early pregnancy and may entail a variety of birth defects. The recurrent dynamics of rubella are relatively poorly resolved, and appear to show considerable diversity globally. Here, we investigate the behaviour of a stochastic seasonally forced susceptible–infected–recovered model to characterize the determinants of these dynamics and illustrate patterns by comparison with measles. We perform a systematic analysis of spectra of stochastic fluctuations around stable attractors of the corresponding deterministic model and compare them with spectra from full stochastic simulations in large populations. This approach allows us to quantify the effects of demographic stochasticity and to give a coherent picture of measles and rubella dynamics, explaining essential differences in the recurrent patterns exhibited by these diseases. We discuss the implications of our findings in the context of vaccination and changing birth rates as well as the persistence of these two childhood infections.

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Section: Discussionmentioning

confidence: 99%

Characterizing the dynamics of rubella relative to measles: the role of stochasticity

Rozhnova

Metcalf

Grenfell

2013

J. R. Soc. Interface.

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“…In the former, the adiabatic limit, ω δ, we find ∆S = −ε 1−δ 2 arcsin(δ) 1− ω 2 6δ 2 π 2 −arcsin 2 (δ) . (33) From this expression it is evident that the leading-order term is constant with respect to ω, see Fig. 6 and the next subsection.…”

Section: Weak Perturbation -Linear Theorymentioning

confidence: 82%

“…One can see that the linear term in ε agrees with ∆S [Eq. (33)] in the leading order in ω δ. Note that the adiabatic mean rate of establishment (36) has a maximum cutoff value at ε c = δ 2 /(1 − δ 2 ) (for which κ * = 0) beyond which the action becomes negative, and therefore the rate has no physical meaning.…”

Section: Adiabatic Approximationmentioning

confidence: 99%

Population switching under a time-varying environment

Israeli

Assaf

2020

Phys. Rev. E

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We examine the switching dynamics of a stochastic population subjected to a deterministically time-varying environment. Our approach is demonstrated in the realm of ecology on a problem of population establishment. Here, by assuming a constant immigration pressure along with a strong Allee effect, at the deterministic level one obtains a critical population size beyond which the system experiences establishment. Notably the latter has been shown to be strongly influenced by the interplay between demographic and environmental noise. We consider two prototypical examples for environmental variations: a temporary environmental change, and a periodically-varying environment. By employing a semi-classical approximation we compute, within exponential accuracy, the change in the establishment probability and mean establishment time of the population, due to the environmental variability. Our analytical results are verified by using a modified Gillespie algorithm which accounts for explicitly time-dependent reaction rates. Finally, our theoretical approach can also be useful in studying switching dynamics in gene regulatory networks under extrinsic variations.

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“…, are respectively solutions of system (6) and system (3). It is now possible to compute the number R ℓ C (1).…”

Section: Consistencymentioning

confidence: 99%

A Consistent Discrete Version of a Nonautonomous SIRVS Model

Mateus

Silva

Vaz

2018

Discrete Dynamics in Nature and Society

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A family of discrete non-autonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens non-standard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. The consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (respectively permanence) for the continuous model we also have extinction (respectively permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.

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On the probability of extinction in a periodic environment

Cited by 25 publications

References 31 publications

Characterizing the dynamics of rubella relative to measles: the role of stochasticity

Characterizing the dynamics of rubella relative to measles: the role of stochasticity

Population switching under a time-varying environment

A Consistent Discrete Version of a Nonautonomous SIRVS Model

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