Analysis of a model for the dynamics of prions II

A new mathematical model for the dynamics of prion proliferation involving an ordinary differential equation coupled with a partial integro-differential equation is analyzed, continuing the work in [J. Prüss, L. Pujo-Menjouet, G.F. Webb, R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. 6 (2006) 225-235]. We show the well-posedness of this problem in its natural phase space , i.e., there is a unique global semiflow on Z₊ associated to the problem.A theorem of threshold type is derived for this model which is typical for mathematical epidemics. If a certain combination of kinetic parameters is below or at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable in Z₊; above the threshold it is unstable, and there is another unique steady state, the disease equilibrium, which inherits that property.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Connectivity threshold of Bluetooth graphs

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well‐known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these. The randomness comes from the underlying distribution of vertices in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 45–66, 2014     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.     

Turing pattern of the Oregonator model

The Oregonator model is the mathematical dynamics which describes the Field-Körös-Noyes mechanics of the famous Belousov-Zhabotinskii? reaction. In this work, we establish some fundamental analytic properties of this dynamics and its corresponding steady state. Under various conditions on the parameters and the size of the reactor, we examine the existence and non-existence of non-constant steady states. In particular, for some properly chosen parameter ranges, we prove the occurrence of the Turing pattern generated by this Oregonator model. Our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the Turing pattern. Our mathematical analysis mainly relies on a priori estimates and the topological degree argument.     

Is prediction possible? Chaotic behavior of Multiple Equilibria Regulation Model in cellular automata topology

In this article, we present the Multiple Equilibria Regulation (MER) Model in cellular automata topology. As argued in previous explorations of the model, for certain parameter values, the behavior of the system exhibits transient chaos (namely, the system is unpredictable but ends in a final steady state). In order to approach empirical reality, we introduce a cellular automata topology. Examining the outcome of the simulations leads us to conclude that for certain parameter values tested, the system yields chaotic behavior. Thus, cellular automata contribution has proven crucial, because the introduced topology converts the behavior of the system from transient chaos to “pure” chaos, i.e., the system is not only unpredictable on the long run but, in addition, it will never rest in a final steady state. According to these findings, authors argue the theoretical hypothesis that the urge for “prediction” in social sciences should be reconsidered in terms of “predictability horizon”. © 2004 Wiley Periodicals, Inc. Complexity 10: 23–36, 2004     

Continuous-time Markov chain models to estimate the premium for extended hedge fund lockups

A lockup period for investment in a hedge-fund is a time period after making the investment during which an investor cannot freely redeem his investment. Since long lockup periods have recently been imposed, it is important to estimate the premium an investor should expect from extended lockups. For this, Derman et al. (Wilmott J. 1(5–6):263–293, 2009) proposed a parsimonious three-state discrete-time Markov Chain (DTMC) to model the state of a hedge fund, allowing the state to change randomly among the states “good,” “sick” and “dead” every year. In this paper, we propose an alternative three-state absorbing continuous-time Markov Chain (CTMC) model, which allows state changes continuously in time instead of yearly. Allowing more dynamic state changes is more realistic, but the CTMC model requires new techniques for parameter fitting. We employ nonlinear programming to solve the new calibration equations. We show that the more realistic CTMC model is a viable alternative to the previous DTMC model for estimating the premium for extended hedge fund lockups.     

Multiplicity results for the unstirred chemostat model with general response functions

We investigate the multiplicity of positive steady state solutions to the unstirred chemostat model with general response functions. It turns out that all positive steady state solutions to this model lie on a single smooth solution curve, whose properties determine the multiplicity of positive steady state solutions. The key point of our analysis is to study the “turning points” on this positive steady state solution curve, and to prove that any nontrivial solution to the associated linearized problem is one of sign by constructing a suitable test function. The main tools used here include bifurcation theory, monotone method, mountain passing lemma and Sturm comparison theorem.     

Modeling the role of the cell cycle in regulating Proteus mirabilis swarm-colony development

We present models and computational results which indicate that the spatial and temporal regularity seen in Proteus mirabilis swarm-colony development is largely an expression of a specific, nearly precise age of dedifferentiation in the cell cycle from motile swarmer cells to immotile dividing cells. This contrasts strongly with reaction–diffusion models of Proteus behavior that ignore or average out the age structure of the cell population and instead use only density-dependent mechanisms. We argue the necessity of retaining this known biological feature using explicit age structure in the model, and suggest that certain experiments may validate this underlying mechanism empirically.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Markov random fields,Markov cocycles and the 3-colored chessboard

The well-known Hammersley–Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley–Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce “Markov cocycles”, reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley–Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview where the underlying graph is Z_d. This family includes all Markov random fields whose support is the d-dimensional “3-colored chessboard”. On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.     

The mechanics of contentious politics: an agent-based modeling approach

ABSTRACT

“Contentious politics” has become the main label to define a wide range of previously separated fields of research encompassing topics such as collective action, radicalization, armed insurgencies, and terrorism. Over the past two decades, scholars have tried to bring these various strands together into a unified field of study. In so doing, they have developed a methodology to isolate and analyze the common social and cognitive mechanisms underlying several diverse historical phenomena such as “insurgencies,” “revolutions,” “radicalization,” or “terrorism.” A multidisciplinary approach was adopted open to contributions from diverse fields such as economics, sociology, and psychology. The aim of this paper is to add to the multidisciplinarity of the field of Contentious Politics (CP) and introduce the instruments of Agent-Based Modeling and network game-theory to the study of some fundamental mechanisms analyzed within this literature. In particular, the model presented in this paper describes the dynamics of one process, here defined as “the radicalization of politics,” and its main underlying mechanisms. Their mechanics are analyzed in diverse social contexts differentiated by the values of four parameters: the extent of repression, inequality, social tolerance, and interconnectivity. The model can be used to explain the basic dynamics underlying different phenomena such as the development of radicalization, populism, and popular rebellions. In the final part, different societies characterized by diverse values of the aforementioned four parameters are tested through Python simulations, thereby offering an overview of the different outcomes that the mechanics of our model can shape according to the contexts in which they operate.     

Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Global stability and periodic solution of the viral dynamics

It is well known that the mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection-free steady state. We also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Stability criteria for the cure state in a cancer model with radiation treatment

In the present paper, a model for treatment by radiation of cancer is developed, where the radiation affects normal cells proportionately. We consider only the case where the radiation delivery is constant. In particular, we are interested in the existence and stability properties of a “cure state”, i.e. a steady state in which the cancer is absent. We use ordinary differential equations to model the growth and interactions. We utilize both analytical and numerical analysis to obtain our results.     

Analytical transient-time solution for temperature in non perfused tissue during radiofrequency ablation

Radiofrequency ablation (RFA) with internally cooled needle-like electrodes is a technique widely used to destroy cancer cells. In a previous study we obtained the analytical solution of the biological heat equation associated with the RFA problem in perfused tissue, i.e. when the governing equation which models the temperature distribution in tissue includes the blood perfusion therm. We also found that under these circumstances the temperature profiles always reach a steady state (limit temperature). However, the analytical solution of the RFA thermal problem without perfusion (e.g. conducted on an organ in which atraumatic vascular clamping is performed to temporally interrupt blood perfusion), cannot be directly obtained by setting the blood perfusion term to zero in the previously obtained solution. In fact, it is necessary to address the mathematical resolution in a totally different way. Our goal was to obtain the analytical expression of the temperature distribution in an RFA process with internally cooled needle-like electrodes when the biological tissue is not perfused. We consider two spatial domains: A finite domain which represents the real situation, and an infinite domain, which only makes sense from a mathematical point of view and which has been traditionally employed in analytical studies. Even though considering infinite time is not realistic, these approaches are surely worth considering in order to understand what happens “far from the electrode” or for “very long periods of time.” The results indicate that the temperature value is finite both when the spatial domain is finite (which implies that a steady state is reached), and when time is finite for any spatial domain. From this it can be concluded that a steady state is never reached if the spatial domain is infinite.     

Almost classical solutions to the total variation flow

The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of “almost classical” solutions we are able to determine evolution of facets – flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x ² is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.     

A model-data weak formulation for simultaneous estimation of state and model bias

We introduce a Petrov–Galerkin regularized saddle approximation which incorporates a “model” (partial differential equation) and “data” (M experimental observations) to yield estimates for both state and model bias. We provide an a priori theory that identifies two distinct contributions to the reduction in the error in state as a function of the number of observations, M: the stability constant increases with M; the model-bias best-fit error decreases with M. We present results for a synthetic Helmholtz problem and an actual acoustics system.