Predicting the natural history of metabolic syndrome with a Markov-system dynamic model: a novel approach

Background Markov system dynamic (MSD) model has rarely been used in medical studies. The aim of this study was to evaluate the performance of MSD model in prediction of metabolic syndrome (MetS) natural history. Methods Data gathered by Tehran Lipid & Glucose Study (TLGS) over a 16-year period from a cohort of 12,882 people was used to conduct the analyses. First, transition probabilities (TPs) between 12 components of MetS by Markov as well as control and failure rates of relevant interventions were calculated. Then, the risk of developing each component by 2036 was predicted once by a Markov model and then by a MSD model. Finally, the two models were validated and compared to assess their performance and advantages by using mean differences, mean SE of matrices, fit of the graphs, and Kolmogorov-Smirnov two-sample test as well as R2 index as model fitting index. Results Both Markov and MSD models were shown to be adequate for prediction of MetS trends. But the MSD model predictions were closer to the real trends when comparing the output graphs. The MSD model was also, comparatively speaking, more successful in the assessment of mean differences (less overestimation) and SE of the general matrix. Moreover, the Kolmogorov-Smirnov two-sample showed that the MSD model produced equal distributions of real and predicted samples (p = 0.808 for MSD model and p = 0.023 for Markov model). Finally, R2 for the MSD model was higher than Markov model (73% for the Markov model and 85% for the MSD model). Conclusion The MSD model showed a more realistic natural history than the Markov model which highlights the importance of paying attention to this method in therapeutic and preventive procedures.

On rational Abel – Poisson means on a segment and approximations of Markov functions

Approximations on the segment [−1, 1] of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure µ satisfies the conditions suppµ = [1, a], a > 1, dµ(t) = φ(t)dt and φ(t) ≍ (t − 1)α on [1, a], a are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment [−1, 1] are given by the method of rational approximation of some elementary Markov functions under study.

DEVELOPMENT OF A MODEL FOR THE DYNAMICS OF PROBABILITIES OF STATES OF SEMI-MARKOV SYSTEMS

The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.

Rate of Convergence and Periodicity of the Expected Population Structure of Markov Systems that Live in a General State Space

In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population structure. We conclude with the study of total variability from the invariant measure in the periodic case for the expected population structure of an MSGS.

Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$ .

Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.