In the current work, a stochastic epidemic model for the dynamics of transmission of Human Immunodeficiency Virus (HIV) infection is developed and analysed. The construction considers nonlinear modes of disease transmission and incorporation of environmental variability in order to improve the reality of the modelling approach beyond deterministic hypotheses. The parameter R0 is obtained and applied to establish sufficient criteria for the existence and uniqueness of a stationary distribution of the system through the Markov semigroup theory and Lyapunov function methods. In addition, sufficient criteria for stochastic elimination of the disease are given under some mild conditions. Finally, numerical simulations are conducted using stochastic schemes to analyse system dynamics under different noise intensities and average transmission rates. The results imply that stochastic disturbances play a fundamental role in describing the evolution of HIV infection. Sensitivity analysis shows that transmission parameters are the key driving factors of disease spread, whereas stochastic impacts significantly affect variability without changing the stability structure. This work advances a better quantitative understanding of disease transmission mechanisms and offers predictive insights into epidemic dynamics and supports the design of intervention strategies.
Continuous state tree-indexed Markov chains provide a flexible framework for modelling and analyzing stochastic processes with continuous state spaces and hierarchical dependencies, making them valuable tools in information theory and related disciplines. In this paper, we mainly study the strong law of large numbers and the asymptotic equipartition property for continuous state Markov chains indexed by a Cayley tree. We first present the formal definition of continuous state Markov chains indexed by a tree. Then, we establish a strong law of large numbers for functions of these chains. Finally, we prove the asymptotic equipartition property for continuous state Markov chains indexed by a Cayley tree, thereby extending classical information theoretic results to a broader class of structured stochastic processes.
In this paper, complete convergence and complete moment convergence for maximal weighted sums and product sums of arrays of rowwise mn -extended negatively dependent random variables are investigated, and some sufficient conditions for convergence are provided. A discussion of how the moment condition can be reduced in the Baum-Katz type theorem with certain weight condition is given. Besides, rather than assuming stochastic domination, we turn to a weaker assumption of uniformly bounded expectations to describe the moment conditions. Additionally, a Marcinkiewicz-Zygmund type strong law of large numbers for product sums of widely orthant dependent random variables is established. The results obtained in the paper extend the corresponding ones for independent and some dependent structures. In addition, the results are applied to establishing strong consistency for estimators in nonparametric regression model and statistics in bootstrap.
In this paper, we establish existence and uniqueness results for generalized backward doubly stochastic differential equations (GRBDSDEs) with a right upper semi-continuous and left-limited (r.u.s.c.l.l.) barrier and stochastic Lipschitz coefficients. The analysis is based on Mertens' decomposition, tools from optimal stopping theory, and a Picard iteration scheme. In addition, we prove a comparison theorem and use it to establish the existence of a minimal solution when the coefficients satisfy a stochastic linear growth condition.
Stochastic primitive equations are fundamental for modelling geophysical fluid flows in the ocean and atmosphere. In large-scale systems, where horizontal motion dominates, we consider only horizontal viscosity to capture the anisotropic nature of the flow. The nonlinear convection term possesses an intrinsic anisotropic structure, which, combined with the anisotropic viscosity, presents substantial analytical challenges. In this paper, we establish a moderate deviation principle through a careful decomposition of the nonlinear terms, together with anisotropic estimates and the weak convergence method. This work provides a refined asymptotic description that bridges the central limit and large deviation regimes.
