A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.     

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

THE THEORY OF DIFFUSION IN CELL MODELS : II. SOLUTION OF THE STEADY STATE FOR THREE DIFFUSING SUBSTANCES

The differential equations describing diffusion in cell models have been extended to include the simultaneous penetration of water and two salts. These equations have been solved for the steady state. Values for the concentrations in the steady state which may be computed from the equations compare favorably with the experimental values obtained by Osterhout, Kamerling, and Stanley. Moreover, it has been shown elsewhere that the solution for the steady state is essential to a discussion of the volume change or "growth" of phase C in the models and, by analogy, in living cells.     

Stochastic difference equation predictors of population fluctuations.

This paper studies the effectiveness of a class of linear statistical estimators called autoregressive and autoregressive-moving averages equations for mimicking and predicting the abundance fluctuations of three species of Drosophila censused at a pine plantation near Bogota, Colombia. A short introductory justification for the use of linear estimators is given followed by a brief discussion of the theoretical basis of statistical prediction. The assumptions of the method, fitting techniques, and use of the equations in forecasting are discussed. The mimicking ability of the equations is tested by comparing monte carlo simulations employing the fitted models to the observed fluctuations of the three species. Of the autoregressive equations fitted to each species two are judged successful and one less than successful. Autoregressive-moving averages models were found to be significantly worse predictors than the simpler autoregressive equations for these three species. The parameter estimates given by the preliminary estimation techniques are compared with the statistically efficient least squares estimates. The estimates compare well for most of the autoregressive models, but the parameter estimates for the autoregressive-moving averages models were misleading.     

Philosophical Aspects of Measurements, Equations and Inferences in Plant Growth Studies

Growth analysis is based on equations that are ‘identities’because they are algebraically self-evident, whereas the moredeterministic models of plant growth are based on ‘conditionalequations’ that represent quantitative hypotheses. Growthanalytical studies consequently focus on the values of the componentsand not on the validity of the equations, whereas ‘validation’is a prime concern for other growth models. Implications ofmeasurement theory, of dependent and independent variables andof compensating components arise in the use of both types ofequation for quantifying growth. There is now available a rangeof approaches, from traditional growth analysis, through variousdevelopments of growth analysis including light conversion analysis,to complex mechanistic models of growth. Growth analysis, yield component analysis, light conversion analysis, mathematical models, measurement theory, derived quantities, independent variables, equations of growth     

Evaluation of enteric methane prediction equations for dairy cows used in whole farm models

The importance of evaluating greenhouse gas (GHG) emissions from dairy cows within the whole farm setting is being realized as more important than evaluating these emissions in isolation. Current whole farm models aimed at evaluating GHG emissions make use of simple regression equations to predict enteric methane (CH₄) production. The objective of the current paper is to evaluate the performance of nine CH₄ prediction equations that are currently being used in whole farm GHG models. Data used to evaluate the prediction equations came from a collection of individual (IND) and treatment averaged (TRT) data. Equations were compared based on mean square prediction error (MSPE) and concordance correlation coefficient (CCC) analysis. In general, predictions were poor, with root MSPE (as a percentage of observed mean) values ranging from 20.2 to 52.5 for the IND database and from 24.0 to 38.2 for the TRT database and CCC values ranging from 0.009 to 0.493 for the IND database and from 0.000 to 0.271 for the TRT database. Overall, the equations of Moe & Tyrrell and IPCC Tier II performed best on the IND dataset, and the equations of Moe & Tyrrell and Kirchgeßner et al., performed best on the TRT dataset. Results show that the simple more generalized equations performed worse than those that attempted to represent important aspects of diet composition, but in general significant amounts of bias and deviation of the regression slope from unity existed for all equations. The low prediction accuracy of CH₄ equations in whole farm models may introduce substantial error into inventories of GHG emissions and lead to incorrect mitigation recommendations.     

Rotational distortion in conventional allometric analyses

Three data sets from the recent literature were submitted to new analyses to illustrate the rotational distortion that commonly accompanies traditional allometric analyses and that often causes allometric equations to be inaccurate and misleading. The first investigation focused on the scaling of evaporative water loss to body mass in passerine birds; the second was concerned with the influence of body size on field metabolic rates of rodents; and the third addressed interspecific variation in kidney mass among primates. Straight lines were fitted to logarithmic transformations by Ordinary Least Squares and Generalized Linear Models, and the resulting equations then were re-expressed as two-parameter power functions in the original arithmetic scales. The re-expressed models were displayed on bivariate graphs together with tracings for equations fitted directly to untransformed data by nonlinear regression. In all instances, models estimated by back-transformation failed to describe major features of the arithmetic distribution whereas equations fitted by nonlinear regression performed quite well. The poor performance of equations based on models fitted to logarithms can be traced to the increased weight and leverage exerted in those analyses by observations for small species and to the decreased weight and leverage exerted by large ones. The problem of rotational distortion can be avoided by performing exploratory analysis on untransformed values and by validating fitted models in the scale of measurement.     

Intelligent modelling of bioprocesses: a comparison of structured and unstructured approaches

This contribution moves in the direction of answering some general questions about the most effective and useful ways of modelling bioprocesses. We investigate the characteristics of models that are good at extrapolating. We trained three fully predictive models with different representational structures (differential equations, differential equations with inheritance of rates and a network of reactions) on Saccharopolyspora erythraea shake flask fermentation data using genetic programming. The models were then tested on unseen data outside the range of the training data and the resulting performances were compared. It was found that constrained models with mathematical forms analogous to internal mass balancing and stoichiometric relations were superior to flexible unconstrained models, even though no a priori knowledge of this fermentation was used.Paper presented at the international conference on trends in monitoring and control of life science applications, 7–8 October 2002, Lyngby, Denmark.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Interdependency and hierarchy of exact and approximate epidemic models on networks

Over the years numerous models of \(SIS\) (susceptible \(\rightarrow \) infected \(\rightarrow \) susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure.     

Dielectric permittivity of quantum plasma. Part I

Collisionless quantum plasma models based on the Schröbinger, Klein-Gordon, Dirac, and Pauli equations are considered. The transverse and longitudinal dielectric permittivities of isotropic quantum plasma are calculated in the frameworks of the models based on the Schröbinger and Klein-Gordon equations without allowance for the particle spin. Dispersion relations for transverse-longitudinal waves in beams of spinless quantum particles are derived, and the simplest quantum waves are analyzed.     

On the Equivalence of Mathematical Models For Cell Proliferation Kinetics

Various types of mathematical models, such as partial differential equations, ordinary differential equations and difference equations, are available in the literature to describe the kinetics of cell proliferation, and different studies of cell kinetic phenomena have been conducted using these models. This paper discusses the equivalence between the different models identifying the conditions and approximations under which one type of models may be derived from another. Such an equivalence study is highly useful for an integration of the diverse results that have been obtained using different models in order to gain a more complete understanding of cell kinetic phenomena.     

Deterministic epidemic models on contact networks: correlations and unbiological terms

The relationship between system-level and subsystem-level master equations is investigated and then utilised for a systematic and potentially automated derivation of the hierarchy of moment equations in a susceptible-infectious-removed (SIR) epidemic model. In the context of epidemics on contact networks we use this to show that the approximate nature of some deterministic models such as mean-field and pair-approximation models can be partly understood by the identification of implicit anomalous terms. These terms describe unbiological processes which can be systematically removed up to and including the nth order by nth order moment closure approximations. These terms lead to a detailed understanding of the correlations in network-based epidemic models and contribute to understanding the connection between individual-level epidemic processes and population-level models. The connection with metapopulation models is also discussed. Our analysis is predominantly made at the individual level where the first and second order moment closure models correspond to what we term the individual-based and pair-based deterministic models, respectively. Matlab code is included as supplementary material for solving these models on transmission networks of arbitrary complexity.     

Variability,vagueness and comparison methods for ecological models

A unified approach is presented for the construction and analysis of models for the dynamics of populations and communities in the presence of temporal variability, vague density dependence, chaos or analytical intractability. The approach is based on comparisons involving simpler models which provide ceilings and floors to the densities predicted by the full models. The method is applied to examples of several types of models, including difference equations, ordinary differential equations, non-linear Leslie matrices and reaction-diffusion equations. The models treated describe various ecological phenomena including self-regulation, competition, predator-prey interactions, age structure and spatial structure. Some results needed for the analysis of matrix models and patch models are given in the Appendix. Research partially supported by NSF grant DMS-93-03708.     

A review of recent developments in modeling of microbial growth kinetics and intraparticle phenomena in solid-state fermentation

Mathematical models are important tools for optimizing the design and operation of solid-state fermentation (SSF) bioreactors. Such models must describe the kinetics of microbial growth, how this is affected by the environmental conditions and how this growth affects the environmental conditions. This is done at two levels of sophistication. In many bioreactor models the kinetics are described by simple empirical equations. However, other models that address the interaction of growth with intraparticle diffusion of enzymes, hydrolysis products and O₂ with the use of mechanistic equations have also been proposed, and give insights into how these microscale processes can potentially limit the overall performance of a bioreactor. The current article reviews the advances that have been made in both the empirical- and mechanistic-type kinetic models and discusses the insights that have been achieved through the modeling work and the improvements to models that will be necessary in the future.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999     

Rapid estimation of nutrients in chicken manure during plant-field composting using physicochemical properties

Regression equations which relate livestock and poultry manure nutrient content to its several physicochemical properties have been reported by previous researchers. This study explores the feasibility and efficiency to determine the nutrients (TN; TP; TK; Cu and Zn) in chicken manure during composting using physicochemical properties (pH, EC and DM), and compares the performances of regression equations in this study with those in the literature. The results show that DM is the best predictor to construct the single linear regressions for all the nutrients (R2≥0.84, p<0.001). In addition, the multiple linear regression equations based on DM and pH are all notable. These findings show the potential of physicochemical models for TN, TP, TK, Cu and Zn with more convenience and rapidness, but further research is needed to develop better models with higher accuracy for the above and other more nutrients.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.