Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

A genetic estimation algorithm for parameters of stochastic ordinary differential equations

A generic method for the estimation of parameters for Stochastic Ordinary Differential Equations (SODEs) is introduced and developed. This algorithm, called the GePERs method, utilises a genetic optimisation algorithm to minimise a stochastic objective function based on the Kolmogorov-Smirnov statistic. Numerical simulations are utilised to form the KS statistic. Further, the examination of some of the factors that improve the precision of the estimates is conducted. This method is used to estimate parameters of diffusion equations and jump-diffusion equations. It is also applied to the problem of model selection for the Queensland electricity market. (C) 2003 Elsevier B.V. All rights reserved.

Gauge Poisson representations for birth/death master equations

Poisson representation techniques provide a powerful method for mapping master equations for birth/death processes -- found in many fields of physics, chemistry and biology -- into more tractable stochastic differential equations. However, the usual expansion is not exact in the presence of boundary terms, which commonly occur when the differential equations are nonlinear. In this paper, a gauge Poisson technique is introduced that eliminates boundary terms, to give an exact representation as a weighted rate equation with stochastic terms. These methods provide novel techniques for calculating and understanding the effects of number correlations in systems that have a master equation description. As examples, correlations induced by strong mutations in genetics, and the astrophysical problem of molecule formation on microscopic grain surfaces are analyzed. Exact analytic results are obtained that can be compared with numerical simulations, demonstrating that stochastic gauge techniques can give exact results where standard Poisson expansions are not able to.

Mine waste beach profile and flow resistance equations

Four mine waste beach longitudinal profile equations are compared theoretically and in statistical analyses of profile data from 64 field and laboratory beaches formed by mine tailings, co-disposed coal mine wastes, and sand. All four equations fit the profile data well. The best performing equation both accounts for particle sorting and satisfies hydraulic constraints, and the combination of assumptions underlying it is considered to best represent the processes occurring on mine waste beaches. Combining these assumptions with the Lacey normal equation leads to a variant of the Manning resistance equation. Features that it is desirable to incorporate in theoretical and numerical models of mine waste beaches are listed.

Predictive regression equations and clinical uses of peripheral pulse timing characteristics in children

Studies have shown that increased arterial stiffening can be an indication of cardiovascular diseases like hypertension. In clinical practice, this can be detected by measuring the blood pressure (BP) using a sphygmomanometer but it cannot be used for prolonged monitoring. It has been established that pulse wave velocity (PWV) is a direct measure of arterial stiffening but its usefulness is hampered by the absence of non-invasive techniques to estimate it. Pulse transit time (PTT) is a simple and non-invasive method derived from PWV. However, limited knowledge of PTT in children is found in the present literature. The aims of this study are to identify independent variables that confound PTT measure and describe PTT regression equations for healthy children. Therefore, PTT reference values are formulated for future pathological studies. Fifty-five Caucasian children (39 male) aged 8.4 +/- 2.3 yr (range 5-12 yr) were recruited. Predictive equations for PTT were obtained by multiple regressions with age, vascular path length, BP indexes and heart rate. These derived equations were compared in their PWV equivalent against two previously reported equations and significant agreement was obtained (p < 0.05). Findings herein also suggested that PTT can be useful as a continuous surrogate BP monitor in children.

Effective cell-centred time-domain Maxwell's equations numerical solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed. © 2004 Elsevier Inc. All rights reserved.

Quantum computing and polynomial equations over the finite field Z(2)

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z(2). This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP subset of P-#P and BQP subset of PP.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

Optical Bloch equations and enhanced decay of Rabi oscillations in strong driving fields

Optical Bloch equations are widely used for describing dynamics in a system consisting molecules, electromagnetic waves, and a thermal bath. We analyze applicability of these equations to a single molecule imbedded in a solid matrix. Classical Bloch equations and the limits of their applicability are derived from more general master equations. Simple and intuitively appealing picture based on stochastic Bloch equations shows that at low temperatures, contrary to common believes, a strong driving field can not only suppress but can also increase decay rates of Rabi oscillations. A physical system where predicted effects can be observed experimentally is suggested. (c) 2005 Elsevier B.V. All rights reserved.

Predictive validity of three ActiGraph energy expenditure equations for children

Purpose: This Study evaluated the predictive validity of three previously published ActiGraph energy expenditure (EE) prediction equations developed for children and adolescents. Methods: A total of 45 healthy children and adolescents (mean age: 13.7 +/- 2.6 yr) completed four 5-min activity trials (normal walking. brisk walking, easy running, and fast running) in ail indoor exercise facility. During each trial, participants were all ActiGraph accelerometer oil the right hip. EE was monitored breath by breath using the Cosmed K4b(2) portable indirect calorimetry system. Differences and associations between measured and predicted EE were assessed using dependent t-tests and Pearson correlations, respectively. Classification accuracy was assessed using percent agreement, sensitivity, specificity, and area under the receiver operating characteristic (ROC) curve, Results: None of the equations accurately predicted mean energy expenditure during each of the four activity trials. Each equation, however, accurately predicted mean EE in at least one activity trial. The Puyau equation accurately predicted EE during slow walking. The Trost equation accurately predicted EE during slow running. The Freedson equation accurately predicted EE during fast running. None of the three equations accurately predicted EE during brisk walking. The equations exhibited fair to excellent classification accuracy with respect to activity intensity. with the Trost equation exhibiting the highest classification accuracy and the Puyau equation exhibiting the lowest. Conclusions: These data suggest that the three accelerometer prediction equations do not accurately predict EE on a minute-by-minute basis in children and adolescents during overground walking and running. The equations maybe, however, for estimating participation in moderate and vigorous activity.

A windows program for the derivation of steady-state equations in enzyme systems

Despite the number of computer-assisted methods described for the derivation of steady-state equations of enzyme systems, most of them are focused on strict steady-state conditions or are not able to solve complex reaction mechanisms. Moreover, many of them are based on computer programs that are either not readily available or have limitations. We present here a computer program called WinStes, which derives equations for both strict steady-state systems and those with the assumption of rapid equilibrium, for branched or unbranched mechanisms, containing both reversible and irreversible conversion steps. It solves reaction mechanisms involving up to 255 enzyme species, connected by up to 255 conversion steps. The program provides all the advantages of the Windows programs, such as a user-friendly graphical interface, and has a short computation time. WinStes is available free of charge on request from the authors. (c) 2006 Elsevier Inc. All rights reserved.