Moment-closure approximations for mass-action models

Although stochastic population models have proved to be a powerful tool in the study of process generating mechanisms across a wide range of disciplines, all too often the associated mathematical development involves nonlinear mathematics, which immediately raises difficult and challenging analytic problems that need to be solved if useful progress is to be made. One approximation that is often employed to estimate the moments of a stochastic process is moment closure. This approximation essentially truncates the moment equations of the stochastic process. A general expression for the marginal- and joint-moment equations for a large class of stochastic population models is presented. The generalisation of the moment equations allows this approximation to be applied easily to a wide range of models. Software is available from http://pysbml.googlecode.com/ to implement the techniques presented here.     

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.     

Accuracy of the Michaelis–Menten approximation when analysing effects of molecular noise

Quantitative biology relies on the construction of accurate mathematical models, yet the effectiveness of these models is often predicated on making simplifying approximations that allow for direct comparisons with available experimental data. The Michaelis–Menten (MM) approximation is widely used in both deterministic and discrete stochastic models of intracellular reaction networks, owing to the ubiquity of enzymatic activity in cellular processes and the clear biochemical interpretation of its parameters. However, it is not well understood how the approximation applies to the discrete stochastic case or how it extends to spatially inhomogeneous systems. We study the behaviour of the discrete stochastic MM approximation as a function of system size and show that significant errors can occur for small volumes, in comparison with a corresponding mass-action system. We then explore some consequences of these results for quantitative modelling. One consequence is that fluctuation-induced sensitivity, or stochastic focusing, can become highly exaggerated in models that make use of MM kinetics even if the approximations are excellent in a deterministic model. Another consequence is that spatial stochastic simulations based on the reaction–diffusion master equation can become highly inaccurate if the model contains MM terms.     

Stochastic control of hysteretic structural systems

A method of stochastic optimal control of hysteretic structural systems under earthquake excitations is presented. Stochastic estimation and control problems are formulated in the form of Itô stochastic differential equations on the basis of the theory of continuous Markov processes. The conditional moment equations given observation data are derived for nonlinear filtering, and are closed by introducing appropriate analytical form of the conditional probability density functions of the state variables. Under the assumption that the admissible controls are expressed as functions of the conditional moment functions the Bellman equation is derived. If the spatial variables of the Bellman equation are defined by a part of the full set of conditional moment functions appearing in the closed moment equations, the resulting Bellman equation is coupled with conditional moment equations both for filtering and for prediction. The Gaussian and non-Gaussian stochastic linearization techniques combined with simple solution techniques to the Bellman equation are examined to solve the Bellman equation or extended Riccati equations without prediction procedures.     

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.     

Insights from unifying modern approximations to infections on networks

Networks are increasingly central to modern science owing to their ability to conceptualize multiple interacting components of a complex system. As a specific example of this, understanding the implications of contact network structure for the transmission of infectious diseases remains a key issue in epidemiology. Three broad approaches to this problem exist: explicit simulation; derivation of exact results for special networks; and dynamical approximations. This paper focuses on the last of these approaches, and makes two main contributions. Firstly, formal mathematical links are demonstrated between several prima facie unrelated dynamical approximations. And secondly, these links are used to derive two novel dynamical models for network epidemiology, which are compared against explicit stochastic simulation. The success of these new models provides improved understanding about the interaction of network structure and transmission dynamics.     

Acoustic wave propagation in disordered microscale granular media under compression

We investigate the effect of dynamic and uniaxial static loading on the wave speeds and rise times of laser generated acoustic waves traveling through a disordered, multilayer aggregate of 2 \(\mu {\mathrm {m}}\) diameter silica microspheres, where the excited dynamic amplitudes are estimated to approach the level of the static overlap between the particles caused by adhesion and externally applied loads. Two cases are studied: a case where the as-fabricated particle network is retained, and a case where the static load has been increased to the point where the aggregate collapses and a rearrangement of the particle network occurs. We observe increases in wave speeds with static loading significantly lower than, and in approximate agreement with, predictions from models based on Hertzian contact mechanics for the pre- and post-collapse states, respectively. The measured rise time of the leading pulse is found to decrease with increasing static load in both cases, which we attribute to decreased scattering and stiffening of the contact network. Finally, we observe an increase in wave speed with increased excitation amplitude that depends on static loading, and whether the system is in the pre- or post-collapse state. The wave speed dependence on amplitude and static load is found to be in qualitative agreement with a one-dimensional discrete model of adhesive spheres, although the observed difference between pre- and post-collapse states is not captured. This investigation, and the approach presented herein, may find use in future studies of the contact mechanics and dynamics of adhesive microgranular systems.     

Integrating stochasticity and network structure into an epidemic model

While the foundations of modern epidemiology are based upon deterministic models with homogeneous mixing, it is being increasingly realized that both spatial structure and stochasticity play major roles in shaping epidemic dynamics. The integration of these two confounding elements is generally ascertained through numerical simulation. Here, for the first time, we develop a more rigorous analytical understanding based on pairwise approximations to incorporate localized spatial structure and diffusion approximations to capture the impact of stochasticity. Our results allow us to quantify, analytically, the impact of network structure on the variability of an epidemic. Using the susceptible–infectious–susceptible framework for the infection dynamics, the pairwise stochastic model is compared with the stochastic homogeneous-mixing (mean-field) model—although to enable a fair comparison the homogeneous-mixing parameters are scaled to give agreement with the pairwise dynamics. At equilibrium, we show that the pairwise model always displays greater variation about the mean, although the differences are generally small unless the prevalence of infection is low. By contrast, during the early epidemic growth phase when the level of infection is increasing exponentially, the pairwise model generally shows less variation.     

Networks and epidemic models.

Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a 'mixing network'. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks-and how these deviate from the random-mixing norm-have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.     

基于Markov链的重组细胞恒化培养的随机建模分析

本文研究了一类重组细胞恒化培养的连续时间Markov链模型.首先利用累积母函数表示出数字特征所满足的矩方程,然后通过对数正态分布近似的矩封闭技术得到了封闭后的矩方程,最后运用Euler-Maruyama方法构建了时间和状态都是连续的It随机微分方程.为了验证矩封闭的合理性,利用数值模拟给出了确定模型、随机模型和矩封闭后的方程的比较,并分析了重组细胞的变化趋势,结果表明其随机游走趋势与相应确定性模型是一致的.     

Profit-oriented shift scheduling of inbound contact centers with skills-based routing, impatient customers, and retrials

This paper presents a profit-oriented shift scheduling approach for inbound contact centers. The focus is on systems in which multiple agent classes with different qualifications serve multiple customer classes with different needs. We assume that customers are impatient, abandon if they have to wait, and that they may retry. A discrete-time modeling approach is used to capture the dynamics of the system due to time-dependent arrival rates. Staffing levels and shift schedules are simultaneously optimized over a set of different approximate realizations of the underlying stochastic processes to consider the randomness of the system. The numerical results indicate that the presented approach works best for medium-sized and large contact centers with skills-based routing of customers for which stochastic queueing models are rarely applicable.     

Stochastic amplification in epidemics

The role of stochasticity and its interplay with nonlinearity are central current issues in studies of the complex population patterns observed in nature, including the pronounced oscillations of wildlife and infectious diseases. The dynamics of childhood diseases have provided influential case studies to develop and test mathematical models with practical application to epidemiology, but are also of general relevance to the central question of whether simple nonlinear systems can explain and predict the complex temporal and spatial patterns observed in nature outside laboratory conditions. Here, we present a stochastic theory for the major dynamical transitions in epidemics from regular to irregular cycles, which relies on the discrete nature of disease transmission and low spatial coupling. The full spectrum of stochastic fluctuations is derived analytically to show how the amplification of noise varies across these transitions. The changes in noise amplification and coherence appear robust to seasonal forcing, questioning the role of seasonality and its interplay with deterministic components of epidemiological models. Childhood diseases are shown to fall into regions of parameter space of high noise amplification. This type of "endogenous" stochastic resonance may be relevant to population oscillations in nonlinear ecological systems in general.     

On the accuracy of the polynomial chaos approximation

Polynomial chaos representations for non-Gaussian random variables and stochastic processes are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. Finite truncations of these series are referred to as polynomial chaos (PC) approximations. This paper explores features and limitations of PC approximations. Metrics are developed to assess the accuracy of the PC approximation. A collection of simple, but relevant examples is examined in this paper. The number of terms in the PC approximations used in the examples exceeds the number of terms retained in most current applications. For the examples considered, it is demonstrated that (1) the accuracy of the PC approximation improves in some metrics as additional terms are retained, but does not exhibit this behavior in all metrics considered in the paper, (2) PC approximations for strictly stationary, non-Gaussian stochastic processes are initially nonstationary and gradually may approach weak stationarity as the number of terms retained increases, and (3) the development of PC approximations for certain processes may become computationally demanding, or even prohibitive, because of the large number of coefficients that need to be calculated. However, there have been many applications in which PC approximations have been successful.     

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schl?gl model revisited

Schlögl''s model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using Matlab. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.     

The effect of population structure on the emergence of drug resistance during influenza pandemics.

The spread of H5N1 avian influenza and the recent high numbers of confirmed human cases have raised international concern about the possibility of a new pandemic. Therefore, antiviral drugs are now being stockpiled to be used as a first line of defence. The large-scale use of antivirals will however exert a strong selection pressure on the virus, and may lead to the emergence of drug-resistant strains. A few mathematical models have been developed to assess the emergence of drug resistance during influenza pandemics. These models, however, neglected the spatial structure of large populations and the stochasticity of epidemic and demographic processes. To assess the impact of population structure and stochasticity, we modify and extend a previous model of influenza epidemics into a metapopulation model which takes into account the division of large populations into smaller units, and develop deterministic and stochastic versions of the model. We find that the dynamics in a fragmented population is less explosive, and, as a result, prophylaxis will prevent more infections and lead to fewer resistant cases in both the deterministic and stochastic model. While in the deterministic model the final level of resistance during treatment is not affected by fragmentation, in the stochastic model it is. Our results enable us to qualitatively extrapolate the prediction of deterministic, homogeneous-mixing models to more realistic scenarios.     

On methods for studying stochastic disease dynamics.

Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.     

Performance evaluation of a multi-product system under CONWIP control

Analytical models of multi-product manufacturing systems operating under CONWIP control are composed of closed queuing networks with synchronization stations. Under general assumptions, these queuing networks are hard to analyze exactly and therefore approximation methods must be used for performance evaluation. This research proposes a new approach based on parametric decomposition. Two-moment approximations are used to estimate the performance measures at individual stations. Subsequently, the traffic process parameters at the different stations are linked using stochastic transformation equations. The resulting set of non-linear equations is solved using an iterative algorithm to obtain estimates of key performance measures such as throughput, and mean queue lengths. Numerical studies indicate that the proposed method is computationally efficient and yields fairly accurate results when compared to simulation.     

Gradient-based criteria for sequential experiment design

Computer experiments are often used as inexpensive alternatives to real-world experiments. Statistical metamodels of the computer model's input-output behavior can be constructed to serve as approximations of the response surface of the real-world system. The suitability of a metamodel depends in part on its intended use. While decision makers may want to understand the entire response surface, they may be particularly keen on finding interesting regions of the design space, such as where the gradient is steep. We present an adaptive, value-enhanced batch sequential algorithm that samples more heavily in such areas while still providing an understanding of the entire surface. The design points within each batch can be run in parallel to leverage modern multi-core computing assets. We illustrate our approach for deterministic computer models, but it has potential for stochastic simulation models as well.     

Crack Closure in Spherical Shells

This article presents an analytical and numerical study of the influence of crack face closure on the stress intensity factor of a crack in a spherical shell subjected to membrane force and bending moment. The formulation developed by Delale and Erdogan based on linearised shallow shell theories is extended to account for crack surface closure in cracked spherical shells, assuming a line contact at the compressive edge of the crack face. It is found that, due to the curvature effects, the closure behaviour of cracks in spherical shell differs significantly from that in flat plates. Depending on the nature of the membrane force and bending moment, crack closure may occur over the entire crack length or only some segments. In the special case of full length crack closure, the coupled integral equations reduce to a single integral equation, leading to a simpler solution. The theoretical results are compared against finite element results of cracked shells compressed between two rigid plates, indicating that the shallow shell theory is valid for a shallowness parameter up to 1/5. This revised version was published online in July 2006 with corrections to the Cover Date.