Population projections for AIDS using an actuarial model

This paper gives details of a model for forecasting AIDS, developed for actuarial purposes, but used also for population projections. The model is only appropriate for homosexual transmission, but it is age-specific, and it allows variation in the transition intensities by age, duration in certain states and calendar year. The differential equations controlling transitions between states are defined, the method of numerical solution is outlined, and the parameters used in five different Bases of projection are given in detail. Numerical results for the population of England and Wales are shown.     

Computational Approaches to Solving Equations Arising from Wound Healing

In the wound healing process, the cell movement associated with chemotaxis generally outweighs the movement associated with random motion, leading to advection-dominated mathematical models of wound healing. The equations in these models must be solved with care, but often inappropriate approaches are adopted. Two one-dimensional test problems arising from advection-dominated models of wound healing are solved using four algorithms—MATLAB’s inbuilt routine pdepe.m, the Numerical Algorithms Group routine d03pcf.f, and two finite volume methods. The first finite volume method is based on a first-order upwinding treatment of chemotaxis terms and the second on a flux limiting approach. The first test problem admits an analytic solution which can be used to validate the numerical results by analyzing two measures of the error for each method: the average absolute difference and a mass balance error. These criteria as well as the visual comparison between the numerical methods and the exact solution lead us to conclude that flux limiting is the best approach to solving advection-dominated wound healing problems numerically in one dimension. The second test problem is a coupled nonlinear three species model of wound healing angiogenesis. Measurement of the mass balance error for this test problem further confirms our hypothesis that flux limiting is the most appropriate method for solving advection-dominated governing equations in wound healing models. We also consider two two-dimensional test problems arising from wound healing, one that admits an analytic solution and a more complicated problem of blood vessels growth into a devascularized wound bed. The results from the two-dimensional test problems also demonstrate that the flux limiting treatment of advective terms is ideal for an advection-dominated problem.     

Effectiveness factor for spherical biofilm catalysts

Immobilization is a method of avoiding wash-out of biocatalyst from a reactor system. For the modelling of these biocatalysts slab, cylinder, sphere and biofilm geometries are frequently used. A biofilm particle consists of an inert core which is used as a carrier for a layer that contains the enzymes or micro-organisms. This paper deals with the modelling and effectiveness factor calculations for such a biofilm particle and a general model for an immobilized, non-growing biocatalyst is presented. The model includes internal and external mass transfer resistance, the partitioning effect and inhibition or reversible reaction kinetics. Due to the non-linear reaction rate equations of the Michaelis Menten type, numerical techniques must be used for the solution of the combined diffusion reaction equation and calculation of the effectiveness factor. In this work we have used two different methods, orthogonal collocation and a method based on Runge-Kutta integration. Comparable use of CPU-time was found for these methods, but numerical stability and accuracy favour the Runge-Kutta method. In the case of Michaelis Menten kinetics (irreversible and without inhibition effects), an analytical expression for an approximate solution is presented. This method, which has an acceptable accuracy, takes far less CPU-time than the fore-mentioned numerical techniques.     

Modeling capsule tissue growth around disk-shaped implants: a numerical and in vivo study

We propose a new mathematical model that describes the growth of fibrous tissue around rigid, disk-shaped implants. A solution methodology based on an efficient regularized iterative method is presented to calibrate the model from some measurements of the capsule tissue concentration. Numerical results obtained with synthetic data are presented to demonstrate the ability of the proposed solution methodology to determine the model parameters corresponding to a given implant. In addition, numerical results obtained with experimental data are presented to illustrate the validity of the proposed model.     

Numerical methods and parameter estimation of a structured population model with discrete events in the life history

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator-prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.     

The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. I. Solution in transform space

Transient mass transfer in a Krogh tissue cylinder is described by a model taking into account axial diffusion in both blood and tissue, a localized permeability barrier at the capillary membrane and a diffusion barrier on the outer surface and at the ends of the cylinder. Radial diffusion in both blood and tissue is assumed to be infinitely fast. In contrast to previous work, which has usually relied on numerical methods for solving the equations, an exact solution is presented here in Laplace transform space. This allows calculation of the moments of the concentration at any point in the cylinder. Numerical results indicate that the moments of the residence time distribution are affected by the boundary conditions used, and that the discrepancies between the predictions using different conditions may be large in some physiological situations. Order-of-magnitude calculations are used to estimate when the use of simpler models may be feasible. The transform space solution may also be useful for parameter estimation, but it seems preferable to extend the present results to a time-domain solution for this purpose.     

Numerical solution of structured population models

Numerical methods are presented for a general age-structured population model with demographic rates depending on age and the total population size. The accuracy of these methods is established by solving problems for which alternate solution techniques are available and are used for comparison. The methods reliably solve test problems with a variety of dynamic behavior. Simulations of a blowfly population exhibit cyclic fluctuations, whereas a simulated squirrel population reaches a stable age distribution and stable equilibrium population size. Life-history attributes are easily studied from the computed solutions, and are discussed for these examples. Recovery of a stressed population back to equilibrium is examined by computing the transition in age structure, and the transient behavior of other properties of the population such as the per capita growth rate, the average age, and the generation length.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed model.     

Numerical integration of the master equation in some models of stochastic epidemiology

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation--up to a desired precision--in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.     

High Algebraic Order Methods for the Numerical Solution of the Schrödinger Equation

Abstract

A family of new hybrid four-step tenth algebraic order methods with phase-lag of order 16(2)22 is developed for the numerical solution of the Schrödinger equation. Based on the new methods a variable-step procedure is introduced. Numerical illustrations obtained for the approximation of the phase shift problem for the well known case of the Lenard-Jones potential and for the numerical solution of the coupled equations arising from the Schrödinger equation show that these new methods are better than other finite difference methods.     

A nonlinear numerical model of percutaneous drug absorption.

A nonlinear mathematical model developed by Chandrasekaran et al. is examined to monitor pharmacokinetic profiles in percutaneous drug absorption and is addressed to several associated problems that could occur in the data analysis of in vitro experiments. The formulation of the model gives rise to a nonlinear partial differential equation (PDE) of parabolic type, and a family of finite-difference methods is developed for the numerical solution of the associated initial/boundary-value problem. The value given to a parameter in this family determines the stability properties of the resulting method and whether the solution is obtained explicitly or implicitly. In the case of implicit members of the family it is seen that the solution of the nonlinear PDE is obtained by solving a linear algebraic system, the coefficient matrix of which is tridiagonal. The behaviors of two methods of the family are examined in a series of numerical experiments. Numerical differentiation and integration procedures are combined to monitor the cumulative amount of drug eliminated into the receptor cell per unit area as time increases. It is found that the use of the equation for the simple membrane model to estimate the permeability coefficient and lag time is warranted even if the system should be described by the dual-sorption model, provided cumulative amount versus time data collected for a sufficiently long time are used. However, being different from the behavior in the simple membrane model, the lag time, which can be estimated in this way, is dose-dependent and decreases with increasing donor cell concentration. On the other hand, the permeability coefficient in the dual-sorption model remains constant irrespective of the donor cell concentrations as in the simple membrane model.     

Comparison of analytically and numerically derived hydrogen exchange-rate distribution functions

Hydrogen exchange-rate probability density functions for lysozyme have been derived by numerical Laplace inversion with the computer program CONTIN. The resulting solution set includes a smooth bimodal solution in agreement with previous analytical results together with a smooth three-peak solution. Numerical analysis of lysozyme hydrogen-exchange data in glycerol/water cosolvent mixtures confirms the previous assignment of the slow-exchange peak to an exchange mechanism involving reversible unfolding. Physicochemical constrations that can reduce the size of the solution set are described. The results are compared with those obtained from previous analytical methods and the limitations of the discrete class and analytical appraches are discussed.     

A modified phase-fitted and amplification-fitted Runge-Kutta-Nyström method for the numerical solution of the radial Schrödinger equation

A new Runge-Kutta-Nyström method, with phase-lag and amplification error of order infinity, for the numerical solution of the Schrödinger equation is developed in this paper. The new method is based on the Runge-Kutta-Nyström method with fourth algebraic order, developed by Dormand, El-Mikkawy and Prince. Numerical illustrations indicate that the new method is much more efficient than other methods derived for the same purpose.     

Stochastic juvenile-adult models with application to a green tree frog population

We derive several stochastic models from a deterministic population model that describes the dynamics of age-structured juveniles coupled with size-structured adults. Numerical simulation results of the stochastic models are compared with the solution of the deterministic model. These models are then used to understand the effect of demographic stochasticity on the dynamics of an urban green tree frog (Hyla cinerea) population.     

根表面养分吸收通量和根围溶质浓度的近似解析解

该文用Nye-Tinker-Barber模型来研究植物根系表面的养分吸收通量和根围溶质浓度的近似解析解。将根围区域分为远场区域和近场区域, 在远场用相似变量, 在近场用尺度变换, 将远场解在根表面展开并与近场解进行待定函数的匹配, 从而获得对流扩散方程根表面通量和浓度的一阶近似解析解, 该解能够简化到扩散方程的解的形式。对氮、钾、硫、磷、镁、钙的养分吸收通量和氮、钾的浓度分别进行数值模拟, 比较模型的数值解、Roose的近似解析解和该文的近似解析解。结果表明: 在扩散方程中, 6种元素通量的解析解与Roose解析解相近, 但均高于数值解, 钾和磷的通量在短时间内迅速衰减; 钾和氮浓度的全局近似解析解与Roose解析解接近, 并与数值解的变化趋势一致。在对流扩散方程中, 除氮外的5种元素通量的近似解较Roose的解析解更接近于数值解, 且没有奇性。     

Methodology for predicting oxygen transport on an intravenous membrane oxygenator combining computational and analytical models

A computational methodology for accurately predicting flow and oxygen-transport characteristics and performance of an intravenous membrane oxygenator (IMO) device is developed, tested, and validated. This methodology uses extensive numerical simulations of three-dimensional computational models to determine flow-mixing characteristics and oxygen-transfer performance, and analytical models to indirectly validate numerical predictions with experimental data, using both blood and water as working fluids. Direct numerical simulations for IMO stationary and pulsating balloons predict flow field and oxygen transport performance in response to changes in the device length, number of and balloon pulsation frequency. Multifiber models are used to investigate interfiber interference and length effects for a stationary balloon whereas a single fiber model is used to analyze the effect of balloon pulsations on velocity and oxygen concentration fields and to evaluate oxygen transfer rates. An analytical lumped model is developed and validated by comparing its numerical predictions with experimental data. Numerical results demonstrate that oxygen transfer rates for a stationary balloon regime decrease with increasing number of fibers, independent of the fluid type. The oxygen transfer rate ratio obtained with blood and water is approximately two. Balloon pulsations show an effective and enhanced flow mixing, with time-dependent recirculating flows around the fibers regions which induce higher oxygen transfer rates. The mass transfer rates increase approximately 100% and 80%, with water and blood, respectively, compared with stationary balloon operation. Calculations with combinations of frequency, number of fibers, fiber length and diameter, and inlet volumetric flow rates, agree well with the reported experimental results, and provide a solid comparative base for analysis, predictions, and comparisons with numerical and experimental data.     

A numerical and experimental study of mass transfer in the artificial kidney

To develop a more efficient and optimal artificial kidney, many experimental approaches have been used to study mass transfer inside, outside, and cross hollow fiber membranes with different kinds of membranes, solutes, and flow rates as parameters. However, these experimental approaches are expensive and time consuming. Numerical calculation and computer simulation is an effective way to study mass transfer in the artificial kidney, which can save substantial time and reduce experimental cost. This paper presents a new model to simulate mass transfer in artificial kidney by coupling together shell-side, lumen-side, and transmembrane flows. Darcy's equations were employed to simulate shell-side flow, Navier-Stokes equations were employed to simulate lumen-side flow, and Kedem-Katchalsky equations were used to compute transmembrane flow. Numerical results agreed well with experimental results within 10% error. Numerical results showed the nonuniform distribution of flow and solute concentration in shell-side flow due to the entry/exit effect and Darcy permeability. In the shell side, the axial velocity in the periphery is higher than that in the center. This numerical model presented a clear insight view of mass transfer in an artificial kidney and may be used to help design an optimal artificial kidney and its operation conditions to improve hemodialysis.     

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Numerical and exact solutions for continuum of alleles models

 Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method. Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002 Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection