2010
DOI: 10.1080/07362990903415882
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Comparison of Markov Chain and Stochastic Differential Equation Population Models Under Higher-Order Moment Closure Approximations

Abstract: Continuous time Markov chain (CTMC) and Itô stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model). Differential equations are derived for the moments of the distribution for each stochastic model. Each moment differential equation depends on higher-order moments. Assumptions are made regarding higher-order moments to form a finite, solvable system. Conditions are given under which the CTMC and SDE BID models have the same moment solution or t… Show more

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Cited by 16 publications
(9 citation statements)
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“…It is worth mentioning that, given the large number of random variables, time-nonhomogeneity, and nonlinearity, exact analytical results for our models are not feasible. Certainly, in the case of time-homogeneous stochastic processes such as continuous-time Markov chain and SDE models, methods to approximate the mean and higher-order moments of the random variables, the final size, the duration until disease extinction, and the variability about the endemic state have been applied, e.g., Britton (2010) ; Ekanayake and Allen (2010) ; Krishnarajah et al (2005) ; Lloyd (2004) ; Ovaskainen and Meerson (2010) ; Van Kampen (1992) . Unfortunately, many of these methods do not extend to the time-nonhomogeneous models with mean-reverting processes.…”
Section: Discussionmentioning
confidence: 99%
“…It is worth mentioning that, given the large number of random variables, time-nonhomogeneity, and nonlinearity, exact analytical results for our models are not feasible. Certainly, in the case of time-homogeneous stochastic processes such as continuous-time Markov chain and SDE models, methods to approximate the mean and higher-order moments of the random variables, the final size, the duration until disease extinction, and the variability about the endemic state have been applied, e.g., Britton (2010) ; Ekanayake and Allen (2010) ; Krishnarajah et al (2005) ; Lloyd (2004) ; Ovaskainen and Meerson (2010) ; Van Kampen (1992) . Unfortunately, many of these methods do not extend to the time-nonhomogeneous models with mean-reverting processes.…”
Section: Discussionmentioning
confidence: 99%
“…The Gaussian closure principle is one choice of a wide variety of distributional closures. For example, one could assume the moments of a lognormal distribution [35] instead…”
Section: Stochastic Closuresmentioning
confidence: 99%
“…However, these ODEs do not form a closed system; they depend on successively higher-order moments so that approximation methods for the higher-order moments are needed to form a finite system of ODEs [11]. For the case of the scalar equation (15), the moment differential equations form a closed system that can be solved.…”
Section: Stochastic Mathematical Modelsmentioning
confidence: 99%