A new weighted infected-block M-matrix method for extinction thresholds in a stochastic Itô-Lévy HIV/AIDS model with real-data validation

Deriving sharp extinction criteria for realistic multi-stage Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome (HIV/AIDS) models under environmental uncertainty remains technically difficult, especially when both correlated continuous fluctuations and abrupt disruptive events are present. In such settings, classical deterministic thresholds may be misleading. Parameter regimes that predict persistence at the Ordinary Differential Equation (ODE) level can nevertheless exhibit extinction once stochastic effects are accounted for. In this work, we developed a new weighted infected-block $M$-matrix methodology to obtain an explicit, computable, almost-sure exponential extinction threshold for a six-compartment Itô-Lévy HIV/AIDS model with correlated diffusions and dual jump mechanisms. The proposed framework constructs positive weights directly from the infected-treatment transition block, and combines these weights with refined diffusion corrections and jump-compensator bounds to produce a precise extinction indicator that quantifies stochastic damping beyond deterministic invasion pressure. To support practical relevance, we calibrated the deterministic core to real monthly HIV-AIDS and ART data from Pakistan (2016–2021) via multi-start nonlinear least squares, computed all threshold objects from the fitted parameters, and then tuned the Itô-Lévy perturbations to match the extinction regime predicted by the theory. Numerical simulations validated the theoretical predictions and illustrated how sufficiently strong fluctuations and rare shocks can drive the system toward extinction even when ${\mathcal{R}}_0>1$ deterministically, thereby providing a data-informed tool to delineate extinction-persistence boundaries in complex stochastic HIV/AIDS dynamics.