Dynamic analysis of an HIV stochastic time-delay differential equations incorporating two infection pathways and CTL immune response.

Dynamical analysis of a delayed ratio-dependent prey–predator model within fluctuating environment

Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Background and ObjectivesLevodopa concentration in patients with Parkinson’s disease is frequently modelled with ordinary differential equations (ODEs). Here, we investigate a pharmacokinetic model of plasma levodopa concentration in patients with Parkinson’s disease by introducing stochasticity to separate the intra-individual variability into measurement and system noise, and to account for auto-correlated errors. We also investigate whether the induced stochasticity provides a better fit than the ODE approach.MethodsIn this study, a system noise variable is added to the pharmacokinetic model for duodenal levodopa/carbidopa gel (LCIG) infusion described by three ODEs through a standard Wiener process, leading to a stochastic differential equations (SDE) model. The R package population stochastic modelling (PSM) was used for model fitting with data from previous studies for modelling plasma levodopa concentration and parameter estimation. First, the diffusion scale parameter (σw), measurement noise variance, and bioavailability are estimated with the SDE model. Second, σw is fixed to certain values from 0 to 1 and bioavailability is estimated. Cross-validation was performed to compare the average root mean square errors (RMSE) of predicted plasma levodopa concentration.ResultsBoth the ODE and the SDE models estimated bioavailability to be approximately 75%. The SDE model converged at different values of σw that were significantly different from zero. The average RMSE for the ODE model was 0.313, and the lowest average RMSE for the SDE model was 0.297 when σw was fixed to 0.9, and these two values are significantly different.ConclusionsThe SDE model provided a better fit for LCIG plasma levodopa concentration by approximately 5.5% in terms of mean percentage change of RMSE.

We develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy

Continuous time Markov chain (CTMC) and It stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model). Differential equations are derived for the moments of the distribution for each stochastic model. Each moment differential equation depends on higher-order moments. Assumptions are made regarding higher-order moments to form a finite, solvable system. Conditions are given under which the CTMC and SDE BID models have the same moment solution or the same stationary solution. The close agreement between the CTMC and SDE models is illustrated in three numerical examples based on normal or log-normal moment closure assumptions.

Stochastic modeling plays an essential role in the study of noise and random fluctuations in biochemical signaling of neural systems. Here, we study numerically noisy fluctuations produced by two different stochastic differential equation models, the chemical Langevin equation (CLE) model [1] and the rate constant stochastic differential equation (rcSDE) model [2], using a biologically realistic neuronal protein kinase C activation pathway [3] as a case study. We compare the CLE and rcSDE models by studying the noise power levels in different system volumes and with different model parameters. In this context, we also assess the problem of negative concentrations by computing the average number of state vectors containing negative concentrations within a single simulation run. Based on the information obtained from the examination of noise power levels, we then choose appropriate model parameters for the rcSDE model and the corresponding system volume for the CLE model. Using these parameters and the volume, we finally simulate the models and compare the results using multiresolution analysis. In the simulations we present here, the rcSDE model is capable of producing the same level of the average power of noise with a notably smaller amount of negative concentrations than the CLE model. This difference is considerable and it suggests that in certain simulation settings, it might be reasonable to use rcSDE model as an approximate simulation approach if it is feasible to use the slightly different interpretation of the nature of noise. A more detailed multiresolution analysis reveals that the resulting noise processes differ at the higher frequencies. In many studies these differences at high frequencies are not of interest as neural biochemical systems often produce relatively stronger fluctuations at low frequencies. In some cases, however, noise might be important for the system function and, in such situations, it is crucial to make sure that the noise is modeled in an appropriate way. Whether the appropriate way is the CLE, the rcSDE model, or some other model is dictated by the application. Different models are based on different premises and it is modeler's task to pick up an appropriate model. We conclude that introduction of techniques from the field of signal processing may help assessing the role of noise in neural systems and what type of model to use. This study shows how the orthogonal wavelet representation or similar multiscale decompositions can be successfully applied to the study of noise in biochemical processes of neural systems.

Stochastic differential equation (SDE) models are useful in describing complex dynamical systems in science and engineering. In this study, we consider a monitoring procedure for an early detection of dispersion parameter change in SDE models. The proposed scheme provides a useful diagnostic analysis for phase I retrospective study and develops a flexible and effective control chart for phase II prospective monitoring. A standardized control chart is constructed, and a bootstrap method is used to estimate the mean and variance of the monitoring statistic. The control limit is obtained as an upper percentile of the maximum value of a standard Wiener process. The proposed procedure appears to have a manageable computational complexity for online implementation and also to be effective in detecting changes. We also investigate the performance of the exponentially weighted mean squared control charts for the continuous SDE processes. A simulation method is used to study the empirical sizes and the average run length characteristics of the proposed scheme, which also demonstrates the effectiveness of our method. Finally, we provide an empirical example for illustration. Copyright © 2014 John Wiley & Sons, Ltd.

Nonlinear Model Predictive Control (NMPC) based on stochastic differential equation (SDE) models offers a systematic method for implementation of NMPC. We demonstrate how the filtering and prediction with SDE models can be used for experimental design (input design), parameter estimation and system identification as well as monitoring and fault detection. Also, we address the importance of disturbance modelling for SDE systems and provide a potential based approach inspired by barrier functions. Efficient numerical procedures for the optimal control problem (OCP) based on the SDE-based identified filtering and prediction model are described. High performance scientific computing is used for uncertainty quantification of the closed-loop system. We illustrate the methods using a number of examples from reactive systems as they arise in chemical engineering, biotechnology, biomedicine, and energy systems. The range and broadness of these systems demonstrates the power of a uniform system approach based on merging systems and control, high-performance scientific computing, and chemical engineering science.

Stochastic Differential Equation (SDE) models are often used to model the dynamics of complex biological systems. The stochastic nature of these models means that some behaviors are more likely than others. It is often the case that a model's primary purpose is to study rare but interesting or important behaviors, such as the formation of a tumor, or the failure of a cyber-physical system. Unfortunately, due to the limited availability of analytic methods for SDEs, stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviors. Naturally, the cost of stochastic simulations increases with the rarity of the behavior under consideration. To address this problem, we introduce a new algorithm, RESERCHE, that is specifically designed to quantify the likelihood of rare but interesting behaviors in SDE models. Our approach relies on the use of temporal logics for specifying rare behaviors of possible interest, and on the ability of bit-vector decision procedures to reason exhaustively about fixed precision arithmetic. We also compute the probability of an observed behavior under the assumption of Gaussian noise.

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Background: Growth trajectories are highly variable between children, making epidemiological analyses challenging both to the identification of malnutrition interventions at the population level and also risk assessment at individual level. We introduce stochastic differential equation (SDE) models into child growth research. SDEs describe flexible dynamic processes comprising: drift - gradual smooth changes – such as physiology or gut microbiome, and diffusion - sudden perturbations, such as illness or infection. Methods: We present a case study applying SDE models to child growth trajectory data from the Haydom, Tanzania and Venda, South Africa sites within the MAL-ED cohort. These data comprise n=460 children aged 0-24 months. A comparison with classical curve fitting (linear mixed models) is also presented. Results: The SDE models offered a wide range of new flexible shapes and parameterizations compared to classical additive models, with performance as good or better than standard approaches. The predictions from the SDE models suggest distinct longitudinal clusters that form distinct ‘streams’ hidden by the large between-child variability. Conclusions: Using SDE models to predict future growth trajectories revealed new insights in the observed data, where trajectories appear to cluster together in bands, which may have a future risk assessment application. SDEs offer an attractive approach for child growth modelling and potentially offer new insights.

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

The chance that a freeway will breakdown, transition from a free-flow to a congested state, is normally assumed to increase with an increase in traffic volume V (vehicles per unit time). In this paper, this assumption is challenged. Traffic density K (vehicles per unit length) proves to be a better predictor. Diffusion or stochastic differential equation (SDE) modeling is used to substantiate the claim. SDE modeling is especially useful in explaining the role that traffic noise (volatility) plays in breakdown. The SDE models take advantage of the unique properties of the geometric Brownian motion (gBM) and Ornstein-Uhlenbeck (OU) model structures. The breakdown probability model of π (K) and delay models provide accurate forecasts.