Opinion dynamics with continuous age structure

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.

Two algorithms that combine Brownian dynami cs (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface, which partitions the domain, and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that the overlap region is required to accurately compute variances using PBD simulations. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented. © 2013 Society for Industrial and Applied Mathematics.

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.

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

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.

Stochastic differential equation (SDE) models are formulated for intra-host virus-cell dynamics during the early stages of viral infection, prior to activation of the immune system. The SDE models incorporate more realism into the mechanisms for viral entry and release than ordinary differential equation (ODE) models and show distinct differences from the ODE models. The variability in the SDE models depends on the concentration, with much greater variability for small concentrations than large concentrations. In addition, the SDE models show significant variability in the timing of the viral peak. The viral peak is earlier for viruses that are released from infected cells via bursting rather than via budding from the cell membrane.

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.

Opinion dynamics in relative agreement models seen as an extension of bounded confidence ones, involve a new agents’ variable usually called opinion uncertainty and have higher level of complexity than that of bounded confidence models. After revising the meaning of the opinion uncertainty variable we conclude that it has to be interpreted as the agent’s opinion toleration, that changes the type of the variable from the social to the psychological one. Since the convergence rates to the stationary states in dynamics of sociological and psychological variables are in general different, we study the effect of agents’ psychology and social environment interaction on the opinion dynamics, using concord and partial antagonism relative agreement model in small-world and scale-free societies. The model considers agents of two psychological types, concord and partial antagonism, that differs it from other relative agreement models. The analysis of opinion dynamics in particular scenarios was used in this work. Simulation results show the importance of this approach, in particular, the effect of small variations in initial conditions on the final state. We found significant mutual influence of opinion and toleration resulting in a variety of statistically stationary states such as quasi consensus, polarization and fragmentation of society into opinion and toleration groups of different configurations. Consensus was found to be rather rare state in a wide range of model parameters, especially in scale-free societies. The model demonstrates different opinion and toleration dynamics in small-world and scale-free societies.

We consider an optimal harvesting problem for a population with continuous age and time structure. On the basis of a version of Pontryagin's principle, we investigate the form of an optimal solution in a special situation..

Author response: Collective dynamics support group drumming, reduce variability, and stabilize tempo drift

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.

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.

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.

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.

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.