Effects of Immigration on Some Stochastic Logistic Models: A Cumulant Truncation Analysis

Matis, James H.; Kiffe, Thomas R.

doi:10.1006/tpbi.1999.1420

Cited by 41 publications

(31 citation statements)

References 9 publications

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“…Stochastic models are useful in epidemiology and ecology and are used widely (Isham, 1991;Allen and Cormier, 1996;Bolker and Pacala, 1997;Filipe and Gibson, 1998;Marion et al, 1998;Matis and Kiffe, 1999;Bauch and Rand, 2000;Keeling, 2000). Usually, the transition probabilities exhibit non-linear dependence on population size or number of infectives which makes the resultant stochastic processes analytically intractable.…”

Section: Introductionmentioning

confidence: 99%

Novel moment closure approximations in stochastic epidemics

Krishnarajah

Cook

Marion

et al. 2005

Bulletin of Mathematical Biology

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Moment closure approximations are used to provide analytic approximations to nonlinear stochastic models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and thirdorder moment closure approximations which we apply to the stochastic SI and SIS models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial. In addition, a novel closure approximation is developed in order to capture the behaviour of the stochastic SIS model at the critical point of persistence or extinction of the process. This mixture approximation, is a third-order approximation and comprises a probability distribution designed to capture the behaviour of the system conditioned on non-extinction (quasi-equilibrium) and a probability mass at 0 which represents the probability of extinction. Two versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results show: 1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and 2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model in marked contrast with existing approaches.

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Section: Introductionmentioning

confidence: 99%

Novel moment closure approximations in stochastic epidemics

Krishnarajah

Cook

Marion

et al. 2005

Bulletin of Mathematical Biology

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“…This question was already posed by MacArthur and Wilson [7,8] who discussed how the equilibrium number of species (the sum of species-specific occupancies), found on an island, depends on the balance of colonization and extinction rates. More recently, Matis and Kiffe [9] considered a nonlinear stochastic population model with immigration and suggested a cumulant truncation procedure to approximate the PDF of the population size in this model. We will deal with a similar model here, but solve it exactly.…”

Section: Introductionmentioning

confidence: 99%

Immigration-extinction dynamics of stochastic populations

Meerson

Ovaskainen²

2013

Phys. Rev. E

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How high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population extinct? Is there any relation between the population size distributions with and without immigration? Under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch, and treat a patch simply as either occupied or empty? We address these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the Verhulst model: a prototypical model of an isolated stochastic population.

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“…A fundamental paper on this truncation approach is given by Renshaw 9 , who subsequently extended this work to bivariate populations in Reference 23 . Recent applications of TS approximation include logistic and power-law logistic processes widely used in population modeling of biological systems 24,25 , with several other popular applications given by Butler 26 . Let K (j) (t) be a jth order truncated cgf for X defined by…”

Section: Ts Approximationsmentioning

confidence: 99%

Improved one‐sided control charts for the mean of a positively skewed population using truncated saddlepoint approximations

Chen

Matis

Benneyan

2011

Quality & Reliability Eng

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This paper develops a truncated saddlepoint (TS) control chart for the mean of a population with a skewed density. This chart requires the estimation of only a few population cumulants from sample data and is relatively straightforward in construction. Performance is assessed at various levels of truncation in terms of type 1 and 2 error rates and compared with other heuristic control charts via the Monte Carlo simulation. The truncated saddlepoint (TS) chart is shown to generally outperform these charts, especially in cases with small sample sizes or high levels of population skewness.

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Effects of Immigration on Some Stochastic Logistic Models: A Cumulant Truncation Analysis

Cited by 41 publications

References 9 publications

Novel moment closure approximations in stochastic epidemics

Novel moment closure approximations in stochastic epidemics

Immigration-extinction dynamics of stochastic populations

Improved one‐sided control charts for the mean of a positively skewed population using truncated saddlepoint approximations

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