Nonlocal Transmission Research Articles

Exponential stability and optimal control of a stochastic brucellosis model with spatial diffusion and nonlocal transmission

In this paper, a stochastic brucellosis model with nonlocal transmission and spatial diffusion is established. Existence, uniqueness and positivity of mild solution to the model are obtained by adopting a truncation method. To ensure the Mean Square Exponential Stability (MSES) of the solution, sufficient conditions are given by employing the inequality techniques and the stability implies that the brucellosis die out in a short period of time. Moreover, we introduce elimination of infected animals and disinfection of brucella to the model as control strategies. Necessary conditions are given to obtain the optimal control which best balance the outcomes and costs of the control by applying Pontryagin’s maximum principle. Finally, the theoretical results are demonstrated by the numerical simulations.

Modeling nonlocal behavior in epidemics via a reaction–diffusion system incorporating population movement along a network

The outbreak of COVID-19, beginning in 2019 and continuing through the time of writing, has led to renewed interest in the mathematical modeling of infectious disease. Recent works have focused on partial differential equation (PDE) models, particularly reaction–diffusion models, able to describe the progression of an epidemic in both space and time. These studies have shown generally promising results in describing and predicting COVID-19 progression. However, people often travel long distances in short periods of time, leading to nonlocal transmission of the disease. Such contagion dynamics are not well-represented by diffusion alone. In contrast, ordinary differential equation (ODE) models may easily account for this behavior by considering disparate regions as nodes in a network, with the edges defining nonlocal transmission. In this work, we attempt to combine these modeling paradigms via the introduction of a network structure within a reaction–diffusion PDE system. This is achieved through the definition of a population-transfer operator, which couples disjoint and potentially distant geographic regions, facilitating nonlocal population movement between them. We provide analytical results demonstrating that this operator does not disrupt the physical consistency or mathematical well-posedness of the system, and verify these results through numerical experiments. We then use this technique to simulate the COVID-19 epidemic in the Brazilian region of Rio de Janeiro, showcasing its ability to capture important nonlocal behaviors, while maintaining the advantages of a reaction–diffusion model for describing local dynamics.

Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization

This paper proposes a level set-based topology optimization method for acoustic metasurfaces consisting of multiple types of unit cells. As a type of acoustic metasurface, we focus on a two-layered metasurface, where each layer contains various types of unit cell designs. To handle this metasurface, we first propose a two-scale homogenization method based on previous studies targeting metasurfaces composed of a single type of unit cell structure. As a result of homogenization, each layer of the array of unit cells is replaced with an interface with a nonlocal transmission condition. Next, an optimization problem is formulated. We set an objective function to achieve negative refraction, and macroscopic responses obtained by the homogenization method are used to define the objective function. The material distributions in each unit cell of the metasurface are set as the design variables. Based on these settings, we perform a sensitivity analysis based on the concept of the topological derivative. A level set-based topology optimization method is introduced to solve the optimization problem, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. An optimal design of a two-layered acoustic metasurface that induces negative refraction is proposed, and a discussion of its mechanism is provided.

Wave propagation in a diffusive SEIR epidemic model with nonlocal transmission and a general nonlinear incidence rate

We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R_{0}=partial _{I}F(S_{0},0)/gamma of the corresponding ordinary differential equations and the minimal wave speed c^{*}. The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.

Topology optimization of acoustic metasurfaces by using a two-scale homogenization method

In this paper, we propose a level set-based topology optimization method for the unit-cell design of acoustic metasurfaces by using a two-scale homogenization method. Based on previous works, we first propose a homogenization method for acoustic metasurfaces that can be combined with topology optimization. In this method, a nonlocal transmission condition depending on the unit cell of the metasurface appears in a macroscale problem. Next, we formulate an optimization problem within the framework of a level set-based topology optimization method, wherein an objective functional is expressed as the macroscopic responses obtained through the homogenization, and material distributions in the unit cell are set as design variables. A sensitivity analysis is conducted based on the concept of the topological derivative. To confirm the validity of the proposed method, two-dimensional numerical examples are provided. First, we provide a numerical example that supports the validity of the homogenization method, and we then perform optimization calculations based on the waveguide settings of the acoustic metasurfaces. In addition, we discuss the mechanism of the obtained optimized structures.

Linear and nonlinear transport across a finite Kitaev chain: An exact analytical study

We present exact analytical results for the differential conductance of a finite Kitaev chain in an N-S-N configuration, where the topological superconductor is contacted on both sides with normal leads. Our results are obtained with the Keldysh non-equilibrium Green's functions technique, using the full spectrum of the Kitaev chain without resorting to minimal models. A closed formula for the linear conductance is given, and the analytical procedure to obtain the differential conductance for the transport mediated by higher excitations is described. The linear conductance attains the maximum value of $e^2/h$ only for the exact zero energy states. Also the differential conductance exhibits a complex pattern created by numerous crossings and anticrossings in the excitation spectrum. We reveal the crossings to be protected by the inversion symmetry, while the anticrossings result from a pairing-induced hybridization of particle-like and hole-like solutions with the same inversion character. Our comprehensive treatment of the Kitaev chain allows us also to identify the contributions of both local and non-local transmission processes to transport at arbitrary bias voltage. Local Andreev reflection processes dominate the transport within the bulk gap and diminish for higher excited states, but reemerge when the bias voltage probes the avoided crossings. The non-local direct transmission is enhanced above the bulk gap, but contributes also to the transport mediated by the topological states.

A parsimonious approach for spatial transmission and heterogeneity in the COVID-19 propagation.

Raw data on the number of deaths at a country level generally indicate a spatially variable distribution of COVID-19 incidence. An important issue is whether this pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies and sanitary measures overwhelms the effect of local factors. Additionally, this approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease.

Topological valley currents via ballistic edge modes in graphene superlattices near the primary Dirac point

Graphene on hexagonal boron nitride (hBN) can exhibit a topological phase via mutual crystallographic alignment. Recent measurements of nonlocal resistance (Rnl) near the secondary Dirac point (SDP) in ballistic graphene/hBN superlattices have been interpreted as arising due to the quantum valley Hall state. We report hBN/graphene/hBN superlattices in which Rnl at SDP is negligible, but below 60 K approaches the value of h/2e2 in zero magnetic field at the primary Dirac point with a characteristic decay length of 2 μm. Furthermore, nonlocal transport transmission probabilities based on the Landauer-Büttiker formalism show evidence for spin-degenerate ballistic valley-helical edge modes, which are key for the development of valleytronics.

Dynamics of a seasonal brucellosis disease model with nonlocal transmission and spatial diffusion

Brucellosis has been increasingly concerned with the animal health and a huge loss of economics. Multiple transmission routes, seasonal pattern and spatial diffusion have been identified in brucellosis propagations. Meanwhile, infectious ability strongly depends on the distance of each adjacent infectious individual. Such property can be captured by a nonlocal infection with some integrable functions. In this paper, we first propose a seasonal brucellosis epidemic model with nonlocal transmissions and spatial diffusions. We calculate the next generation operator R(x) by a renewal process. The basic reproduction number R0 is defined as the spectral radius of R(x), which plays an equivalent role in a principal eigenvalue of a linear operator. It is shown that the proposed model exhibits the threshold dynamics in terms of R0, i.e, if R0<1, then the brucella-free steady state is globally asymptotically stable; otherwise, the model admits at least one positively periodic steady state. Numerical simulations elucidate that time heterogeneity enhances magnitudes of oscillations of animal population and brucella. Moreover, enlarging effective infection radii increases the risk of brucellosis propagations. As a result, we suggest that herdsmen should improve the sanitation of animals’ environment and isolate infected animals instantly to control brucellosis prevalence.

A Physical Optics Approach to the Analysis of Metascreens

Artificial screens based on metasurfaces (MTSs), also called metascreens (MetSs) are becoming a very popular tool for electromagnetic field manipulation. While significant research efforts have been devoted to the development of synthesis methodologies, less work has been done for the accurate modeling of real structures based on this concept. This paper presents a method based on a Physical Optics (PO) for the efficient description of the scattered field of metascreens consisting of a stack of MTS separated by dielectric layers. The derivation of the PO currents is based on the definition of proper transmisssion coefficients for the electric and magnetic fields which, in turn, relies on an equivalent transmission line model of the multilayer structure. In this model, the MTSs are represented through homogenized equivalent surface impedances. The proposed model takes into account the non-local transmission properties and the finite thickness and size of the MetS. The accuracy in the scattered field prediction has been verified through comparison with full-wave simulations.

A biologically plausible supervised learning method for spiking neural networks using the symmetric STDP rule

Spiking neural networks (SNNs) possess energy-efficient potential due to event-based computation. However, supervised training of SNNs remains a challenge as spike activities are non-differentiable. Previous SNNs training methods can be generally categorized into two basic classes, i.e., backpropagation-like training methods and plasticity-based learning methods. The former methods are dependent on energy-inefficient real-valued computation and non-local transmission, as also required in artificial neural networks (ANNs), whereas the latter are either considered to be biologically implausible or exhibit poor performance. Hence, biologically plausible (bio-plausible) high-performance supervised learning (SL) methods for SNNs remain deficient. In this paper, we proposed a novel bio-plausible SNN model for SL based on the symmetric spike-timing dependent plasticity (sym-STDP) rule found in neuroscience. By combining the sym-STDP rule with bio-plausible synaptic scaling and intrinsic plasticity of the dynamic threshold, our SNN model implemented SL well and achieved good performance in the benchmark recognition task (MNIST dataset). To reveal the underlying mechanism of our SL model, we visualized both layer-based activities and synaptic weights using the t-distributed stochastic neighbor embedding (t-SNE) method after training and found that they were well clustered, thereby demonstrating excellent classification ability. Furthermore, to verify the robustness of our model, we trained it on another more realistic dataset (Fashion-MNIST), which also showed good performance. As the learning rules were bio-plausible and based purely on local spike events, our model could be easily applied to neuromorphic hardware for online training and may be helpful for understanding SL information processing at the synaptic level in biological neural systems.

Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

ABSTRACTWe consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.

Fast multidimensional ensemble empirical mode decomposition for the analysis of big spatio-temporal datasets.

In this big data era, it is more urgent than ever to solve two major issues: (i) fast data transmission methods that can facilitate access to data from non-local sources and (ii) fast and efficient data analysis methods that can reveal the key information from the available data for particular purposes. Although approaches in different fields to address these two questions may differ significantly, the common part must involve data compression techniques and a fast algorithm. This paper introduces the recently developed adaptive and spatio-temporally local analysis method, namely the fast multidimensional ensemble empirical mode decomposition (MEEMD), for the analysis of a large spatio-temporal dataset. The original MEEMD uses ensemble empirical mode decomposition to decompose time series at each spatial grid and then pieces together the temporal–spatial evolution of climate variability and change on naturally separated timescales, which is computationally expensive. By taking advantage of the high efficiency of the expression using principal component analysis/empirical orthogonal function analysis for spatio-temporally coherent data, we design a lossy compression method for climate data to facilitate its non-local transmission. We also explain the basic principles behind the fast MEEMD through decomposing principal components instead of original grid-wise time series to speed up computation of MEEMD. Using a typical climate dataset as an example, we demonstrate that our newly designed methods can (i) compress data with a compression rate of one to two orders; and (ii) speed-up the MEEMD algorithm by one to two orders.

Regularity for a Local–Nonlocal Transmission Problem

We formulate and study an elliptic transmission-like problem combining local and nonlocal elements. Let \({\mathbb{R}^n}\) be separated into two components by a smooth hypersurface Γ. On one side of Γ, a function satisfies a local second-order elliptic equation. On the other, it satisfies a nonlocal one of lower order. In addition, a nonlocal transmission condition is imposed across the interface Γ, which has a natural variational interpretation. We deduce the existence of solutions to the corresponding Dirichlet problem, and show that under mild assumptions they are Hölder continuous, following the method of De Giorgi. The principal difficulty stems from the lack of scale invariance near Γ, which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter. We then turn to the question of optimal regularity and qualitative properties of solutions, and show that (in the case of constant coefficients and flat Γ) they satisfy a kind of transmission condition, where the ratio of “fractional conormal derivatives” on the two sides of the interface is fixed. A perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface. Throughout, we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution, which has an interpretation in the context of quasigeostrophic dynamics.

Multiplicity results for nonlocal elliptic transmission problem with variable exponent

In this paper, a transmission problem given by a system of two nonlinear equations of p(x)-Kirchho type with nonstandard growth conditions are studied. Using the mountain pass theorem combined with the Ekeland's variational principle, we obtain at least two distinct, non-trivial weak solutions.

Variational theory and domain decomposition for nonlocal problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.

The effectiveness of fallowing strategies in disease control in salmon aquaculture assessed with an SIS model

Salmon production is an important industry in Scotland, with an estimated retail value >£1 billion. However, this salmon industry can be threatened by the invasion and spread of diseases. To reduce this risk, the industry is divided into management areas that are physically separated from each other. Pathogens can spread between farms by local processes such as water movement or by long-distance processes such as live fish movements. Here, network modelling was used to investigate the importance of transmission routes at these two scales. We used different disease transmission rates ( β), where infected farms had the probability of 0.10, 0.25 or 0.50 per month to infect each contacted farm. Interacting farms were modelled in such a way that neighbours within a management area could infect each other, resulting in two contacts per farm per month. In addition, non-local transmission occurred at random. Salmon are input to marine sites where they are raised to harvest size, the site is then fallowed; in the model the effects of different fallowing strategies (synchronised, partial synchronised and unsynchronised fallowing at the management area level) on the emergence of diseases were investigated. Synchronised fallowing was highly effective at eradicating epidemics when transmission rate is low ( β = 0.10) even when long distance contacts were fairly common (up to 1.5 farm −1 month −1). However for higher transmission rates, long distance contacts have to be kept at much lower levels (0.15 contacts month −1 where β = 0.25) when synchronised fallowing was applied. If fallowing was partially synchronised or unsynchronised then low rates of long-distance contact are required (0.75 or 0.15 farm −1 month −1) even if β = 0.10. These results demonstrate the potential benefits of having epidemiologically isolated management areas and applying synchronised fallowing.

Homogenization of [formula omitted]-Laplacian in perforated domains with a nonlocal transmission condition

We study the asymptotic behavior, as ε → 0 , of u ε solutions to a nonlinear elliptic equation with nonstandard growth condition in domains containing a grid-type microstructure F ε that is concentrated in an arbitrary small neighborhood of a given hypersurface Γ. We assume that u ε = A ε on ∂ F ε , where A ε is an unknown constant. The macroscopic equation and a nonlocal transmission condition on Γ are obtained by the variational homogenization technique in the framework of Sobolev spaces with variables exponents. This result is then illustrated by a periodic example. To cite this article: B. Amaziane et al., C. R. Mecanique 337 (2009).

Difficulties with Collapse Interpretations of Quantum Mechanics

Quantum mechanics gives many versions of reality but we perceive only one. One potential explanation for this, the one considered here, is that the wave function collapses down to just one version. The experimental situation is briefly reviewed, with no evidence found for collapse. The theoretical position is also reviewed and found wanting. Collapse-by-observation schemes are logically untenable. A mathematical theory of collapse must be nonlinear, a significant departure from current quantum theory. In addition, the primary candidate theory, the GRW-Pearle model, requires instantaneous, non-local transmission of information. It also requires transmission of information across versions of reality, which is forbidden in current quantum mechanics. And there is no apparent physical quantity, such as particle number, that can provide the mechanism for collapse in all cases. Further, these aspects, which are in disagreement with current theory, seem to be necessary for any mathematical theory of collapse. The conclusion is that the outlook for mathematical collapse schemes is not encouraging.

On the Tricomi Problem for the Lavrent'ev-Bitsadze Equation with a Nonlocal Transmission Condition

On the Tricomi Problem for the Lavrent'ev-Bitsadze Equation with a Nonlocal Transmission Condition