Discrete Space Model Research Articles

Choice of landscape discretisation method affects the inferred rate of spread in wildlife disease spread models

Disease modelling at the livestock-wildlife interface is an important topic for which discrete-space models are used for the wildlife component. One prominent example is African Swine Fever, where wild boar play an influential role as reservoirs of disease spillover into domestic pig farms. In this paper, we present a simulation study that demonstrates the impact of seemingly arbitrary choices of landscape discretisation method on the inferred rate of spread within the model. We use an ordinary differential equation model to implement a simplified model of disease transmission between discrete groups of wild boar with spillover into domestic pig farms contained within a homogeneous landscape. We examine a range of scenarios whereby the landscape is discretised into wild boar patches of varying size and shape, and compare the rate of spread between domestic pig farms placed at fixed points on the landscape. Our results demonstrate a non-monotonic relationship between patch size and rate of spread, which is particularly unstable and unpredictable for square and triangular shaped patches. Discretisation of the landscape into hexagons appears to produce a more stable relationship between patch size and rate of spread for the three types of transmission kernel we investigated. Although the rate of disease spread does converge to a stable value, this occurs at patch sizes that are much smaller than would be used in practice for wild boar. We conclude that outputs of disease models containing a wildlife component should not be considered to be robust to arbitrary choices for patch size and placement, but rather as a source of uncertainty to be examined using sensitivity analysis. Furthermore, we strongly recommend the use of hexagons rather than squares or right triangles for landscape discretisation.

Modelling daisy quorum drive: A short-term bridge across engineered fitness valleys.

Engineered gene-drive techniques for population modification and/or suppression have the potential for tackling complex challenges, including reducing the spread of diseases and invasive species. Gene-drive systems with low threshold frequencies for invasion, such as homing-based gene drive, require initially few transgenic individuals to spread and are therefore easy to introduce. The self-propelled behavior of such drives presents a double-edged sword, however, as the low threshold can allow transgenic elements to expand beyond a target population. By contrast, systems where a high threshold frequency must be reached before alleles can spread-above a fitness valley-are less susceptible to spillover but require introduction at a high frequency. We model a proposed drive system, called "daisy quorum drive," that transitions over time from a low-threshold daisy-chain system (involving homing-based gene drive such as CRISPR-Cas9) to a high-threshold fitness-valley system (requiring a high frequency-a "quorum"-to spread). The daisy-chain construct temporarily lowers the high thresholds required for spread of the fitness-valley construct, facilitating use in a wide variety of species that are challenging to breed and release in large numbers. Because elements in the daisy chain only drive subsequent elements in the chain and not themselves and also carry deleterious alleles ("drive load"), the daisy chain is expected to exhaust itself, removing all CRISPR elements and leaving only the high-threshold fitness-valley construct, whose spread is more spatially restricted. Developing and analyzing both discrete patch and continuous space models, we explore how various attributes of daisy quorum drive affect the chance of modifying local population characteristics and the risk that transgenic elements expand beyond a target area. We also briefly explore daisy quorum drive when population suppression is the goal. We find that daisy quorum drive can provide a promising bridge between gene-drive and fitness-valley constructs, allowing spread from a low frequency in the short term and better containment in the long term, without requiring repeated introductions or persistence of CRISPR elements.

Mean-squared exponential stability of distributed almost periodic sequence for nonlocal space-time discrete shunting inhibitory cellular neural networks with Brownian motions

The discrete-time and discrete-space model for stochastic nonlocal shunting inhibitory cellular neural networks with reaction diffusions are modelled in the first time. Owing to the consideration of spatial variables, the discrete model derived in this article is more complex than the traditional ordinary difference model. In accordance with the constant-variation-formula for discrete-time and discrete-space model, Banach contracting mapping principle, the method of proof by contradiction and stochastic calculus, we also obtain the existence of a unique bounded almost periodic sequence for the discrete model, which is exponentially stable in the mean-square sense. Noting that it is the first time to consider the dynamics of almost periodicity in distribution of time-space discrete neural network models. The practicability of the present results is demonstrated by means of an illustration.

Self-Optimizing Control Strategy for Distributed Parameter Systems

Real-time optimization (RTO) has been a popular approach for addressing optimal operation by repetitively optimizing online. However, this approach is impractical for distributed parameter systems (DPSs) due to its expensive computational cost. Self-optimizing control (SOC) has emerged as a promising alternative strategy, which achieves acceptable optimality by controlling the designed controlled variables (CVs) at constant set points without reoptimizing. Motivated by this, the existing lumped parameter SOC method is for the first time expanded to a typical class of DPSs based on a high-fidelity discrete-space model. Starting from an infinite-order system itself, the SOC problem is constructed as the minimal loss functional regarding the CVs through the operator operation and variational method. To obtain an operational analytical solution of CVs, the optimization problem is reformulated based on the discrete-space system derived by the finite difference. Considering the variety of sensor and actuator placements, the CVs are analytically designed with the minimal static loss through the stacking matrix approach. Finally, a case study of a plug-flow reactor is presented to demonstrate the feasibility of the proposed approach to achieve optimal operation for DPSs.

Time-discretization approximation enriches continuous-time discrete-space models for animal movement

Continuous time discrete state models are a valuable tool for explaining animal movement. However, data collection to fit such models over a specified window of time can be misaligned with the actual realization of the movement process. This necessitates approximate model fitting, at present, through approximate imputation distributions (AIDs). Here, we propose a direct time-discretization approximation to the likelihood. The approach employs familiar ideas from hidden Markov modeling. Computation is implemented through the induced infinitesimal generator matrix. Linearization of this matrix expedites computation time. Through simulation and a real data application involving whale movement, we demonstrate that this model fitting strategy can outperform AID approaches.

A Deep Learning Approach to Dynamic Interbank Network Link Prediction

Lehman Brothers’ failure in 2008 demonstrated the importance of understanding interconnectedness in interbank networks. The interbank market plays a significant role in facilitating market liquidity and providing short-term funding for each other to smooth liquidity shortages. Knowing the trading relationship could also help understand risk contagion among banks. Therefore, future lending relationship prediction is important to understand the dynamic evolution of interbank networks. To achieve the goal, we apply a deep learning framework model of interbank lending to an electronic trading interbank network for temporal trading relationship prediction. There are two important components of the model, which are the Graph convolutional network (GCN) and the Long short-term memory (LSTM) model. The GCN and LSTM components together capture the spatial–temporal information of the dynamic network snapshots. Compared with the Discrete autoregressive model and Dynamic latent space model, our proposed model achieves better performance in both the precrisis and the crisis period.

Lower Bound on the Capacity of the Continuous-Space SSFM Model of Optical Fiber

The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the signal-to-noise ratio SNR and the number of segments in distance K is considered. It is shown that if K ≥ SNR2/3 and SNR → ∞, the capacity of the resulting continuous-space lossless model is lower bounded by 1/2 log2(1 + SNR) - 1/2 + o(1), where o(1) tends to zero with SNR. As K → ∞, the inter-symbol interference (ISI) averages out to zero due to the law of large numbers and the SSFM model tends to a diagonal phase noise model. It follows that, in contrast to the discrete-space model where there is only one signal degree-of-freedom (DoF) at high powers, the number of DoFs in the continuous-space model is at least half of the input dimension n. Intensity-modulation and direct detection achieves this rate. The pre-log in the lower bound when K=δ√SNR is generally characterized in terms of δ. It is shown that if the nonlinearity parameter γ → ∞, the capacity of the continuous-space model is 1/2 log2(1 + SNR) + o(1). The SSFM model when the dispersion matrix does not depend on K is considered. It is shown that the capacity of this model when K=δ√SNR, δ > 3, and SNR → ∞ is 1/2n log2(1 + SNR) + O(1). Thus, there is only one DoF in this model. Finally, it is found that the maximum achievable information rates (AIRs) of the SSFM model with back-propagation equalization obtained using numerical simulation follows a double-ascent curve. The AIR characteristically increases with SNR, reaching a peak at a certain optimal power, and then decreases as SNR is further increased. The peak is attributed to a balance between noise and stochastic ISI. However, if the power is further increased, the AIR will increase again, approaching the lower bound 1/2 log(1 + SNR) - 1/2 + o(1). The second ascent is because the ISI averages out to zero with K → ∞ sufficiently fast.

A Multiresolution Approach for Large Real-World Camera Placement Optimization Problems

There have been numerous attempts at solving the optimal camera placement problem across multiple applications. Exact linear programming-based, as well as, heuristic combinatorial optimization methods were shown to be successful in providing optimal or near-optimal solutions to this problem. Working over a discrete space model is the general practice when solving the camera placement problem. However, discretized environments often limit the methods’ usage only to small-scale datasets due to resource and time constraints that grow exponentially with the number of 3D points collected from the discrete space. We propose a multi-resolution approach that enables the usage of existing optimization algorithms on large real-world problems modelled using high resolution 3D grids. Our method works by grouping together the given discrete set of possible camera locations into clusters of points, multiple times, resulting in multiple resolution levels. Camera placement optimization is repeated for all resolution levels while propagating the optimized solution from low to high resolutions. Our experiments on both simulated and real data with grids of varying sizes show that using our multi-resolution approach, existing camera placement optimization methods can be used even on high resolution grids consisting of hundreds of thousands of points. Our results also show that the strategy of grouping points together by exploiting underlying 3D geometry to optimize camera poses is not only significantly faster than optimizing on the entire set of samples but, it also provides better camera coverage.

PD-Type Iterative Learning Control for Uncertain Spatially Interconnected Systems

This paper puts forward a PD-type iterative learning control algorithm for a class of discrete spatially interconnected systems with unstructured uncertainty. By lifting and changing the variable of discrete space model, the uncertain spatially interconnected systems is converted into equivalent singular system, and the general state space model is derived in view of singular system theory. Then, the state error and output error information are used to design the iterative learning control law, transforming the controlled system into an equivalent repetitive process model. Based on the stability theory of repetitive process, sufficient condition for the stability of the system along the trial is given in the form of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed algorithm is verified by the simulation of ladder circuits.

Current inversion in a periodically driven two-dimensional Brownian ratchet

It is well known that Brownian ratchets can exhibit current reversals, wherein the sign of the current switches as a function of the driving frequency. We introduce a spatial discretization of such a two-dimensional Brownian ratchet to enable spectral methods that efficiently compute those currents. These discrete-space models provide a convenient way to study the Markovian dynamics conditioned upon generating particular values of the currents. By studying such conditioned processes, we demonstrate that low-frequency negative values of current arise from typical events and high-frequency positive values of current arises from rare events. We demonstrate how these observations can inform the sculpting of time-dependent potential landscapes with a specific frequency response.

A quantum walk with both a continuous-time limit and a continuous-spacetime limit

Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a quantum walks or quantum cellular automata based) enjoying a relativistic continuous-spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice fermions in curved spacetime with synchronous coordinates.

Some remarks on mean field games

In this article, we study three aspects of mean field games (MFG). The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of “noise” in discrete space models and the formulation of the Master Equation in this case. Finally, we show how MFG reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

Spatial vs. non-spatial transboundary pollution control in a class of cooperative and non-cooperative dynamic games

We analyze a transboundary pollution differential game where, in addition to the standard temporal dimension, a spatial dimension is introduced to capture the geographical relationships among regions. Each region behaves strategically and maximizes its welfare net of environmental damage caused by the pollutant stock. The emission-output ratio is reduced by investment in region specific clean technology which evolves over time. The spatio-temporal dynamics of the pollutant stock is described by a parabolic partial differential equation. Using aggregate variables we study the feedback Nash equilibrium of a discrete-space model which could be seen as a space discretization of the continuous-space model. The discrete-space model presents the three main features of the original formulation: the model is truly dynamic; the agents behave strategically; and the model incorporates spatial aspects. For special functional forms previously used in the literature we analytically characterize the feedback Nash equilibrium and evaluate the impact of the introduction of the spatial dimension in the economic-environmental model. We show that our spatial model is a generalization of the model that disregards the spatial aspects. We analytically show that as the parameter describing how pollution diffuses among regions tends to infinity the equilibrium policies converge to those in the non-spatial setting. In the non-cooperative framework the spatially non-myopic behavior prescribes lower equilibrium emission rates, and consequently a lower global pollution stock. This is compatible with greater long-run welfares. In the cooperative framework, although the strategic interaction among the players does not exist, the only decision-maker still makes spatially strategic decisions.

Stochastic Model Checking for Predicting Component Failures and Service Availability

When a component fails in a critical communications service, how urgent is a repair? If we repair within 1 hour, 2 hours, or $n$ hours, how does this affect the likelihood of service failure? Can a formal model support assessing the impact, prioritisation, and scheduling of repairs in the event of component failures, and forecasting of maintenance costs? These are some of the questions posed to us by a large organisation and here we report on our experience of developing a stochastic framework based on a discrete space model and temporal logic to answer them. We define and explore both standard steady-state and transient temporal logic properties concerning the likelihood of service failure within certain time bounds, forecasting maintenance costs, and we introduce a new concept of envelopes of behaviour that quantify the effect of the status of lower level components on service availability. The resulting model is highly parameterised and user interaction for experimentation is supported by a lightweight, web-based interface.

Dynamic traffic assignment using the macroscopic fundamental diagram: A Review of vehicular and pedestrian flow models

Link-level Dynamic Traffic Assignment (DTA) models of large cities may suffer from prohibitive computation times, and calibration/validation can become the major challenge faced in their real-time implementation. The empirical evidence in 2008 in support of the existence of a Macroscopic Fundamental Diagram (MFD) on urban networks led to the formulation of discrete-space DTA models, where the link-level representation of networks replaced by a zone-based representation. In parallel, a large body of DTA models based on Hughes’ pedestrian model have been formulated in continuum space as 2-dimensional conservation laws, where the speed-density relationship has been recently interpreted as a MFD. Unfortunately, today it is not clear whether the continuum-space and discrete-space approaches are consistent with each other, in the sense that a well-defined discrete-space model should correspond to the numerical solution of a continuum model. This paper aims to shed light on this matter by performing a review of the literature in both modeling approaches. We found that (i) the discrete-space DTA models are prone to stalling problem leading to unrealistic gridlocks, (ii) with the exception of Hänseler’s pedestrian models, existing discrete-space models are not guaranteed to be consistent with a continuum formulation, which presents the potential for significant biases between the two approaches although they are trying to answer the same question, (iii) the speed-density relationships used in the continuum-space literature are very simplistic and do not incorporate the effects of network, particularly the effects of intersections, which is the strong suit of the MFD, (iv) the models mostly have been applied to the cases with only one or a few destinations and do not seem to be capable of modeling large-scale networks. We also found that further research is needed to (i) incorporate departure time choice, (ii) improve existing numerical methods, possibly extending recent advances on the one-dimensional kinematic wave (LWR) model, (iii) study the properties of system optimum solutions, (iv) examine the real-time applicability of current continuum-space models compared to traditional DTA methods, and (v) formulate anisotropic models for the interaction of intersecting flows.

Efficient Reconfigurable Architecture for Pricing Exotic Options

This article presents a new method for Monte Carlo (MC) option pricing using field-programmable gate arrays (FPGAs), which use a discrete-space random walk over a binomial lattice, rather than the continuous space-walks used by existing approaches. The underlying hypothesis is that the discrete-space walk will significantly reduce the area needed for each MC engine, and the resulting increase in parallelisation and raw performance outweighs any accuracy losses introduced by the discretisation. Experimental results support this hypothesis, showing that for a given MC simulation size, there is no significant loss in accuracy by using a discrete space model for the path-dependent exotic financial options. Analysis of the binomial simulation model shows that only limited-precision fixed-point arithmetic is needed, and also shows that pairs of MC kernels are able to share RAM resources. When using realistic constraints on pricing problems, it was found that the size of a discrete-space MC engine can be kept to 370 Flip-Flops and 233 Lookup Tables, allowing up to 3,000 variance-reduced MC cores in one FPGA. The combination of a highly parallelisable architecture and model-specific optimisations means that the binomial pricing technique allows for a 50× improvement in throughput compared to existing FPGA approaches, without any reduction in accuracy.

Spatial effects and strategic behavior in a multiregional transboundary pollution dynamic game

We analyze a transboundary pollution differential game where pollution control is spatially distributed among a number of agents with predetermined spatial relationships. The analysis emphasizes, first, the effects of the different geographical relationships among decision makers; and second, the strategic behaviour of the agents. The dynamic game considers a pollution stock (the state variable) distributed among one large region divided in subregions which control their own emissions of pollutants. The emissions are also represented as distributed variables. The dynamics of the pollution stock is defined by a parabolic partial differential equation. We numerically characterize the feedback Nash equilibrium of a discrete-space model that still captures the spatial interactions among agents. We evaluate the impact of the strategic and spatially dynamic behaviour of the agents on the design of equilibrium environmental policies.

Band selection algorithm based on information entropy for hyperspectral image classification

A band selection algorithm based on information entropy is proposed for hyperspectral image classification. First, original spectral features are transformed into discrete features and represented by a discrete space model. Then, the band selection algorithm based on information entropy is adopted to reduce feature dimensionality. The bands with weak class separability are effectively abandoned by the band selection algorithm. Moreover, support vector machine classifiers with composite kernels are employed to incorporate spatial features into spectral features, reducing speckle errors in the classification maps. The proposed methods are applied to three benchmark hyperspectral data sets for classification. The performance of the proposed methods is compared with a band selection algorithm based on mutual information. The experimental results demonstrate that the band selection algorithm based on information entropy can effectively reduce feature dimensionality and improve classification accuracy.

Patches Approach to Investigate the Populational Dynamics in Dengue

In areas where resources are located in patches or discrete locations, human dispersal is more conveniently modeled, in which the population is divided into discrete patches. In this work we develop a general discrete model to analyze the spread of Dengue disease. In the process of mathematical modeling we take into account the human populations and the circulation of a single serotype of dengue mosquitoes. The movements of susceptible, infected and recovered humans among all patches are considered. Aquatic phases with different carrying capacities are considered within the patches. Also an arbitrary number of patches can be used to simulate the spread of dengue disease. In this paper we performed numerical experiments to show the applicability of this methodology to investigate the dengue disease problem. The general discrete space model was developed for solving epidemiological problems whereas the human-vector interactions and human mobilities play an important role. Based on our numerical results, we may recommend the general patches model for solving epidemiological problems in Population Dynamics.

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.