Long-run analysis of a perturbed HIV/AIDS model with antiretroviral therapy and heavy-tailed increments performed by tempered stable Lévy jumps

The novelty of this article lies in providing a new framework for modeling the dynamics of HIV/AIDS infection under antiretroviral therapy (ART) which aims to reduce a person's viral load to an undetectable level. Since HIV/AIDS transmission can exhibit very tailed behavior (unpredictable jumps) due to the fact that the risk of spreading can vary greatly depending on the behavior and characteristics of individuals, we probe the effect of α-stable jumps on its asymptotic dynamics. Our proposed model is a compartmental step-wise formulation that takes the form of a system of Itô-Lévy differential equations with α-stable process. First, we check its well-posedness and we give the necessary assumptions needed for the analysis. Then, and based on a new approach, we prove two main asymptotic scenarios: extinction and stationary persistence of HIV/AIDS infection. The theoretical outcomes show that the kinetic behavior of our HIV/AIDS system is principally induced by certain underlying parameters, which are accurately correlated to the Lévy noise amplitudes. Finally, we reinforce our study with two numerical experiments to test the impact of the innovative mathematical techniques adopted to obtain the main results. Moreover, we numerically highlight the influence of α-stable jumps in epidemic state transition and dynamic behavior change.