On exact solutions to epidemic dynamic models

In this study, we address an SIR (susceptible-infected-recovered) model that is given as a system of first order differential equations and propose the SIR model on time scales which unifies and extends continuous and discrete models. More precisely, we derive the exact solution to the SIR model and discuss the asymptotic behavior of the number of susceptibles and infectives. Next, we introduce an SIS (susceptible-infected-susceptible) model on time scales and find the exact solution. We solve the models by using the Bernoulli equation on time scales which provides an alternative method to the existing methods. Having the models on time scales also leads to new discrete models. We illustrate our results with examples where the number of infectives in the population is obtained on different time scales.     

High-Dimensional Mixed Graphical Models

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.     

Pattern discrete and mixed Hit-and-Run for global optimization

We develop new Markov chain Monte Carlo samplers for neighborhood generation in global optimization algorithms based on Hit-and-Run. The success of Hit-and-Run as a sampler on continuous domains motivated Discrete Hit-and-Run with random biwalk for discrete domains. However, the potential for efficiencies in the implementation, which requires a randomization at each move to create the biwalk, lead us to a different approach that uses fixed patterns in generating the biwalks. We define Sphere and Box Biwalks that are pattern-based and easily implemented for discrete and mixed continuous/discrete domains. The pattern-based Hit-and-Run Markov chains preserve the convergence properties of Hit-and-Run to a target distribution. They also converge to continuous Hit-and-Run as the mesh of the discretized variables becomes finer, approaching a continuum. Moreover, we provide bounds on the finite time performance for the discrete cases of Sphere and Box Biwalks. We embed our samplers in an Improving Hit-and-Run global optimization algorithm and test their performance on a number of global optimization test problems.     

An investigation of the internal structure of shock profiles for shock capturing schemes

The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.     

Elastic lattices: equilibrium, invariant laws and homogenization

In the recent years, lattice modelling proved to be a topic of renewed interest. Indeed, fields as distant as chemical modelling and biological tissue modelling use network models that appeal to similar equilibrium laws. In both cases, obtaining an equivalent continuous model allows to simplify numerical procedures. We define the basic properties of lattices: elasticity, frame-indifference, hyperelasticity. We determine rigorously the form that constitutive laws undertake under frame-indifference and hyperelasticity assumptions. Finally, we describe an homogenization technique designed for discrete structures that provides a limit continuum mechanics model and, in the special case of hexagonal lattices, we investigate the symmetry properties of the limit constitutive law.      

From discrete to continuum: A Young measure approach

The passage from atomistic to continuum models is usually done via G\Gamma-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions. We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the G\Gamma-limit with respect to the weak * convergence of measures. The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a ``multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the ``microstructure" at two different levels.     

Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models

In this paper, the size-effects in the torsional and axial response of microtubules by using the nonlocal continuum rod model is investigated. To this end, continuous and discrete rod models are performed for modeling of microtubules. A simple finite element procedure is used for modeling and solution of nonlocal discrete system equation for microtubules. The influence of the small length scale on the vibration frequencies is examined both torsional and axial vibration cases. Some parametric results are also presented for examination of the accuracy and performances of discrete and continuous models.     

Limiting Connection between Discrete and Continuous Time Forward Interest Rate Curve Models

We study discrete time Heath–Jarrow–Morton (HJM) type of interest rate curve models, where the forward interest rates – in contrast to the classical HJM models – are driven by a random field. Our main aim is to investigate the relationship between the discrete time forward interest rate curve model and its continuous time counterpart. We derive a general result on the convergence of discrete time models and we give special focus on the nearly unit root spatial autoregression model.     

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.     

A spatially continuous max-flow and min-cut framework for binary labeling problems

We propose and investigate novel max-flow models in the spatially continuous setting, with or without i priori defined supervised constraints, under a comparative study of graph based max-flow/min-cut. We show that the continuous max-flow models correspond to their respective continuous min-cut models as primal and dual problems. In this respect, basic conceptions and terminologies from discrete max-flow/min-cut are revisited under a new variational perspective. We prove that the associated nonconvex partitioning problems, unsupervised or supervised, can be solved globally and exactly via the proposed convex continuous max-flow and min-cut models. Moreover, we derive novel fast max-flow based algorithms whose convergence can be guaranteed by standard optimization theories. Experiments on image segmentation, both unsupervised and supervised, show that our continuous max-flow based algorithms outperform previous approaches in terms of efficiency and accuracy.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

Estimation of time-varying mortality rates using continuous models for Daphnia magna

Structured population models that make the assumption of constant demographic rates do not accurately describe the complex life histories seen in many species. We investigated the accuracy of using constant versus time-varying mortality rates within discrete and continuously structured models for Daphnia magna. We tested the accuracy of the models we considered using density-independent survival data for 90 daphnids. We found that a continuous differential equation model with a time-varying mortality rate was the most accurate model for describing our experimental D. magna survival data. Our results suggest that differential equation models with variable parameters are an accurate tool for estimating mortality rates in biological scenarios in which mortality might vary significantly with age.     

A nonautonomous epidemic model on time scales

We obtain conditions for permanence and extinction of the infection for a nonautonomous SIQR model defined on an arbitrary time scale. The threshold conditions are given by some numbers that play the role of the basic reproduction number in this setting. As a particular case of our result, we recover several threshold conditions obtained in the literature, on discrete or continuous time, for autonomous, periodic models and general nonautonomous models and we also discuss some new situations, including an aperiodic time scale.     

Bayesian Inference for the One-Factor Copula Model

We develop efficient Bayesian inference for the one-factor copula model with two significant contributions over existing methodologies. First, our approach leads to straightforward inference on dependence parameters and the latent factor; only inference on the former is available under frequentist alternatives. Second, we develop a reversible jump Markov chain Monte Carlo algorithm that averages over models constructed from different bivariate copula building blocks. Our approach accommodates any combination of discrete and continuous margins. Through extensive simulations, we compare the computational and Monte Carlo efficiency of alternative proposed sampling schemes. The preferred algorithm provides reliable inference on parameters, the latent factor, and model space. The potential of the methodology is highlighted in an empirical study of 10 binary measures of socio-economic deprivation collected for 11,463 East Timorese households. The importance of conducting inference on the latent factor is motivated by constructing a poverty index using estimates of the factor. Compared to a linear Gaussian factor model, our model average improves out-of-sample fit. The relationships between the poverty index and observed variables uncovered by our approach are diverse and allow for a richer and more precise understanding of the dependence between overall deprivation and individual measures of well-being.     

Learning the Structure of Mixed Graphical Models

We consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parameterization of the model. Supplementary materials for this article are available online.     

Euler's problem,Euler's method,and the standard map; or,the discrete charm of buckling

Summary We explore the relation between the classical continuum model of Euler buckling and an iterated mapping which is not only a mathematical discretization of the former but also has an exact, discrete mechanical analogue. We show that the latter possesses great numbers of “parasitic” solutions in addition to the natural discretizations of classical buckling modes. We investigate this rich bifurcational structure using both mechanical analysis of the boundary value problem and dynamical studies of the initial value problem, which is the familiar standard map. We use this example to explore the links between discrete initial and boundary value problems and, more generally, to illustrate the complex relations among physical systems, continuum and discrete models and the analytical and numerical methods for their study.     

Dynamics of a continuous Hénon model

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular, high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations.     

Discrete and continuous time representations and mathematical models for large production scheduling problems: A case study from the pharmaceutical industry

The underlying time framework used is one of the major differences in the basic structure of mathematical programming formulations used for production scheduling problems. The models are either based on continuous or discrete time representations. In the literature there is no general agreement on which is better or more suitable for different types of production or business environments. In this paper we study a large real-world scheduling problem from a pharmaceutical company. The problem is at least NP-hard and cannot be solved with standard solution methods. We therefore decompose the problem into two parts and compare discrete and continuous time representations for solving the individual parts. Our results show pros and cons of each model. The continuous formulation can be used to solve larger test cases and it is also more accurate for the problem under consideration.     

Memory effects and random walks in reaction-transport systems

In this article, we study continuous and discrete models to describe reaction transport systems with memory and long range interaction. In these models the transport process is described by a non-Brownian random walk model and the memory is induced by a waiting time distribution of the gamma type. Numerical results illustrating the behavior of the solution of discrete models are also included.