Global Stability Analysis of a Mathematical Model for the "Shock-and-Kill" Strategy in HIV-1/SIV Brain Infection

This paper provides a rigorous mathematical resolution of the open global stability problem for a "shock-and-kill" model of HIV-1/SIV infection in brain reservoirs recently formulated by Roda et al. (2021). The model explicitly incorporates the effects of latency-reversing agents and enhanced immune clearance of reactivated cells. We derive an explicit formula for the basic reproduction number $R_0$ , which serves as the sole threshold parameter governing viral eradication versus persistence and integrates infection pathways from both productive and latent compartments. By combining the next-generation matrix approach with an extended graph-theoretic Lyapunov method for multigraphs with parallel arcs, we rigorously establish that the disease-free equilibrium is globally asymptotically stable when $R_0\le 1$ , whereas a unique productive equilibrium exists and is globally asymptotically stable when $R_0>1$ . To resolve the sign-indefinite quadratic perturbations induced by structurally distinct parallel transmission arcs-a fundamental bottleneck of classical graph-theoretic Lyapunov schemes-we develop a refined composite Lyapunov framework equipped with hierarchically calibrated parameters. Systematic asymptotic scaling and multi-parameter tuning eliminate indefinite cyclic quadratic interactions, securing strict negative definiteness of the Lyapunov derivative and overcoming key limitations of conventional graph-based methods. These global stability results provide a definitive mathematical answer to whether therapeutic interventions guarantee viral eradication or lead to persistent brain-reservoir infection. Furthermore, they furnish a rigorous theoretical foundation for the "shock-and-kill" strategy and establish mathematically precise conditions to guide the design of safe, effective interventions for eliminating HIV-1/SIV from CNS reservoirs.