Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

Capturing the start point of the virus-cell interaction with high-speed 3D single-virus tracking

The early stages of the virus-cell interaction have long evaded observation by existing microscopy methods due to the rapid diffusion of virions in the extracellular space and the large 3D cellular structures involved. Here we present an active-feedback single-virus tracking method with simultaneous volumetric imaging of the live cell environment to address this knowledge gap to present unprecedented detail to the extracellular phase of the infectious cycle. We report previously unobserved phenomena in the early stages of the virus-cell interaction, including skimming contact events at the millisecond timescale, orders of magnitude change in diffusion coefficient upon binding, and cylindrical and linear diffusion modes along filopodia. Finally, we demonstrate how this new method can move single-virus tracking from simple monolayer culture towards more tissue-like conditions by tracking single virions in tightly packed epithelial cells. This multi-resolution method presents new opportunities for capturing fast, 3D processes in biological systems.

Graphically interpreting how incision thresholds influence topographic and scaling properties of modeled landscapes

We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes but not in landscapes with diffusion. We illustrate the response of these curvature–steepness index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

Early-Time Non-Equilibrium Pitch Angle Diffusion of Electrons by Whistler-Mode Hiss in a Plasmaspheric Plume Associated with BARREL Precipitation

In August 2015, the Balloon Array for Radiation belt Relativistic Electron Losses (BARREL) observed precipitation of energetic (<200 keV) electrons magnetically conjugate to a region of dense cold plasma as measured by the twin Van Allen Probes spacecraft. The two spacecraft passed through the high density region during multiple orbits, showing that the structure was spatial and relatively stable over many hours. The region, identified as a plasmaspheric plume, was filled with intense hiss-like plasma waves. We use a quasi-linear diffusion model to investigate plume whistler-mode hiss waves as the cause of precipitation observed by BARREL. The model input parameters are based on the observed wave, plasma and energetic particle properties obtained from Van Allen Probes. Diffusion coefficients are found to be largest in the same energy range as the precipitation observed by BARREL, indicating that the plume hiss waves were responsible for the precipitation. The event-driven pitch angle diffusion simulation is also used to investigate the evolution of the electron phase space density (PSD) for different energies and assumed initial pitch angle distributions. The results show a complex temporal evolution of the phase space density, with periods of both growth and loss. The earliest dynamics, within the ∼5 first minutes, can be controlled by a growth of the PSD near the loss cone (by a factor up to ∼2, depending on the conditions, pitch angle, and energy), favored by the absence of a gradient at the loss cone and by the gradients of the initial pitch angle distribution. Global loss by 1-3 orders of magnitude (depending on the energy) occurs within the first ∼100 min of wave-particle interaction. The prevalence of plasmaspheric plumes and detached plasma regions suggests whistler-mode hiss waves could be an important driver of electron loss even at high L-value (L ∼6), outside of the main plasmasphere.

Autoencoder-Based Reduced Order Observer Design for a Class of Diffusion-Convection-Reaction Systems

The application of autoencoders in combination with Dynamic Mode Decomposition for control (DMDc) and reduced order observer design as well as Kalman Filter design is discussed for low order state reconstruction of a class of scalar linear diffusion-convection-reaction systems. The general idea and conceptual approaches are developed following recent results on machine-learning based identification of the Koopman operator using autoencoders and DMDc for finite-dimensional discrete-time system identification. The resulting linear reduced order model is combined with a classical Kalman Filter for state reconstruction with minimum error covariance as well as a reduced order observer with very low computational and memory demands. The performance of the two schemes is evaluated and compared in terms of the approximated L2 error norm in a numerical simulation study. It turns out, that for the evaluated case study the reduced-order scheme achieves comparable performance with significantly less computational load.

Inverse cascade anomalies in fourth-order Leith models

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for wave turbulence arising in the Nonlinear Schrödinger model and for the gravitational waves in the Einstein’s vacuum field model. We show that inverse transfer is not described by a scaling of the constant-flux solution but has an anomalous scaling. We compute the anomalous exponents and analyze their origin using the theory of dynamical systems.

Test-retest reproducibility of in vivo oscillating gradient and microscopic anisotropy diffusion MRI in mice at 9.4 Tesla

Background and purpose Microstructure imaging with advanced diffusion MRI (dMRI) techniques have shown increased sensitivity and specificity to microstructural changes in various disease and injury models. Oscillating gradient spin echo (OGSE) dMRI, implemented by varying the oscillating gradient frequency, and microscopic anisotropy (μA) dMRI, implemented via tensor valued diffusion encoding, may provide additional insight by increasing sensitivity to smaller spatial scales and disentangling fiber orientation dispersion from true microstructural changes, respectively. The aims of this study were to characterize the test-retest reproducibility of in vivo OGSE and μA dMRI metrics in the mouse brain at 9.4 Tesla and provide estimates of required sample sizes for future investigations. Methods Twelve adult C57Bl/6 mice were scanned twice (5 days apart). Each imaging session consisted of multifrequency OGSE and μA dMRI protocols. Metrics investigated included μA, linear diffusion kurtosis, isotropic diffusion kurtosis, and the diffusion dispersion rate (Λ), which explores the power-law frequency dependence of mean diffusivity. The dMRI metric maps were analyzed with mean region-of-interest (ROI) and whole brain voxel-wise analysis. Bland-Altman plots and coefficients of variation (CV) were used to assess the reproducibility of OGSE and μA metrics. Furthermore, we estimated sample sizes required to detect a variety of effect sizes. Results Bland-Altman plots showed negligible biases between test and retest sessions. ROI-based CVs revealed high reproducibility for most metrics (CVs < 15%). Voxel-wise CV maps revealed high reproducibility for μA (CVs ~ 10%), but low reproducibility for OGSE metrics (CVs ~ 50%). Conclusion Most of the μA dMRI metrics are reproducible in both ROI-based and voxel-wise analysis, while the OGSE dMRI metrics are only reproducible in ROI-based analysis. Given feasible sample sizes (10–15), μA metrics and OGSE metrics may provide sensitivity to subtle microstructural changes (4–8%) and moderate changes (> 6%), respectively.

On numerical solution of nonlinear parabolic multicomponent diffusion-reaction problems

This work continues our previous analysis concerning the numerical solution of the multi-component mass transfer equations. The present test problems are two-dimensional, parabolic, non-linear, diffusion- reaction equations. An implicit finite difference method was used to discretize the mathematical model equations. The algorithm used to solve the non-linear system resulted for each time step is the modified Picard iteration. The numerical performances of the preconditioned conjugate gradient algorithms (BICGSTAB and GMRES) in solving the linear systems of the modified Picard iteration were analysed in detail. The numerical results obtained show good numerical performances.

Effect of Human Mobility On The Spatial Spread of Airborne Diseases: An Epidemic Model With Indirect Transmission

We extended a class of coupled PDE-ODE models for studying the spatial spread of airborne diseases by incorporating human mobility. Human populations are modeled with patches, and a Lagrangian perspective is used to keep track of individuals’ places of residence. The movement of pathogens in the air is modeled with linear diffusion and coupled to the SIR dynamics of each human population through an integral of the density of pathogen around the population patch. In the limit of fast diffusion pathogens, the method of matched asymptotic analysis is used to reduce the coupled PDE-ODE model to a nonlinear system of ODEs for the average density of pathogens in the air. The reduced system of ODEs is used to derive the basic reproduction number and the final size relation for the model. Numerical simulations of the full PDE-ODE model and the reduced system of ODEs are used to assess the impact of human mobility, together with the diffusion of pathogens on the dynamics of the disease. Results from the two models are consistent and show that human mobility significantly affects disease dynamics. In addition, we show that an increase in the diffusion rate of pathogen leads to a smaller epidemic.