Stability and Hopf bifurcation analysis of an HIV infection model with latent reservoir, immune impairment and delayed CTL immune response

TL;DR

This work extends an HIV infection model by incorporating latent reservoirs, saturating CTL responses, dual transmission pathways, and three time delays ($\tau_1$, $\tau_2$, $\tau_3$). It establishes nonnegativity and boundedness of solutions, derives the basic and immune reproduction numbers $\mathcal{R}_0$ and $\mathcal{R}_1$, and classifies three equilibria $E_0$, $E_1$, and $E_2$ with explicit conditions for their global stability. A Hopf bifurcation analysis shows that increasing the CTL-delay $\tau_3$ can destabilize the endemic equilibrium $E_2$, giving rise to stable or unstable periodic solutions depending on center-manifold coefficients. Numerical simulations corroborate the analytical results, revealing robust global stability regimes and delay-induced oscillations. The findings highlight how latency and immune-response timing regulate HIV persistence and rebound, offering insights for timing-based therapeutic strategies, while acknowledging the limitations of a deterministic framework and proposing future stochastic extensions.

Abstract

In this paper, we develop a dynamic model of HIV infection that incorporates latent hosts, cytotoxic T lymphocyte (CTL) immunity, saturated incidence rates, and two transmission mechanisms: virus-to-cell and cell-to-cell transmission. The model has three kinds of delays: intracellular delay, replication of viruses delay, immune response delay. Initially, the model's solutions are confirmed to be both nonnegative and bounded for nonnegative initial values. Subsequently, two biologically critical parameters were identified: the virus reproduction number $\mathcal{R}_0$ and the immune reproduction number $\mathcal{R}_1$. Thereafter, by invoking LaSalle's principle of invariance alongside Lyapunov functionals, we establish stability criteria for each equilibrium. The results indicate that the stability of the endemic equilibrium may be altered by a positive immune delay, whereas intracellular and viral replication delays do not affect the equilibria. By considering the delay in the immune response as a bifurcation-inducing threshold, we derive the exact conditions necessary for these stability transitions. Further analysis shows that increasing the immune delay destabilizes the endemic equilibrium, inducing a Hopf bifurcation. Additionally, using the center manifold theorem and normal form theory, we explored the direction and stability of Hopf bifurcations in detail. To corroborate these theoretical results, numerical simulations are systematically conducted.

Figures (13)

• Figure 1: The pictorial representation of proposed model
• Figure 2: The curves of functions $z$ and $y$
• Figure 3: Simulation result of $E_0$ while $\tau_1=1$, $\tau_2=1$ and$\tau_3=2$
• Figure 4: Simulation result of $E_1$ while $\tau_1=1$, $\tau_2=1$ and$\tau_3=2$
• Figure 5: Simulation result of $E_2$ while $\tau_1=0.25$, $\tau_2=0.25$ and$\tau_3=0$

Theorems & Definitions (12)

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• Lemma 4.1
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