Stochastic Biological Models Research Articles

On stochastic models in biology and medicine

Stochastic models along with deterministic models are successfully used for mathematical description of biological processes. They apply knowledge from probability theory and mathematical statistics to analyze specific characteristics of living systems. The paper is devoted to some stochastic models of various phenomena in biology and medicine. Basic concepts and definitions used in classical probability models are considered and illustrated by several examples with solutions. The stochastic kinetic modeling approach is described. A new kinetic model of autoimmune disease is presented. It is a system of nonlinear partial integro-differential equations supplemented by corresponding initial conditions. The modeling problem is solved computationally.

Asymptotic behaviors of stochastic epidemic models with jump-diffusion

In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations.

Stochastic models in biology and the invariance problem

Invariance is a crucial property for many mathematical models of biological or biomedical systems,meaning that the solutions necessarily take values in a given range. Thisproperty reflects physical or biological constraints of the system and is independent ofthe model under consideration. While most classical deterministic models respectinvariance, many recent stochastic extensions violate thisfundamental property. Based on an invariance criterion for systems of stochasticdifferential equations we discuss several stochastic models exhibitingthis behavior and propose classes of modified, admissible models as possible resolutions.Numerical simulations are presented to illustrate the model behavior.

Parameter discovery in stochastic biological models using simulated annealing and statistical model checking.

Stochastic models are increasingly used to study the behaviour of biochemical systems. While the structure of such models is often readily available from first principles, unknown quantitative features of the model are incorporated into the model as parameters. Algorithmic discovery of parameter values from experimentally observed facts remains a challenge for the computational systems biology community. We present a new parameter discovery algorithm that uses simulated annealing, sequential hypothesis testing, and statistical model checking to learn the parameters in a stochastic model. We apply our technique to a model of glucose and insulin metabolism used for in-silico validation of artificial pancreata and demonstrate its effectiveness by developing parallel CUDA-based implementation for parameter synthesis in this model.

On designing multicore-aware simulators for systems biology endowed with OnLine statistics.

The paper arguments are on enabling methodologies for the design of a fully parallel, online, interactive tool aiming to support the bioinformatics scientists .In particular, the features of these methodologies, supported by the FastFlow parallel programming framework, are shown on a simulation tool to perform the modeling, the tuning, and the sensitivity analysis of stochastic biological models. A stochastic simulation needs thousands of independent simulation trajectories turning into big data that should be analysed by statistic and data mining tools. In the considered approach the two stages are pipelined in such a way that the simulation stage streams out the partial results of all simulation trajectories to the analysis stage that immediately produces a partial result. The simulation-analysis workflow is validated for performance and effectiveness of the online analysis in capturing biological systems behavior on a multicore platform and representative proof-of-concept biological systems. The exploited methodologies include pattern-based parallel programming and data streaming that provide key features to the software designers such as performance portability and efficient in-memory (big) data management and movement. Two paradigmatic classes of biological systems exhibiting multistable and oscillatory behavior are used as a testbed.

Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models.

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.