Unique stationary distribution for a high–dimensional stochastic HIV model with multi–stage transmission, standard incidence rate and lognormal Ornstein–Uhlenbeck process

Abstract HIV poses a severe threat to global public health. Its transmission and evolutionary processes are complex and difficult to fully characterise through direct observation. Motivated by this challenge, we investigate a seven-dimensional stochastic HIV model with multi-stage and multi-path transmission. The model adopts the standard incidence rate and introduces a lognormal Ornstein–Uhlenbeck process to describe environmental perturbations. Although most existing studies on high-dimensional stochastic epidemic models are limited to proving the existence of stationary distributions, this paper, by applying Markov semigroup theory, for the first time derives sufficient conditions for the existence and uniqueness of an invariant probability measure of the proposed model. Meanwhile, we prove that the density of the solution distribution converges to this unique invariant density in the $L^{1}$ L 1 -norm, providing a more rigorous theoretical guarantee for characterizing the persistent spread of the disease. Subsequently, numerical simulations further verify the validity of the theoretical conclusions.