Добірка наукової літератури з теми "Van Kampen expansion"

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Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible–exposed–infected–recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.

The spatial evolution of the velocity in a turbulent fluid is shown to consist of a mean strain rate plus a noise term whose mean-square amplitude is dependent upon the local velocity. Assuming the independence of different length scales allows one to eliminate the effect of cubic and higher moments of the velocity upon the evolution of the two-point probability density function for the velocity, analogous to the system-size expansion of the master equation (N. G. van Kampen. Adv. Chem. Phys. 34, 245 (1976); R. Kubo, K. Matsuo, and K. Kitahara. J. Stat. Phys. 9, 51 (1973); M. Suzuki. Adv. Chem. Phys. 46, 198 (1981)). Furthermore, for the velocity structures that exhibit the most internal correlation (i.e., the flow fields that are most likely to be observed) the mean strain rate drops out, leaving only the nonlinear diffusion term. In other words, the dependence of the two-point velocity statistics upon length scale or separation is found to be governed by the gradients of the stress (rather than of the velocity, as is usually assumed.) This provides a new approach for predicting the dominant patterns observed in turbulent flows and a new means of characterizing or classifying different structures, namely, by their nonlinear diffusion coefficient. As a physically relevant example, the isotropic case with algebraic nonlinearity is considered.

- Tim Behrend, Nancy K. Florida, Javanese literature in Surakarta manuscripts; Volume 2; Manuscripts of the Mangkunagaran palace. Ithaca, New York: Cornell University Southeast Asia Program, 2000, 575 pp. - Harold Brookfield, Judith M. Heimann, The most offending soul alive; Tom Harrisson and his remarkable life. Honolulu: University of Hawai’i Press, 1998, 468 pp. - Harold Brookfield, Victor T. King, Rural development and social science research; Case studies from Borneo. Phillips, Maine: Borneo Research Council, 1999, xiii + 359 pp. [Borneo Research Council Proceedings Series 6.] - J.G. de Casparis, Roy E. Jordaan, The Sailendras in Central Javanese history; A survey of research from 1950 to 1999. Yogyakarta: Penerbitan Universitas Sanata Dharma, 1999, iv + 108 pp. - H.J.M. Claessen, Francoise Douaire-Marsaudon, Les premiers fruits; Parenté, identité sexuelle et pouvoirs en Polynésie occidentale (Tonga, Wallis et Futuna). 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We compare initial value and eigenvalue problems for two-dimensional perturbations of the inviscid shear flow in a channel. Singular solutions, known in plasma physics as van Kampen (vK) modes, are constructed. They form a complete set of eigenfunctions for decomposition of any initial perturbation for stable wavy perturbations. A pair of discrete modes appears to ensure completeness in the unstable case. Expansion coefficients for eigenmodes are found, and equivalence of temporal evolution obtained with the help of the evolutionary equation for vorticity and expansion over eigenmodes is presented. This alternative description of the evolution using vK-modes is analogous to ones found earlier in plasma and in stellar dynamics. In particular, for stable wavy perturbations, an initial state decays first exponentially due to Landau damping, then algebraically. It has been established (numerically and analytically) that the final decay law is $t^{-1}$.Also we numerically demonstrate that Landau-damped perturbations are not true eigenmodes, but rather a superposition of vK-modes with a real frequency which does not retain its shape over time. However, solution on a contours in the complex plane may exhibit properties of a true eigenmode, i.e., decay without changing its spatial form. Energy redistribution between perturbation and the flow, in stable and unstable regimes, is analyzed.

AbstractInfluenza is an infectious disease with obvious periodic changes over time. It is of great practical significance to explore the non-environment-related factors that cause this regularity for influenza control and individual protection. In this paper, based on the randomness of population number and the heterogeneity of population contact, we have established a stochastic infectious disease model about influenza based on the degree of the network, and obtained the power spectral density function by using the van Kampen expansion method of the master equation. The relevant parameters are obtained by fitting the influenza data of sentinel hospitals. The results of the numerical analysis show that: (1) for the infected, the infection period of patients who go to the sentinel hospitals is particularly different from the others who do not; (2) for all the infected, there is an obvious nonlinear relationship between their infection period and the visiting rate of the influenza sentinel hospitals, the infection rate and the degree. Among them, only the infection period of patients who do not go to the sentinel hospitals decreased monotonously with the infection rate (increased monotonously with the visiting rate), while the rest had a non-monotonic relationship.

Experimental studies have shown that the protein abundance in living cells varies from few tens to several thousands molecules per species. Molecular fluctuations roughly scale as the inverse square root of the number of molecules due to the random timing of reactions. It is hence expected that intrinsic noise plays an important role in the dynamics of biochemical networks. The Chemical Master Equation is the accepted description of these systems under well-mixed conditions. Because analytical solutions to this equation are available only for simple systems, one often has to resort to approximation methods. A popular technique is an expansion in the inverse volume to which the reactants are confined, called van Kampen's system size expansion. Its leading order terms are given by the phenomenological rate equations and the linear noise approximation that quantify the mean concentrations and the Gaussian fluctuations about them, respectively. While these approximations are valid in the limit of large molecule numbers, it is known that physiological conditions often imply low molecule numbers. We here develop systematic approximation methods based on higher terms in the system size expansion for general biochemical networks. We present an asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion. We then derive an analytical approximation of the corresponding time-dependent probability distribution. Finally, we devise a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions. We show through the use of biological examples that the first few terms of the expansion yield accurate approximations even for low number of molecules. The theory is hence expected to closely resemble the outcomes of single cell experiments.

The classical approach to population dynamics is deterministic in nature: the concentrations of the interacting constituents are assumed to be continuous variables. Alternatively, one can proceed with a stochastic modelling, respecting the intrinsic granularity of the scrutinized system. Starting for a microscopic, hence inherently stochastic, formulation of the inspected model, one can rigorously recover the underlying deterministic picture and assess the role played by finite size corrections via the celebrated van Kampen expansion. At the conventional order of approximation, also called linear noise approximation, the van Kampen analysis yields a Fokker-Planck equation for the distribution of fluctuations. This machinery can be for instance applied to explain the emergence of the so-called quasi-cycles, regular time oscillations of the concentrations, which result from a resonant amplification of the stochastic noise. This possibility is discussed in the first part of the thesis with reference to problem of intracellular calcium dynamics. Then, we have considered the van Kampen expansion beyond the Gaussian order of approximation. The method has been challenged for a selection models and its predictive ability tested versus numerical simulations. In practice, when accounting for higher order corrections, one obtains a generalized Fokker-Planck equation which depends explicitly on N, the size of the system. This enables us to write down a system of ODEs for the unknown moments of the distribution of fluctuations. The sought distribution is eventually recovered by Fourier inversion. The analysis that we have carried out testifies on the adequacy of the van Kampen expansion also when the weak noise hypothesis breaks down or, equivalently, when the characteristic size of the population is small. To complete the study we have also drawn a quantitative comparison between the WKB and the generalized van Kampen expansion, working with a stochastic version of the logistic equations. In the second part of the thesis we have studied reaction-diffusion systems of both stochastic and deterministic inspiration. As an interesting ingredient we have considered the effect of the microscopic competition for the available space, imposing a finite carrying capacity constraint into the microscopic formulation of the models. This is presumably relevant when the interacting species are densely packed in space, in a regime that is often referred to as to the molecular crowding. We studied in particular the conditions for the emergence of stochastic spatially extended patterns and the interplay between Turing like patterns and the cross diffusion due to crowding.