A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Bayesian inference for stochastic kinetic models using a diffusion approximation

This article is concerned with the Bayesian estimation of stochastic rate constants in the context of dynamic models of intracellular processes. The underlying discrete stochastic kinetic model is replaced by a diffusion approximation (or stochastic differential equation approach) where a white noise term models stochastic behavior and the model is identified using equispaced time course data. The estimation framework involves the introduction of m- 1 latent data points between every pair of observations. MCMC methods are then used to sample the posterior distribution of the latent process and the model parameters. The methodology is applied to the estimation of parameters in a prokaryotic autoregulatory gene network.     

An approach to the residence time distribution for stochastic multi-compartment models

Stochastic compartmental models are widely used in modeling processes such as drug kinetics in biological systems. This paper considers the distribution of the residence times for stochastic multi-compartment models, especially systems with non-exponential lifetime distributions. The paper first derives the moment generating function of the bivariate residence time distribution for the two-compartment model with general lifetimes and approximates the density of the residence time using the saddlepoint approximation. Then, it extends the distributional approach to the residence time for multi-compartment semi-Markov models combining the cofactor rule for a single destination and the analytic approach to the two-compartment model. This approach provides a complete specification of the residence time distribution based on the moment generating function and thus facilitates an easier calculation of high-order moments than the approach using the coefficient matrix. Applications to drug kinetics demonstrate the simplicity and usefulness of this approach.     

A comparison of three models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We compare three mathematical models of the IPR, fitting each of them to the same data set to determine ranges for the parameter values. Each of the fits indicates that fast activation of the receptor, followed by slow inactivation, is an important feature of the model, and also that the speed of inositol trisphosphate (IP₃) binding cannot necessarily be assumed to be faster than Ca²⁺ activation. In addition, the model which assumed saturating binding rates of Ca²⁺ to the IPR demonstrated the best fit. However, lack of convergence in the fitting procedure indicates that responses to step increases of [Ca²⁺] and [IP₃] provide insufficient data to determine the parameters unambiguously in any of the models.     

Stochastic niche structure and diversity maintenance in the T cell repertoire

The reliability of the immune response to pathogenic challenge depends critically on the size and diversity of the T cell repertoire. We study naïve T cell repertoire diversity maintenance by a stochastic model that incorporates the concept of competition between T cells for survival stimuli emanating from self-antigen presenting cells (APCs). In the mean field approximation we show that clonotype extinction is certain and compute mean extinction times. We introduce the concept of mean niche overlap and show that clones with a mean niche overlap greater than one have a short repertoire lifespan. This selection differential induces minimal recognition commonality between T cell receptors (TCRs) resulting in a diverse T cell repertoire.     

A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods

Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed. These methods rely on Markov chain Monte Carlo methods. Both temporal and non-temporal data are considered. The methods are illustrated with a number of examples featuring different models and datasets.     

A new class of non-linear stochastic population models with mass conservation

We study the effects of random feeding, growing and dying in a closed nutrient-limited producer/consumer system, in which nutrient is fully conserved, not only in the mean, but, most importantly, also across random events. More specifically, we relate these random effects to the closest deterministic models, and evaluate the importance of the various times scales that are involved. These stochastic models differ from deterministic ones not only in stochasticity, but they also have more details that involve shorter times scales. We tried to separate the effects of more detail from that of stochasticity. The producers have (nutrient) reserve and (body) structure, and so a variable chemical composition. The consumers have only structure, so a constant chemical composition. The conversion efficiency from producer to consumer, therefore, varies. The consumers use reserve and structure of the producers as complementary compounds, following the rules of Dynamic Energy Budget theory. Consumers die at constant specific rate and decompose instantaneously. Stochasticity is incorporated in the behaviour of the consumers, where the switches to handling and searching, as well as dying are Poissonian point events. We show that the stochastic model has one parameter more than the deterministic formulation without time scale separation for conversions between searching and handling consumers, which itself has one parameter more than the deterministic formulation with time scale separation for these conversions. These extra parameters are the contributions of a single individual producer and consumer to their densities, and the ratio of the two, respectively. The tendency to oscillate increases with the number of parameters. The focus bifurcation point has more relevance for the asymptotic behaviour of the stochastic model than the Hopf bifurcation point, since a randomly perturbed damped oscillation exhibits a behaviour similar to that of the stochastic limit cycle particularly near this bifurcation point. For total nutrient values below the focus bifurcation point, the system gradually becomes more confined to the direct neighbourhood of the isocline for which the producers do not change.     

A general comparison of relaxed molecular clock models

Several models have been proposed to relax the molecular clock in order to estimate divergence times. However, it is unclear which model has the best fit to real data and should therefore be used to perform molecular dating. In particular, we do not know whether rate autocorrelation should be considered or which prior on divergence times should be used. In this work, we propose a general bench mark of alternative relaxed clock models. We have reimplemented most of the already existing models, including the popular lognormal model, as well as various prior choices for divergence times (birth-death, Dirichlet, uniform), in a common Bayesian statistical framework. We also propose a new autocorrelated model, called the "CIR" process, with well-defined stationary properties. We assess the relative fitness of these models and priors, when applied to 3 different protein data sets from eukaryotes, vertebrates, and mammals, by computing Bayes factors using a numerical method called thermodynamic integration. We find that the 2 autocorrelated models, CIR and lognormal, have a similar fit and clearly outperform uncorrelated models on all 3 data sets. In contrast, the optimal choice for the divergence time prior is more dependent on the data investigated. Altogether, our results provide useful guidelines for model choice in the field of molecular dating while opening the way to more extensive model comparisons.     

A model for the interaction of two chemicals

We describe a method for studying the interaction of two anesthetic agents, Morphine and Midazolam, acting simultaneously in the same individual. Representing the levels of the two chemicals by diffusion processes, we assume their interaction is governed by a linear combination of the separate components. Pharmacological data is used to estimate the model parameters and, in particular, to determine the coefficient in the linear combination. This leads to the conclusion that the two chemicals have a counteractive effect.     

Stochastic von Bertalanffy models, with applications to fish recruitment

We consider three individual-based models describing growth in stochastic environments. Stochastic differential equations (SDEs) with identical von Bertalanffy deterministic parts are formulated, with a stochastic term which decreases, remains constant, or increases with organism size, respectively. Probability density functions for hitting times are evaluated in the context of fish growth and mortality. Solving the hitting time problem analytically or numerically shows that stochasticity can have a large positive impact on fish recruitment probability. It is also demonstrated that the observed mean growth rate of surviving individuals always exceeds the mean population growth rate, which itself exceeds the growth rate of the equivalent deterministic model. The consequences of these results in more general biological situations are discussed.     

Identification and comparison of stochastic metabolic/hemodynamic models (sMHM) for the generation of the BOLD signal

This paper extends a previously formulated deterministic metabolic/hemodynamic model for the generation of blood oxygenated level dependent (BOLD) responses to include both physiological and observation stochastic components (sMHM). This adds a degree of flexibility when fitting the model to actual data by accounting for un-modelled activity. We then show how the innovation method can be used to estimate unobserved metabolic/hemodynamic as well as vascular variables of the sMHM, from simulated and actual BOLD data. The proposed estimation method allowed for doing model comparison by calculating the model’s AIC and BIC. This methodology was then used to select between different neurovascular coupling assumptions underlying sMHM. The proposed framework was first validated on simulations and then applied to BOLD data from a motor task experiment. The models under comparison in the analysis of the actual data considered that vascular response was coupled to: (I) inhibition, (II) excitation, (III) both excitation and inhibition. Data was best described by model II, although model III was also supported.     

Time-varying Markov regression random-effect model with Bayesian estimation procedures: Application to dynamics of functional recovery in patients with stroke

The rates of functional recovery after stroke tend to decrease with time. Time-varying Markov processes (TVMP) may be more biologically plausible than time-invariant Markov process for modeling such data. However, analysis of such stochastic processes, particularly tackling reversible transitions and the incorporation of random effects into models, can be analytically intractable. We make use of ordinary differential equations to solve continuous-time TVMP with reversible transitions. The proportional hazard form was used to assess the effects of an individual’s covariates on multi-state transitions with the incorporation of random effects that capture the residual variation after being explained by measured covariates under the concept of generalized linear model. We further built up Bayesian directed acyclic graphic model to obtain full joint posterior distribution. Markov chain Monte Carlo (MCMC) with Gibbs sampling was applied to estimate parameters based on posterior marginal distributions with multiple integrands. The proposed method was illustrated with empirical data from a study on the functional recovery after stroke.     

A comparison of alternative methods to compute conditional genotype probabilities for genetic evaluation with finite locus models

An increased availability of genotypes at marker loci has prompted the development of models that include the effect of individual genes. Selection based on these models is known as marker-assisted selection (MAS). MAS is known to be efficient especially for traits that have low heritability and non-additive gene action. BLUP methodology under non-additive gene action is not feasible for large inbred or crossbred pedigrees. It is easy to incorporate non-additive gene action in a finite locus model. Under such a model, the unobservable genotypic values can be predicted using the conditional mean of the genotypic values given the data. To compute this conditional mean, conditional genotype probabilities must be computed. In this study these probabilities were computed using iterative peeling, and three Markov chain Monte Carlo (MCMC) methods – scalar Gibbs, blocking Gibbs, and a sampler that combines the Elston Stewart algorithm with iterative peeling (ESIP). The performance of these four methods was assessed using simulated data. For pedigrees with loops, iterative peeling fails to provide accurate genotype probability estimates for some pedigree members. Also, computing time is exponentially related to the number of loci in the model. For MCMC methods, a linear relationship can be maintained by sampling genotypes one locus at a time. Out of the three MCMC methods considered, ESIP, performed the best while scalar Gibbs performed the worst.     

A probabilistic model of cell size reduction in Pseudo-nitzschia delicatissima (Bacillariophyta)

The pennate planktonic diatom Pseudo-nitzschia delicatissima is very common in temperate marine waters and often responsible for blooms. Due to its surrounding rigid silicate frustrule the diatom undergoes successive size reduction as its vegetative reproduction cycle proceeds. Since a long time the life cycle of diatoms has raised scientific interest and some years ago extensive samples of Pseudo-nitzschia have been taken from coastal waters. Mating and cell size reduction experiments were carried out and served us as a data basis for a probabilistic model of cell size reduction.We applied a homogenous non-stationary continuous-time Markov chain to model the development of individual diatoms from an initial size of about until cell death which occurred when the size reached its low at about . In contrast to conventional curve fitting models we are capable of calculating confidence intervals for estimates of the population ages as well as integrate the process of auxospore formation into the model. We thus propose a unique way to describe the stationary size distribution in a diatom population in terms of cell division and auxospore formation probabilities of its individuals.     

Linear discrete models with different time scales

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order.