Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell density.

Abstract

We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state is globally asymptotically stable if . When , then becomes unstable, and another basic reproduction number of CTL response becomes the dynamic threshold in the sense that if , then the CTL-inactivated steady state is globally asymptotically stable; and if , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.