Sampling the Functional Kolmogorov Forward Equations for Nonstationary Queueing Networks

2017 ◽  
Vol 29 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Jamol Pender
Keyword(s):  
Queueing Networks ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

scholarly journals Modelling gambiense human African trypanosomiasis infection in villages of the Democratic Republic of Congo using Kolmogorov forward equations

2021 ◽  
Author(s):  
Christopher N Davis ◽  
Matt J Keeling ◽  
Kat S Rock
Keyword(s):  
Human African Trypanosomiasis ◽  
Democratic Republic ◽  
Democratic Republic Of Congo ◽  
Mechanistic Model ◽  
Stochastic Methods ◽  
African Trypanosomiasis ◽  
Exact Methods ◽  
Republic Of Congo ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

Stochastic methods for modelling disease dynamics enables the direct computation of the probability of elimination of transmission (EOT). For the low-prevalence disease of human African trypanosomiasis (gHAT), we develop a new mechanistic model for gHAT infection that determines the full probability distribution of the gHAT infection using Kolmogorov forward equations. The methodology allows the analytical investigation of the probabilities of gHAT elimination in the spatially-connected villages of the Kwamouth and Mosango health zones of the Democratic Republic of Congo, and captures the uncertainty using exact methods. We predict that, if current active and passive screening continue at current levels, local elimination of infection will occur in 2029 for Mosango and after 2040 in Kwamouth, respectively. Our method provides a more realistic approach to scaling the probability of elimination of infection between single villages and much larger regions, and provides results comparable to established models without the requirement of detailed infection structure. The novel flexibility allows the interventions in the model to be implemented specific to each village, and this introduces the framework to consider the possible future strategies of test-and-treat or direct treatment of individuals living in villages where cases have been found, using a new drug.

scholarly journals On the Kolmogorov forward equations within Caputo and Riemann-Liouville fractions derivatives

2016 ◽  
Vol 24 (3) ◽  
pp. 5-19
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Laplace Transforms ◽  
Caputo Fractional Derivatives ◽  
The Right ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

AbstractIn this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for α ∈ (1, +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.

The transition probabilities of the extended simple stochastic epidemic model and the Haskey model

1972 ◽  
Vol 9 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Epidemic Model ◽  
Transition Probabilities ◽  
Cross Infection ◽  
Special System ◽  
Stochastic Epidemic ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

We present a solution to a special system of Kolmogorov forward equations. We use this result to present a useful expression for the transition probabilities of the extended simple stochastic epidemic model and an epidemic model involving cross-infection between two otherwise isolated groups.

A note on the two-sex population process

1986 ◽  
Vol 23 (A) ◽  
pp. 335-344 ◽  
Author(s):  
J. Gani ◽  
Pyke Tin
Keyword(s):  
Integral Equation ◽  
Generating Function ◽  
Explicit Solution ◽  
Probability Generating Function ◽  
Equation Approach ◽  
Population Process ◽  
Integral Equation Approach ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Birth Death

This paper considers the two-sex birth-death model {X(t), Y(t); t ≧ 0}; an explicit solution is obtained for its probability generating function. It is shown that moments of the process can be found directly from the Kolmogorov forward equations for the probabilities. An integral equation approach is also used to throw light on the structure of the process.

scholarly journals Fractional Discrete Processes: Compound and Mixed Poisson Representations

2014 ◽  
Vol 51 (01) ◽  
pp. 19-36 ◽  
Author(s):  
Luisa Beghin ◽  
Claudio Macci
Keyword(s):  
Nonnegative Integer ◽  
Negative Binomial ◽  
Inverse Gaussian ◽  
Inverse Gaussian Process ◽  
Probability Mass ◽  
Negative Binomial Process ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Mass Functions ◽  
Binomial Process

We consider two fractional versions of a family of nonnegative integer-valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Pólya-Aeppli process, the Poisson inverse Gaussian process, and the negative binomial process. We also define and study some more general fractional versions with two fractional parameters.

Computational and estimation procedures in multidimensional right-shift processes and some applications

1975 ◽  
Vol 7 (2) ◽  
pp. 349-382 ◽  
Author(s):  
Richard J. Kryscio ◽  
Norman C. Severo
Keyword(s):  
Queueing Theory ◽  
Probability Function ◽  
Complete System ◽  
Transition Probability ◽  
Finite State ◽  
Computational Procedures ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Transition Probability Function ◽  
Finite State Space

A right-shift process is a Markov process with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. A multidimensional right-shift process consists of v ≧ 1 concurrent and dependent right-shift processes. In this paper applications of multidimensional right-shift processes to some well-known examples from epidemic theory, queueing theory and the Beetle probblem due to Lucien LeCam are discussed. A transformation which orders the Kolmogorov forward equations into a triangular array is provided and some computational procedures for solving the resulting system of equations are presented. One of these procedures is concerned with the problem of evaluating a given transition probability function rather than obtaining the solution to the complete system of forward equations. This particular procedure is applied to the problem of estimating the parameters of a multidimensional right-shift process which is observed at only a finite number of fixed timepoints.

scholarly journals Modelling gambiense human African trypanosomiasis infection in villages of the Democratic Republic of Congo using Kolmogorov forward equations

2021 ◽  
Vol 18 (183) ◽  
Author(s):  
Christopher N. Davis ◽  
Matt J. Keeling ◽  
Kat S. Rock
Keyword(s):  
Human African Trypanosomiasis ◽  
Democratic Republic ◽  
Democratic Republic Of Congo ◽  
Mechanistic Model ◽  
Stochastic Methods ◽  
African Trypanosomiasis ◽  
Exact Methods ◽  
Republic Of Congo ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

Stochastic methods for modelling disease dynamics enable the direct computation of the probability of elimination of transmission. For the low-prevalence disease of human African trypanosomiasis (gHAT), we develop a new mechanistic model for gHAT infection that determines the full probability distribution of the gHAT infection using Kolmogorov forward equations. The methodology allows the analytical investigation of the probabilities of gHAT elimination in the spatially connected villages of different prevalence health zones of the Democratic Republic of Congo, and captures the uncertainty using exact methods. Our method provides a more realistic approach to scaling the probability of elimination of infection between single villages and much larger regions, and provides results comparable to established models without the requirement of detailed infection structure. The novel flexibility allows the interventions in the model to be implemented specific to each village, and this introduces the framework to consider the possible future strategies of test-and-treat or direct treatment of individuals living in villages where cases have been found, using a new drug.

The transition probabilities of the extended simple stochastic epidemic model and the Haskey model

1972 ◽  
Vol 9 (03) ◽  
pp. 471-485 ◽  
Author(s):  
Richard J. Kryscio
Keyword(s):  
Epidemic Model ◽  
Transition Probabilities ◽  
Cross Infection ◽  
Special System ◽  
Stochastic Epidemic ◽  
Stochastic Epidemic Model ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

We present a solution to a special system of Kolmogorov forward equations. We use this result to present a useful expression for the transition probabilities of the extended simple stochastic epidemic model and an epidemic model involving cross-infection between two otherwise isolated groups.

A note on the two-sex population process

1986 ◽  
Vol 23 (A) ◽  
pp. 335-344 ◽  
Author(s):  
J. Gani ◽  
Pyke Tin
Keyword(s):  
Integral Equation ◽  
Generating Function ◽  
Explicit Solution ◽  
Probability Generating Function ◽  
Equation Approach ◽  
Population Process ◽  
Integral Equation Approach ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Birth Death

This paper considers the two-sex birth-death model {X(t), Y(t); t ≧ 0}; an explicit solution is obtained for its probability generating function. It is shown that moments of the process can be found directly from the Kolmogorov forward equations for the probabilities. An integral equation approach is also used to throw light on the structure of the process.

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