scholarly journals Survival probabilities for HIV infected patients through semi-Markov processes

2014 ◽  
Vol 51 (1) ◽  
pp. 13-36 ◽  
Author(s):  
Giovanni Masala ◽  
Giuseppina Cannas ◽  
Marco Micocci
Keyword(s):  
Markov Processes ◽  
Evolution Equations ◽  
Markov Models ◽  
Transition Probabilities ◽  
Survival Probabilities ◽  
Proposed Model ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Hiv 1 ◽  
Quantities Of Interest

SUMMARY In this paper we apply a parametric semi-Markov process to model the dynamic evolution of HIV-1 infected patients. The seriousness of the infection is rendered by the CD4+ T-lymphocyte counts. For this purpose we introduce the main features of nonhomogeneous semi-Markov models. After determining the transition probabilities and the waiting time distributions in each state of the disease, we solve the evolution equations of the process in order to estimate the interval transition probabilities. These quantities appear to be of fundamental importance for clinical predictions. We also estimate the survival probabilities for HIV infected patients and compare them with respect to certain categories, such as gender, age group or type of antiretroviral therapy. Finally we attach a reward structure to the aforementioned semi-Markov processes in order to estimate clinical costs. For this purpose we generate random trajectories from the semi-Markov processes through Monte Carlo simulation. The proposed model is then applied to a large database provided by ISS (Istituto Superiore di Sanità, Rome, Italy), and all the quantities of interest are computed.

General solidarity theorems for semi-Markov processes

1972 ◽  
Vol 9 (04) ◽  
pp. 789-802
Author(s):  
Choong K. Cheong ◽  
Jozef L. Teugels
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Large Set ◽  
Semi Markov Process ◽  
Regularly Varying ◽  
Specific Pair ◽  
Semi Markov Processes

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij (t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

scholarly journals Testing the Semi Markov Model Using Monte Carlo Simulation Method for Predicting the Network Traffic

2020 ◽  
pp. 713-720
Author(s):  
Shirin Kordnoori ◽  
Hamidreza Mostafaei ◽  
Shaghayegh Kordnoori ◽  
Mohammadmohsen Ostadrahimi
Keyword(s):  
Monte Carlo ◽  
Markov Model ◽  
Markov Processes ◽  
Network Traffic ◽  
Markov Models ◽  
Transition Probabilities ◽  
Simulation Method ◽  
Monte Carlo Algorithm ◽  
Statistical Characteristics ◽  
Semi Markov Processes

Semi-Markov processes can be considered as a generalization of both Markov and renewal processes. One of the principal characteristics of these processes is that in opposition to Markov models, they represent systems whose evolution is dependent not only on their last visited state but on the elapsed time since this state. Semi-Markov processes are replacing the exponential distribution of time intervals with an optional distribution. In this paper, we give a statistical approach to test the semi-Markov hypothesis. Moreover, we describe a Monte Carlo algorithm able to simulate the trajectories of the semi-Markov chain. This simulation method is used to test the semi-Markov model by comparing and analyzing the results with empirical data. We introduce the database of Network traffic which is employed for applying the Monte Carlo algorithm. The statistical characteristics of real and synthetic data from the models are compared. The comparison between the semi-Markov and the Markov models is done by computing the Autocorrelation functions and the probability density functions of the Network traffic real and simulated data as well. All the comparisons admit that the Markovian hypothesis is rejected in favor of the more general semi Markov one. Finally, the interval transition probabilities which show the future predictions of the Network traffic are given.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (03) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij (t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

General solidarity theorems for semi-Markov processes

1972 ◽  
Vol 9 (4) ◽  
pp. 789-802
Author(s):  
Choong K. Cheong ◽  
Jozef L. Teugels
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Large Set ◽  
Semi Markov Process ◽  
Regularly Varying ◽  
Specific Pair ◽  
Semi Markov Processes

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij(t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (3) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

A semi-markov model for clinical trials

1965 ◽  
Vol 2 (02) ◽  
pp. 269-285 ◽  
Author(s):  
George H. Weiss ◽  
Marvin Zelen
Keyword(s):  
Clinical Trials ◽  
Probability Distribution ◽  
Acute Leukemia ◽  
Markov Model ◽  
Markov Models ◽  
Transition Probabilities ◽  
State Transitions ◽  
Semi Markov Processes ◽  
Absorbing States ◽  
Epidemic Diseases

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

Fitting hidden semi-Markov models to breakpoint rainfall data

2001 ◽  
Vol 38 (A) ◽  
pp. 142-157 ◽  
Author(s):  
John Sansom ◽  
Peter Thomson
Keyword(s):  
Dwell Time ◽  
Large Scale ◽  
Markov Models ◽  
Transition Probabilities ◽  
Rainfall Data ◽  
The Em Algorithm ◽  
Semi Markov Process ◽  
The Times ◽  
Log Normal ◽  
Disjoint Domains

The paper proposes a hidden semi-Markov model for breakpoint rainfall data that consist of both the times at which rain-rate changes and the steady rates between such changes. The model builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the rainfall data the observations are modelled as mixtures of log-normal distributions within unobserved states where the states evolve in time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions.Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture of distributions over disjoint domains providing a flexible alternative. The methods are also extended to deal with censored data. An application of the model to a large-scale bivariate dataset of breakpoint rainfall measurements at Wellington, New Zealand, is discussed.

scholarly journals Possibility of Application of the Theory of Semi-Markov Processes to Determine Reliability of Diagnosing Systems / MOŻLIWOŚĆ ZASTOSOWANIA TEORII PROCESÓW SEMI-MARKOWA DO OKREŚLENIA NIEZAWODNOŚCI SYSTEMÓW DIAGNOZUJĄCYCH

2012 ◽  
Vol 24 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Jerzy Girtler
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Diagnostic Tests ◽  
The State ◽  
Actual State ◽  
Reliability Model ◽  
Discrete State ◽  
Semi Markov Process ◽  
Semi Markov Processes

Abstract The paper provides justification for the necessity to define reliability of diagnosing systems (SDG) in order to develop a diagnosis on state of any technical mechanism being a diagnosed system (SDN). It has been shown that the knowledge of SDG reliability enables defining diagnosis reliability. It has been assumed that the diagnosis reliability can be defined as a diagnosis property which specifies the degree of recognizing by a diagnosing system (SDG) the actual state of the diagnosed system (SDN) which may be any mechanism, and the conditional probability p(S*/K*) of occurrence (existence) of state S* of the mechanism (SDN) as a diagnosis measure provided that at a specified reliability of SDG, the vector K* of values of diagnostic parameters implied by the state, is observed. The probability that SDG is in the state of ability during diagnostic tests and the following diagnostic inferences leading to development of a diagnosis about the SDN state, has been accepted as a measure of SDG reliability. The theory of semi-Markov processes has been used for defining the SDG reliability, that enabled to develop a SDG reliability model in the form of a seven-state (continuous-time discrete-state) semi-Markov process of changes of SDG states.

Two Markov Models of Neighborhood Housing Turnover

1972 ◽  
Vol 4 (2) ◽  
pp. 133-146 ◽  
Author(s):  
G Gilbert
Keyword(s):  
Mathematical Models ◽  
Markov Processes ◽  
Renewal Process ◽  
Markov Models ◽  
Transition Probabilities ◽  
Markov Renewal Process ◽  
Turnover Process

This paper develops two mathematical models of housing turnover in a neighborhood. The first of these draws upon the theory of non-homogeneous Markov processes and includes the effects of present neighborhood composition upon future turnover probabilities. The second model considers the turnover process as a Markov renewal process and therefore allows the inclusion of length of occupancy as a determinant of transition probabilities. Example calculations for both models are included, and procedures for using the models are outlined.

scholarly journals Monounireducible Nonhomogeneous Continuous Time Semi-Markov Processes Applied to Rating Migration Models

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Guglielmo D'Amico ◽  
Jacques Janssen ◽  
Raimondo Manca
Keyword(s):  
Distribution Function ◽  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Topological Structure ◽  
Credit Rating ◽  
Fundamental Importance ◽  
Sufficient Condition ◽  
Semi Markov Process ◽  
Semi Markov Processes

Monounireducible nonhomogeneous semi- Markov processes are defined and investigated. The mono- unireducible topological structure is a sufficient condition that guarantees the absorption of the semi-Markov process in a state of the process. This situation is of fundamental importance in the modelling of credit rating migrations because permits the derivation of the distribution function of the time of default. An application in credit rating modelling is given in order to illustrate the results.

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