Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation

Sustained spiral waves, also known as rotors, are pivotal mechanisms in persistent atrial fibrillation (AF). Stochasticity is inevitable in nonlinear biological systems such as the heart; however, it is unclear how noise affects the instability of spiral waves in human AF. This study presents a stochastic two-dimensional mathematical model of human AF and explores how Gaussian white noise affects the instability of spiral waves. In homogeneous tissue models, Gaussian white noise may lead to spiral-wave meandering and wavefront break-up. As the noise intensity increases, the spatial dispersion of phase singularity (PS) points increases. This finding indicates the potential AF-protective effects of cardiac system stochasticity by destabilizing the rotors. By contrast, Gaussian white noise is unlikely to affect the spiral-wave instability in the presence of localized scar or fibrosis regions. The PS points are located at the boundary or inside the scar/fibrosis regions. Localized scar or fibrosis may play a pivotal role in stabilizing spiral waves regardless of the presence of noise. This study suggests that fibrosis and scars are essential for stabilizing the rotors in stochastic mathematical models of AF. Further patient-derived realistic modeling studies are required to confirm the role of scar/fibrosis in AF pathophysiology.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

Bayesian spatial modelling of terrestrial radiation in Switzerland

The geographic variation of terrestrial radiation can be exploited in epidemiological studies of the health effects of protracted low-dose exposure. Various methods have been applied to derive maps of this variation. We aimed to construct a map of terrestrial radiation for Switzerland. We used airborne γ-spectrometry measurements to model the ambient dose rates from terrestrial radiation through a Bayesian mixed-effects model and conducted inference using Integrated Nested Laplace Approximation (INLA). We predicted higher levels of ambient dose rates in the alpine regions and Ticino compared with the western and northern parts of Switzerland. We provide a map that can be used for exposure assessment in epidemiological studies and as a baseline map for assessing potential contamination.

Estimating ambient air pollutant levels in Suzhou through the SPDE approach with R-INLA

Spatio-temporal models of ambient air pollution can be used to predict pollutant levels across a geographical region. These predictions may then be used as estimates of exposure for individuals in analyses of the health effects of air pollution. Integrated nested Laplace approximations is a method for Bayesian inference, and a fast alternative to Markov chain Monte Carlo methods. It also facilitates the SPDE approach to spatial modelling, which has been used for modelling of air pollutant levels, and is available in the R-INLA package for the R statistics software. Covariates such as meteorological variables may be useful predictors in such models, but covariate misalignment must be dealt with. This paper describes a flexible method used to estimate pollutant levels for six pollutants in Suzhou, a city in China with dispersed air pollutant monitors and weather stations. A two-stage approach is used to address misalignment of weather covariate data.

The Microscopic Derivation and Well-Posedness of the Stochastic Keller-Segel Equation

In this paper, we propose and study a stochastic aggregation-diffusion equation of the Keller-Segel (KS) type for modeling the chemotaxis in dimensions d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

A spatial partial differential equation approach to addressing unit misalignments in Bayesian poisson space-time models

Spatial analyses using data from geographic areas that change shape and location over time, like US ZIP codes, produce biased results to the extent that unit misalignments are related to covariate effects. To address this issue, one method has incorporated a fixed effect measure of population shifts and a spatial structure as a block-diagonal neighborhood adjacency matrix within a Besag-York-Mollié (BYM) model. However, this approach assumes that spatial relationships among units change with time and precludes the assessment of temporal dynamic effects. Here, we assume that a continuous Gaussian random field underlies misaligned data and apply a stochastic partial differential equation (SPDE) approach to modeling area outcomes. We compare SPDE and BYM methods and show that both provide similar estimates of covariate effects. Importantly, we demonstrate that the SPDE approach can additionally identify autoregressive processes underlying the development of problematic health outcomes using data observed across Pennsylvania over 11 years.

Evaluating a Bayesian modelling approach (INLA-SPDE) for environmental mapping

Understanding the uncertainty in spatial modelling of environmental variables is important because it provides the end-users with the reliability of the maps. Over the past decades, Bayesian statistics has been successfully used. However, the conventional simulation-based Markov Chain Monte Carlo (MCMC) approaches are often computationally intensive. In this study, the performance of a novel Bayesian inference approach called Integrated Nested Laplace Approximation with Stochastic Partial Differential Equation (INLA-SPDE) was evaluated using independent calibration and validation datasets of various skewed and non-skewed soil properties and was compared with a linear mixed model estimated by residual maximum likelihood (REML-LMM). It was found that INLA-SPDE was equivalent to REML-LMM in terms of the model performance and was similarly robust with sparse datasets (i.e. 40-60 samples). In comparison, INLA-SPDE was able to estimate the posterior marginal distributions of the model parameters without extensive simulations. It was concluded that INLA-SPDE had the potential to map the spatial distribution of environmental variables along with their posterior marginal distributions for environmental management. Some drawbacks were identified with INLA-SPDE, including artefacts of model response due to the use of triangle meshes and a longer computational time when dealing with non-Gaussian likelihood families.