Stability and Hopf Bifurcation Analysis of a Delayed Reaction-Diffusion System for CD8 T Cell Response to Viral Infection

Abstract

In this study, a three-variable time-delay reaction-diffusion system was constructed to characterize the immune response of CD8T
cells to viral infection, covering infected cells, free viruses and CD8T effector cells. Uninfected target cells were approximated as constants
by quasi-steady-state approximation. To reflect the time lag from antigen recognition to function of T cells, the system introduces an immune
activation delay. Under Neumann boundary conditions, the conditions for the existence of equilibrium points and the basic number of regenerates are given first, and then the local asymptotic stability of the system without delay and the Turing instability criterion are analyzed. Taking
the time delay as the bifurcation parameter, the Hopf bifurcation critical threshold is derived from the root distribution of the characteristic
equation, and the bifurcation direction and periodic solution stability are determined with the help of the central manifold theorem and gauge
theory. Numerical experiments show that the increase in time delay leads to steady-state instability and causes periodic oscillations, and the
diffusion effect can expand the stable region and delay the occurrence of bifurcation. This study provides a theoretical perspective for understanding the impact of T-cell response delay on infection persistence.