A cost assignment policy for home care patients

The patient assignment problem in Home care (HC) consists of allocating each newly admitted patient to his/her reference operator, chosen among a set of possible operators. The continuity of care, where pursued, imposes that the assignment is not changed for a long period. The main goal of the assignment is to balance the workload among the operators. In the literature, the problem is usually solved with numerical approaches based on mathematical programming that do not consider the stochastic aspects of the problem. We derive a structural policy to assign a newly admitted patient while balancing the workload among the operators, by minimizing the expected value of a cost function that penalizes the overtime of operators. The workloads already loaded to the operators are assumed to be random variables as they are in the practice, while the demand related to the new patient is considered both deterministic and stochastic. Results show that the variability of the new patient’s demand is negligible with respect to the variability of the already assigned workloads and that similar assignments are obtained both in the presence or in the absence of this demand variability. A numerical comparison with the current practice of assigning the new patient to the operator with the highest expected available capacity shows that better balancings and cost savings can be reached by implementing the proposed policy.     

Undamped oscillations in prey–predator models on a finite size lattice

Sustained oscillation is frequently observed in population dynamics of biospecies. The oscillation comes not only from deterministic but also from stochastic characteristics. In the present article, we deal with a finite size lattice which contains prey and predator. The interaction between a pair of lattice points is carried out by two different methods; local and global interactions. In the former, interaction occurs between adjacent sites, while in the latter interaction takes place between any pair of lattice sites. It is found that both systems exhibit undamped oscillations. The amplitude of oscillation decreases with the increase of the total lattice sites. In the case of global interaction, we can present a stochastic differential equation which is composed of two factors, i.e., the Lotka–Volterra equation with density dependence and noise term. The quantitative agreement between theory and simulation results of global interaction is almost perfect. The stochastic theory qualitatively expresses characteristics of sustainable oscillation for local interaction.     

A mathematical approach to spatial distribution and temporal succession in plant communities

The Volterra equations which represent competitions between two species are utilized to examine the phenomenon of boundary formation between two species of plants. The set of stable stationary points for these equations is determined and is illustrated in a product space of parameters and dynamical variables. The stages of boundary appearance and succession are visualized by considering slow changes of the parameters as functions of time and space.     

Considerations on models of movement detection

A general characterization of multi-input movement detection models is given in terms of the Volterra series formalism. When nonlinearities of order higher than the second are negligible, an n-input system can be decomposed into a set of 2-input systems, summing linearly. For a (symmetrical) 2-input system which has significant nonlinearities only up to the second order, the correlation model is its most general expression, if the infinite time average of the output is taken. Specific observations from optomotor experiments (e.g. phase invariance and contrast frequency dependence) can be interpreted in a general way in terms of properties of the Volterra representation.     

Anticipatory approach for dynamic and stochastic shipment matching in hinterland synchromodal transportation

Flexible Services and Manufacturing Journal - This paper investigates a dynamic and stochastic shipment matching problem faced by network operators in hinterland synchromodal transportation. We...     

Periodicity and boundedness of solutions of generalized differential equations of growth

Sufficient conditions are given for the existence of periodic solutions of differential equations, having as special cases the equations used to describe the competition between two species. The Poincaré bifurcation theory is used to secure one set of conditions, and another set of conditions is secured through a generalization of a method of V. Volterra. The question of boundedness is considered and conditions implying boundedness and conditions implying that populations are bounded away from zero are given. Several integrable classes of systems are discovered and a particular example having periodic solutions is examined in detail. This research was supported by the Air Force Office of Scientific Research under Grant 62-207.     

区间多种群Volterra生态系统的全局稳定性

讨论了区间多种群Volterra生态系统正平衡点的全局渐近稳定性,对于互惠型区间多种群Volterra生态系统不仅给出了正平衡点的全局稳定的充分必要条件,而且给出了系统存在平衡点的充分条件,对于一般的区间多种群Volterra生态系统给出了正平衡点全局稳定的充分条件。     

Sustained oscillations for density dependent Markov processes

Simulations of models of epidemics, biochemical systems, and other bio-systems show that when deterministic models yield damped oscillations, stochastic counterparts show sustained oscillations at an amplitude well above the expected noise level. A characterization of damped oscillations in terms of the local linear structure of the associated dynamics is well known, but in general there remains the problem of identifying the stochastic process which is observed in stochastic simulations. Here we show that in a general limiting sense the stochastic path describes a circular motion modulated by a slowly varying Ornstein–Uhlenbeck process. Numerical examples are shown for the Volterra predator–prey model, Sel’kov’s model for glycolysis, and a damped linear oscillator.     

Sequence-specific binding of arc repressor to DNA. Effects of operator mutations and modifications

A set of arc operators with transition and/or transversion mutations at each operator base pair has been constructed. By determining the ability of Arc to bind these variant operators, the importance of each base pair for Arc recognition has been assessed. Methylation protection experiments have also been used to probe points of close contact between Arc and most of the mutant operators. These data, together with phosphate interference results obtained previously for the wild type operator, provide information about the operator surface that is contacted when Arc binds.     

Interpolating three-dimensional kinematic data using quaternion splines and hermite curves

Kinematic interpolation is an important tool in biomechanics. The purpose of this work is to describe a method for interpolating three-dimensional kinematic data, minimizing error while maintaining ease of calculation. This method uses cubic quaternion and hermite interpolation to fill gaps between kinematic data points. Data sets with a small number of samples were extracted from a larger data set and used to validate the technique. Two additional types of interpolation were applied and then compared to the cubic quaternion interpolation. Displacement errors below 2% using the cubic quaternion method were achieved using 4% of the total samples, representing a decrease in error over the other algorithms.     

Permanence and global attractivity for Lotka–Volterra difference systems

 The permanence and global attractivity for two-species difference systems of Lotka–Volterra type are considered. It is proved that a cooperative system cannot be permanent. For a permanent competitive system, the explicit expression of the permanent set E is obtained and sufficient conditions are given to guarantee the global attractivity of the positive equilibrium of the system. Received: 21 May 1997 / Revised version: 25 November 1998     

密度制约竞争二种群Volterra方程解的有界性及参数估计

本文给出密度制约且相互竞争二种群Volterra方程解的一种有界性及存在唯一性的证明。基于此,参考单种群Logistic方程反问题的方法”,给出了该Volterra方程参数的一种估计。     

Reversible Adaptive Trees

We describe reversible adaptive trees, a class of stochastic algorithms modified from the formerly described adaptive trees. They evolve in time a finite subset of an ambient Euclidean space of any dimension, starting from a seed point and, accreting points to the evolving set, they grow branches towards a target set which can depend on time. In contrast with plain adaptive trees, which were formerly proven to have strong convergence properties to a static target, the points of reversible adaptive trees are removed from the tree when they have not been used recently enough in a path from the root to an accreted point. This, together with a straightening process performed on the branches, permits the tree to follow some moving targets and still remain adapted to it. We then discuss in what way one can see such reversible trees as a model for a qualitative property of resilience, which leads us to discuss qualitative modeling.     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.     

Dynamics of Epidemiological Models

We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.     

On representation and approximation of nonlinear systems

Many methods for the analysis of nonlinear systems rely on a Volterra system-representation in terms of integral kernels. This paper considers two questions: 1) whether Volterra-like representations are possible for all smooth systems (i.e. analytic operators),-2) which classes of systems can be approximated by some interesting subclasses of the smooth systems, e.g. Volterra systems with separable kernels and sandwich systems (cascades of linear systems and nomemory nonlinearities). Whereas the answer to question 1 is positive, the answer to question 2 depends on the type of approximation, i.e. topology, that is used.     

A reinforcement-depletion urn problem—I. Basic theory

An urn contains balls of different colors. Specified numbers of each color are added and form a reinforcement. The total reinforcement is randomly removed, forming a depletion. The process, not necessarily with the same reinforcements, is performed a number of times. The factorial moment generating function of the urn configurations at any stage is given in terms of multivariate difference operators. Cases when the reinforcement vector is defined as a stochastic variable are considered. The problem is a generalization of an urn model associated with radioactive atoms and stable atoms proposed by S. R. Bernard. The solutions given here have a definite application to the problem of modelling tracers in compartmental systems.     

Generic properties of edges and “corners” on smooth greyvalue surfaces

The edge line on a smooth greyvalue surface, defined as locus of maximal slope, is a curve embedded in the negatively curved part of the greyvalue surface. For an open and dense set of greyvalue functions the edge line has transverse double points as its only singular points, meets the parabolic curve tangentially at isolated points, and intersects the zero crossings of the Laplacean of the greyvalue function transversely. Defining a greyvalue corner as a curvature extremum of the edge line one can show that, again for an open and dense set of greyvalue functions, these corners are isolated points in the image corresponding to ordinary curvature extrema of the edge. Detecting such corners in greyvalue images requires differential operators containing partial derivatives of order five, which raises some doubts about the existence of numerically robust algorithms for detecting these features in digital images.