Stochastic models for toxicant-stressed populations

We obtain conditions for the existence of an invariant distribution on (0, ∞) for stochastic growth models of Ito type. We interpret the results in the case where the intrinsic growth rate is adjusted to account for the impact of a toxicant on the population. Comparisons with related results for ODE models by Hallamet al. are given, and consequences of taking the Stratonovich interpretation for the stochastic models are mentioned.     

Populations with Explicit Borders in Space and Time: Concept,Terminology, and Estimation of Characteristic Parameters

Biologists studying short-lived organisms have become aware of the need to recognize an explicit temporal extend of a population over a considerable time. In this article we outline the concept and the realm of populations with explicit spatial and temporary boundaries. We call such populations “temporally bounded populations”. In the concept, time is of the same importance as space in terms of a dimension to which a population is restricted. Two parameters not available for populations that are only spatially defined characterise temporally bounded populations: total population size, which is the total number of individuals present within the temporal borders, and total residence time, which is the sum of the residence times of all individuals. We briefly review methods to estimate these parameters. We illustrate the concept for the large blue butterfly (Maculinea nausithous) and outline insights into ecological and conservation-relevant processes that cannot be gained without the use of the concept.     

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.     

Stoichiometry in producer-grazer systems: Linking energy flow with element cycling

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator—prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer—grazer systems differs radically from nutrient enrichment. Hence, expressing producer—grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.     

Individual-based models for stage structured populations: formulation of “no regression” development equations

Individual-based models describe the growth dynamics of a population by performing numerical simulations of the life histories of its individuals. The life of an individual is determined by the basic processes of development, reproduction and mortality. In this paper the model equations for the development process are stochastic difference equations with discrete time and describe the time evolution of the status of an individual, in terms of a physiological age. We address the formulation of development models, when “regression” effects (defined as negative development) on the status of an individual are forbidden; this is a natural assumption when the physiological age is defined in terms of an abstract non-decreasing indicator measuring the maturity or the percentage of development. Different stochastic models of the development process are presented, and their behaviours are analyzed by varying the stochasticity level, which takes into account the degree of intraspecific variability. Moreover, remarks on the choice of the time step are reported.     

Fides: Reliable trust-region optimization for parameter estimation of ordinary differential equation models

Ordinary differential equation (ODE) models are widely used to study biochemical reactions in cellular networks since they effectively describe the temporal evolution of these networks using mass action kinetics. The parameters of these models are rarely known a priori and must instead be estimated by calibration using experimental data. Optimization-based calibration of ODE models on is often challenging, even for low-dimensional problems. Multiple hypotheses have been advanced to explain why biochemical model calibration is challenging, including non-identifiability of model parameters, but there are few comprehensive studies that test these hypotheses, likely because tools for performing such studies are also lacking. Nonetheless, reliable model calibration is essential for uncertainty analysis, model comparison, and biological interpretation.We implemented an established trust-region method as a modular Python framework (fides) to enable systematic comparison of different approaches to ODE model calibration involving a variety of Hessian approximation schemes. We evaluated fides on a recently developed corpus of biologically realistic benchmark problems for which real experimental data are available. Unexpectedly, we observed high variability in optimizer performance among different implementations of the same mathematical instructions (algorithms). Analysis of possible sources of poor optimizer performance identified limitations in the widely used Gauss-Newton, BFGS and SR1 Hessian approximation schemes. We addressed these drawbacks with a novel hybrid Hessian approximation scheme that enhances optimizer performance and outperforms existing hybrid approaches. When applied to the corpus of test models, we found that fides was on average more reliable and efficient than existing methods using a variety of criteria. We expect fides to be broadly useful for ODE constrained optimization problems in biochemical models and to be a foundation for future methods development.     

Fault tolerance in the cardiac ganglion of the lobster

Canavier et al. (1997) used phase response curves (PRCs) of individual oscillators to characterize the possible modes of phase-locked entrainment of an N-oscillator ring network. We extend this work by developing a mathematical criterion to determine the local stability of such a mode based on the PRCs. Our method does not assume symmetry; neither the oscillators nor their connections need be identical. To use these techniques for predicting modes and determining their stability, one need only determine the PRC of each oscillator in the ring either experimentally or from a computational model. We show that network stability cannot be determined by simply testing the ability of each oscillator to entrain the next. Stability depends on the number of neurons in the ring, the type of mode, and the slope of each PRC at the point of entrainment of the respective neuron. We also describe simple criteria which are either necessary or sufficient for stability and examine the implications of these results. Received: 2 April 1998 / Accepted in revised form: 2 July 1998     

Analysis of a model for macroparasitic infection with variable aggregation and clumped infections

 A model for macroparasitic infection with variable aggregation is considered. The starting point is an immigration-and-death process for parasites within a host, as in [3]; it is assumed however that infections will normally occur with several larvae at the same time. Starting from here, a four-dimensional, where free-living larvae are explicitly considered, and a three-dimensional model are obtained with same methods used in [26]. The equilibria of these models are found, their stability is discussed, as well as some qualitative features. It has been found that the assumption of “clumped” infections may have dramatic effects on the aggregation exhibited by these models. Infections with several larvae at the same time also increases the stability of the endemic equilibria of these models, and makes the occurrence of subcritical bifurcations (and consequently multiple equilibria) slightly more likely. The results of the low-dimensional model have also been compared to numerical simulations of the infinite system that describes the immigration-and-death process. It appears that the results of the systems are, by and large, in close correspondence, except for a parameter region where the four-dimensional model exhibits unusual properties, such as the occurrence of multiple disease-free equilibria, that do not appear to be shared by the infinite system. Received 28 October 1996; in revised form 11 April 1997     

Coexistence of different serotypes of dengue virus

 We formulate a non–linear system of differential equations that models the dynamics of dengue fever. This disease is produced by any of the four serotypes of dengue arbovirus. Each serotype produces permanent immunity to it, but only a certain degree of cross–immunity to heterologous serotypes. In our model we consider the relation between two serotypes. Our interest is to analyze the factors that allow the invasion and persistence of different serotypes in the human population. Analysis of the model reveals the existence of four equilibrium points, which belong to the region of biological interest. One of the equilibrium points corresponds to the disease–free state, the other three equilibria correspond to the two states where just one serotype is present, and the state where both serotypes coexist, respectively. We discuss conditions for the asymptotic stability of equilibria, supported by analytical and numerical methods. We find that coexistence of both serotypes is possible for a large range of parameters. Received: 7 July 1998 / Revised version: 12 July 2002 / Published online: 26 September 2002 Keywords or phrases: Dengue fever – Primary and secondary infections – Serotype – Coexistence – Threshold – Basic reproduction number – Persistence     

Analytic steady-state space use patterns and rapid computations in mechanistic home range analysis

Mechanistic home range models are important tools in modeling animal dynamics in spatially complex environments. We introduce a class of stochastic models for animal movement in a habitat of varying preference. Such models interpolate between spatially implicit resource selection analysis (RSA) and advection-diffusion models, possessing these two models as limiting cases. We find a closed-form solution for the steady-state (equilibrium) probability distribution u* using a factorization of the redistribution operator into symmetric and diagonal parts. How space use is controlled by the habitat preference function w depends on the characteristic width of the animals’ redistribution kernel: when the redistribution kernel is wide relative to variation in w, u* ∝ w, whereas when it is narrow relative to variation in w, u* ∝ w ². In addition, we analyze the behavior at discontinuities in w which occur at habitat type boundaries, and simulate the dynamics of space use given two-dimensional prey-availability data, exploring the effect of the redistribution kernel width. Our factorization allows such numerical simulations to be done extremely fast; we expect this to aid the computationally intensive task of model parameter fitting and inverse modeling.      

The Coalescence of Intrahost HIV Lineages Under Symmetric CTL Attack

Cytotoxic T lymphocytes (CTLs) are immune system cells that are thought to play an important role in controlling HIV infection. We develop a stochastic ODE model of HIV-CTL interaction that extends current deterministic ODE models. Based on this stochastic model, we consider the effect of CTL attack on intrahost HIV lineages assuming that CTLs attack several epitopes with equal strength. In this setting, we introduce a limiting version of our stochastic ODE under which we show that the coalescence of HIV lineages can be described through Poisson-Dirichlet distributions. Through numerical experiments, we show that our results under the limiting stochastic ODE accurately reflect HIV lineages under CTL attack when the HIV population size is on the low end of its hypothesized range. Current techniques of HIV lineage construction depend on the Kingman coalescent. Our results give an explicit connection between CTL attack and HIV lineages.     

Hypothalamus–Pituitary–Thyroid Feedback Control: Implications of Mathematical Modeling and Consequences for Thyrotropin (TSH) and Free Thyroxine (FT4) Reference Ranges

The components of thyrotropic feedback control are well established in mainstream physiology and endocrinology, but their relation to the whole system’s integrated behavior remains only partly understood. Most modeling research seeks to derive a generalized model for universal application across all individuals. We show how parameterizable models, based on the principles of control theory, tailored to the individual, can fill these gaps. We develop a system network describing the closed-loop behavior of the hypothalamus–pituitary (HP)–thyroid interaction and the set point targeted by the control system at equilibrium. The stability of this system is defined by using loop gain conditions. Defined points of homeostasis of the hypothalamus–pituitary–thyroid (HPT) feedback loop found at the intersections of the HP and thyroid transfer functions at the boundaries of normal reference ranges were evaluated by loop gain calculations. At equilibrium, the feedback control approaches a point defined in both dimensions by a [TSH]–[FT4] coordinate for which the loop gain is greater than unity. This model describes the emergence of homeostasis of the HPT axis from characteristic curves of HP and thyroid, thus supporting the validity of the translation between physiological knowledge and clinical reference ranges.     

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible → infected → removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback–Leibler divergence for the two fitted models to these data.     

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

Directed formation of chromosomal deletions in Salmonella typhimurium : targeting of specific genes induced during infection

In vivo expression technology (IVET) has resulted in the isolation of more than 100 Salmonella typhimurium genes that are induced during infection. Many of these in vivo induced (ivi) genes, as well as other virulence genes, are clustered in regions of the chromosome that are specific for Salmonella and are not present in Escherichia coli (e.g., pathogenicity islands). It would be desirable to be able to delete such putative virulence regions of the chromosome, and if the deletion removes genes that play a role in pathogenesis subsequent efforts can then be focused on individual genes that reside within that region. We therefore have developed a strategy for constructing chromosomal deletions which are not limited in size, have defined endpoints with a selectable marker at the joint point, and are not dependent on prior knowledge of sequences contained within the deleted region. Such deletion strategies can be applied to almost any bacterium with homologous recombination and to plasmid-based mutational systems where homologous recombination is not desired or feasible. Received: 6 October 1997 / Accepted: 30 December 1997     

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.      

Bayesian Wombling for Spatial Point Processes

Summary In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point‐referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point‐process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point‐level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point‐referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point‐referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.     

Statistical strategies for avoiding false discoveries in metabolomics and related experiments

Many metabolomics, and other high-content or high-throughput, experiments are set up such that the primary aim is the discovery of biomarker metabolites that can discriminate, with a certain level of certainty, between nominally matched ‘case’ and ‘control’ samples. However, it is unfortunately very easy to find markers that are apparently persuasive but that are in fact entirely spurious, and there are well-known examples in the proteomics literature. The main types of danger are not entirely independent of each other, but include bias, inadequate sample size (especially relative to the number of metabolite variables and to the required statistical power to prove that a biomarker is discriminant), excessive false discovery rate due to multiple hypothesis testing, inappropriate choice of particular numerical methods, and overfitting (generally caused by the failure to perform adequate validation and cross-validation). Many studies fail to take these into account, and thereby fail to discover anything of true significance (despite their claims). We summarise these problems, and provide pointers to a substantial existing literature that should assist in the improved design and evaluation of metabolomics experiments, thereby allowing robust scientific conclusions to be drawn from the available data. We provide a list of some of the simpler checks that might improve one’s confidence that a candidate biomarker is not simply a statistical artefact, and suggest a series of preferred tests and visualisation tools that can assist readers and authors in assessing papers. These tools can be applied to individual metabolites by using multiple univariate tests performed in parallel across all metabolite peaks. They may also be applied to the validation of multivariate models. We stress in particular that classical p-values such as “p < 0.05”, that are often used in biomedicine, are far too optimistic when multiple tests are done simultaneously (as in metabolomics). Ultimately it is desirable that all data and metadata are available electronically, as this allows the entire community to assess conclusions drawn from them. These analyses apply to all high-dimensional ‘omics’ datasets.     

Two Bayesian methods for estimating parameters of the von Bertalanffy growth equation

The von Bertalanffy growth equation (VBGE) is commonly used in ecology and fisheries management to model individual growth of an organism. Generally, a nonlinear regression is used with length-at-age data to recover key life history parameters: L _∞ (asymptotic size), k (the growth coefficient), and t ₀ (a time used to calculate size at age 0). However, age data are often unavailable for many species of interest, which makes the regression impossible. To confront this problem, we have developed a Bayesian model to find L _∞ using only length data. We use length-at-age data for female blue shark, Prionace glauca, to test our hypothesis. Preliminary comparisons of the model output and the results of a nonlinear regression using the VBGE show similar estimates of L _∞ . We also developed a full Bayesian model that fits the VBGE to the same data used in the classical regression and the length-based Bayesian model. Classical regression methods are highly sensitive to missing data points, and our analysis shows that fitting the VBGE in a Bayesian framework is more robust. We investigate the assumptions made with the traditional curve fitting methods, and argue that either the full Bayesian or the length-based Bayesian models are preferable to classical nonlinear regressions. These methods clarify and address assumptions␣made in classical regressions using von Bertalanffy growth and facilitate more detailed stock assessments of species for which data are sparse.