Global dynamics of a delayed viral infection model mediated by pattern recognition receptors with CTL immune response

In this paper, we develop a viral dynamics model incorporating three distinct time delays to characterize the complex interplay between viral infection and host immune responses. In the modeling process, we integrate the innate immune signaling pathway mediated by pattern recognition receptors (PRRs) and couple it with the specific adaptive immune response involving cytotoxic T lymphocytes (CTLs). Three physiologically meaningful delays---namely, the intracellular delay, the virus replication delay, and the CTL immune response delay---are introduced to account for the temporal lags in viral production, de novo cell infection, and adaptive immune recruitment, respectively. By constructing appropriate Lyapunov functionals and applying LaSalle's invariance principle, we rigorously establish the global asymptotic stability conditions for both the infection-free equilibrium and the infected equilibrium, thereby elucidating the influence of delays on the long-term system dynamics. Furthermore, taking the CTL immune response delay as the bifurcation parameter, we analyze the distribution of the characteristic roots and derive sufficient conditions for the existence of Hopf bifurcations. Our results demonstrate that an increase in this delay can induce periodic oscillations, providing a theoretical explanation for the fluctuations in viral load and the dynamic imbalance of immune responses observed in clinical settings. Numerical simulations are performed to corroborate the theoretical findings.