Refining extinction criteria in a complex multi-stage epidemic system with non-Gaussian Lévy noise

This paper proposes a new approach for analyzing extinction conditions in multi-stage epidemic models, incorporating stochastic noises to account for sudden environmental or population-level changes that influence infection transmission. By utilizing an $(n+1)$-dimensional perturbed system that captures both gradual amelioration and proportional jumps, the research establishes sharp extinction criteria, refining the assumptions typically found in existing literature. Moving away from classical methods that rely on ergodicity, this framework employs moment analysis of the average solution time for an auxiliary equation, effectively addressing the challenges arising from the lack of an explicit expression of the invariant measure. The theoretical results are compared with those from previous studies, and numerical simulations, particularly focused on AIDS/HIV, serve to confirm and reinforce the findings.