A Global Lyapunov Resolution of the Zeraoulia–Sprott HIV-Therapy Conjecture

Abstract

We study the autonomous three-dimensional HIV dynamics model formed by the concentrations of CD4 lymphocytes, CD8 lymphocytes, and the HIV-1 viral load. Zeraoulia and Sprott conjectured, on the basis of numerical evidence, that the system is nonchaotic for every positive choice of its parameters. We give a self-contained global proof. A threshold Δ = p₅x₁⁰ − p₆x₂⁰ separates two regimes. If Δ ≤ 0, every nonnegative solution converges to the virus-free equilibrium. If Δ > 0, every solution with positive initial viral load converges to a unique positive equilibrium, whereas the invariant face of zero viral load converges to the virus-free equilibrium. The proof uses explicit relative-entropy Lyapunov functions whose mixed terms cancel exactly, followed by LaSalle’s invariance principle. In particular, every nonnegative trajectory converges to an equilibrium, which excludes periodic orbits and compact chaotic attractors in the biologically relevant state space. MSC 2020: 34D23 (primary); 34C28, 92B05 (secondary).