On the Extinction of the S–I–S stochastic logistic epidemic

1989 ◽  
Vol 26 (4) ◽  
pp. 685-694 ◽  
Author(s):  
Richard J. Kryscio ◽  
Claude Lefèvre
Keyword(s):  
Chemical Reactions ◽  
Logistic Model ◽  
Diffusion Processes ◽  
Birth And Death Processes ◽  
Time To Extinction ◽  
Stochastic Logistic Model ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time ◽  
Quasi Stationary Distribution

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.

On the Extinction of the S–I–S stochastic logistic epidemic

1989 ◽  
Vol 26 (04) ◽  
pp. 685-694
Author(s):  
Richard J. Kryscio ◽  
Claude Lefèvre
Keyword(s):  
Chemical Reactions ◽  
Logistic Model ◽  
Diffusion Processes ◽  
Birth And Death Processes ◽  
Time To Extinction ◽  
Stochastic Logistic Model ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time ◽  
Quasi Stationary Distribution

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.

The quasistationary distribution of the stochastic logistic model

2001 ◽  
Vol 38 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Otso Ovaskainen
Keyword(s):  
Relative Error ◽  
Population Biology ◽  
Logistic Model ◽  
Reproduction Ratio ◽  
Time To Extinction ◽  
Major Interest ◽  
Stochastic Logistic Model ◽  
Mean Time To Extinction ◽  
Number Of Individuals ◽  
Mean Time

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.

The quasistationary distribution of the stochastic logistic model

2001 ◽  
Vol 38 (04) ◽  
pp. 898-907 ◽  
Author(s):  
Otso Ovaskainen
Keyword(s):  
Relative Error ◽  
Population Biology ◽  
Logistic Model ◽  
Reproduction Ratio ◽  
Time To Extinction ◽  
Major Interest ◽  
Stochastic Logistic Model ◽  
Mean Time To Extinction ◽  
Number Of Individuals ◽  
Mean Time

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individualsN→ ∞ (with relative error of the approximations of the order𝒪(1/N)), and at the limit of the basic reproduction ratioR0→ ∞ (with relative error of the approximations of the order𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values ofNandR0> 1.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (2) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (02) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

A bounded growth population subjected to emigrations due to population pressure

1981 ◽  
Vol 18 (03) ◽  
pp. 571-582 ◽  
Author(s):  
A. C. Trajstman
Keyword(s):  
Size Distribution ◽  
Population Level ◽  
Population Pressure ◽  
Population Increase ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
The Times ◽  
Negative Exponential ◽  
Mean Time

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure. The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.

A bounded growth population subjected to emigrations due to population pressure

1981 ◽  
Vol 18 (3) ◽  
pp. 571-582 ◽  
Author(s):  
A. C. Trajstman
Keyword(s):  
Size Distribution ◽  
Population Level ◽  
Population Pressure ◽  
Population Increase ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
The Times ◽  
Negative Exponential ◽  
Mean Time

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure.The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.

scholarly journals Evaluating an icon of population persistence: the Devil's Hole pupfish

2014 ◽  
Vol 281 (1794) ◽  
pp. 20141648 ◽  
Author(s):  
J. Michael Reed ◽  
Craig A. Stockwell
Keyword(s):  
Count Data ◽  
Small Populations ◽  
Vulnerable Species ◽  
Human Intervention ◽  
Microsatellite Data ◽  
Time To Extinction ◽  
Credible Intervals ◽  
Mean Time To Extinction ◽  
Iconic Status ◽  
Mean Time

The Devil's Hole pupfish Cyprinodon diabolis has iconic status among conservation biologists because it is one of the World's most vulnerable species. Furthermore, C. diabolis is the most widely cited example of a persistent, small, isolated vertebrate population; a chronic exception to the rule that small populations do not persist long in isolation. It is widely asserted that this species has persisted in small numbers (less than 400 adults) for 10 000–20 000 years, but this assertion has never been evaluated. Here, we analyse the time series of count data for this species, and we estimate time to coalescence from microsatellite data to evaluate this hypothesis. We conclude that mean time to extinction is approximately 360–2900 years (median 410–1800), with less than a 2.1% probability of persisting 10 000 years. Median times to coalescence varied from 217 to 2530 years, but all five approximations had wide credible intervals. Our analyses suggest that Devil's Hole pupfish colonized this pool well after the Pleistocene Lakes receded, probably within the last few hundred to few thousand years; this could have occurred through human intervention.

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