scholarly journals From ODE to Open Markov Chains, via SDE: an application to models for infections in individuals and populations

2020 ◽  
Vol 8 (1) ◽  
pp. 180-197
Author(s):  
Manuel L. Esquível ◽  
Paula Patrício ◽  
Gracinda R. Guerreiro
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
State Space ◽  
Dynamic Change ◽  
General Purpose ◽  
The State ◽  
Ode Model ◽  
Individual Level ◽  
Ode System ◽  
The Individual

AbstractWe present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.

Criteria for classifying general Markov chains

1976 ◽  
Vol 8 (04) ◽  
pp. 737-771 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
The State ◽  
General State ◽  
Classical Work ◽  
The Status ◽  
General State Space

The aim of this paper is to present a comprehensive set of criteria for classifying as recurrent, transient, null or positive the sets visited by a general state space Markov chain. When the chain is irreducible in some sense, these then provide criteria for classifying the chain itself, provided the sets considered actually reflect the status of the chain as a whole. The first part of the paper is concerned with the connections between various definitions of recurrence, transience, nullity and positivity for sets and for irreducible chains; here we also elaborate the idea of status sets for irreducible chains. In the second part we give our criteria for classifying sets. When the state space is countable, our results for recurrence, transience and positivity reduce to the classical work of Foster (1953); for continuous-valued chains they extend results of Lamperti (1960), (1963); for general spaces the positivity and recurrence criteria strengthen those of Tweedie (1975b).

A New Algorithm for Computing the Ergodic Probability Vector for Large Markov Chains

1990 ◽  
Vol 4 (1) ◽  
pp. 89-116 ◽  
Author(s):  
Ushlo Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
The State ◽  
Special Structure ◽  
Probability Vector ◽  
Rank Reduction ◽  
Replacement Process ◽  
Novel Algorithm

A novel algorithm is developed which computes the ergodic probability vector for large Markov chains. Decomposing the state space into lumps, the algorithm generates a replacement process on each lump, where any exit from a lump is instantaneously replaced at some state in that lump. The replacement distributions are constructed recursively in such a way that, in the limit, the ergodic probability vector for a replacement process on one lump will be proportional to the ergodic probability vector of the original Markov chain restricted to that lump. Inverse matrices computed in the algorithm are of size (M – 1), where M is the number of lumps, thereby providing a substantial rank reduction. When a special structure is present, the procedure for generating the replacement distributions can be simplified. The relevance of the new algorithm to the aggregation-disaggregation algorithm of Takahashi [29] is also discussed.

Team Creativity and Innovation

2017 ◽  
Author(s):  
Roni Reiter-Palmon ◽  
Mackenzie Harms
Keyword(s):  
The State ◽  
Organizational Success ◽  
Team Creativity ◽  
Research And Practice ◽  
Individual Level ◽  
Context Variable ◽  
The Past ◽  
State Of Research ◽  
The Individual ◽  
Creativity And Innovation

For the past two decades, creativity and innovation have been viewed by researchers as critical to organizational success and survival. Understanding the factors that facilitate or inhibit creativity and innovation at the individual level has been the focus of much of the research in the area. In recent years, research in organizational psychology and management has focused on understanding creativity and innovation in teams. However, while earlier work on teams and creativity focused on the team as a context variable, and individual creativity as the outcome, more recent research emphasizes creativity as the outcome. This chapter provides an overview of the state of research and practice as it relates to team creativity and innovation in organizations.

Lumpability for non-irreducible finite markov chains

1984 ◽  
Vol 21 (03) ◽  
pp. 567-574 ◽  
Author(s):  
Atef M. Abdel-Moneim ◽  
Frederick W. Leysieffer
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
The State ◽  
Discrete Time Markov Chain ◽  
Finite Markov Chains

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities

1973 ◽  
Vol 73 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Gerald S. Goodman ◽  
S. Johansen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Continuous Transition ◽  
Countable State Space ◽  
Countable State ◽  
Uniformly Continuous

1. SummaryWe shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

Content, Correlation, and Meaning of Categories ‘Legal Person’ and ‘Legal Personality’: General-Purpose Aspect

2019 ◽  
Vol 10 (7) ◽  
pp. 2156
Author(s):  
Vitalii V. VOLYNETS ◽  
Volodymyr A. SICHEVLIUK ◽  
Ilona V. KAMINSKA
Keyword(s):  
General Theory ◽  
General Purpose ◽  
Legal Person ◽  
Present Stage ◽  
Legal Personality ◽  
Individual Level ◽  
New Knowledge ◽  
Theory Of Law ◽  
The Individual

At the present stage of its development, the general theory of law is aimed at achieving, as far as possible, a greater degree of practical use. Legal theorists seek to answer questions that are devoid of scholastic nature and derive from the practice of real legal relations. The performance of this task involves the movement of fundamental science, which is the general theory of state and law, on the path of ascending from the array of abstract reflections of legal reality, already formed by it, to obtaining its more specific theoretical reproductions. The purpose of this study is to present the correlation of categories such as ‘legal person’ and ‘legal personality’. The relevance of the study lies in the inability to gain new knowledge without the dialectical application of the framework of categories and concepts and methodology of the theory of law to the study of special (e.g., branch) and single (implemented at the individual level) legal phenomena in their relation to the general regularities of the functioning of the state and law. The research presents the content, correlation and meaning of these categories in the theory of law, and demonstrates their diversified use in Ukrainian legislation. The content of the category ‘subject of law’ covers those persons to whom the rights, duties and responsibilities, which are enshrined in the rules of objective law, are addressed. The category of ‘legal personality’ emphasizes the key role of objective law in constituting legal personality.  

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (3) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

The shi, diplomats, and urban expansion during the Warring States Period

2020 ◽  
pp. 656-671
Author(s):  
Andrew Meyer
Keyword(s):  
Urban Expansion ◽  
Social Institutions ◽  
The State ◽  
State Governments ◽  
Technological Advancement ◽  
Warring States Period ◽  
Individual Level ◽  
Warring States ◽  
The Individual ◽  
Economic Specialization

The shi, or “knights,” were not a coherent class during the Warring States period, though figures identified as such were central to the social, political, and cultural dynamism of the era. As the fragmentary states of the early Zhou era politically consolidated, the nature of the aristocracy changed. The aristocracy bifurcated into a steeply divergent populace of “kings” and their kin at the top and the mass of undifferentiated knights far below. Although not exactly a period of shi ascendancy, it was, at the individual level, a time of very fluid social mobility. State governments grew in power to the point of being able to determine the power and status even of hereditary aristocrats. All social positions became gauged in relation to their utility for the state. Low-born knights could rise to positions of high power and status through meritorious service to the state. Diplomacy became a field in which talents for strategy or rhetoric could earn great merit. Some of the most influential figures of the Warring States were humble knights that distinguished themselves as diplomats. Social fluidity was likewise embodied in urbanization. As the Warring States saw rapid population growth, technological advancement, and economic specialization cities grew in size and changed in character. Where they had once been principally military and cult centers, they evolved into centers of commerce and manufacturing in which new communities and social institutions took shape.

scholarly journals Markov Chains Conditioned Never to Wait Too Long at the Origin

2009 ◽  
Vol 46 (03) ◽  
pp. 812-826
Author(s):  
Saul Jacka
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
The State ◽  
Coin Tossing ◽  
First Time ◽  
Vague Convergence ◽  
Irreducible Markov Chain

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain,X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit asT→∞ in the cases where either the state space is finite orXis transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

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