Diffusion with stochastic resetting of interacting particles emerging in a model of population genetics

In this paper we put forward a Generalized Ohta-Kimura ladder model (GOKM) which bears a strong liaison with the so-called jump-type Fleming-Viot process (JFVP). The novelty here, when we compare with the classical Ohta-Kimura model, is that we now have an operator which allows multiple interaction among the individuals. It has to do with a generalized branching mechanism: m individual types extinguish and one individual type splits into m copies. The system of evolution equations arising from GOKM can be seen as a system of n-dimensional Kolmogorov forward equations (or Fokker-Planck equations). Besides the interest in its own right a favorable feature of GOKM, vis-`a-vis JFVP, is that its analysis requires a more amenable armory of concepts and mathematical technique to analyze some relevant issues such as correlation, indistinguishability of individuals and stationarity. In addition, as a by product, we show that the connection between Ohta-Kimura Model and diffusion with resetting, as previously structured in [6], can be extended to our setting.

Modelling gambiense human African trypanosomiasis infection in villages of the Democratic Republic of Congo using 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.

Modelling gambiense human African trypanosomiasis infection in villages of the Democratic Republic of Congo using 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.

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

In 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.

NONSTATIONARY LOSS QUEUES VIA CUMULANT MOMENT APPROXIMATIONS

In this paper, we provide a new technique for analyzing the nonstationary Erlang loss queueing model with abandonment. Our method uniquely combines the use of the functional Kolmogorov forward equations with the well-known Gram-Charlier series expansion from the statistics literature. Using the Gram-Charlier series expansion, we show that we can estimate salient performance measures of the loss queue such as the mean, variance, skewness, kurtosis, and blocking probability. Lastly, we provide numerical examples to illustrate the effectiveness of our approximations.

Fractional Discrete Processes: Compound and Mixed Poisson Representations

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.

Fractional Discrete Processes: Compound and Mixed Poisson Representations

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.

A note on the two-sex population process

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.