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

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 \(R_0\) $R_0$ and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state \(E_0\) $E_0$ is globally asymptotically stable if \(R_0<1\) $R_0<1$ . When \(R_0>1\) $R_0>1$ , then \(E_0\) $E_0$ becomes unstable, and another basic reproduction number of CTL response \(R_1\) $R_1$ becomes the dynamic threshold in the sense that if \(R_1<1\) $R_1<1$ , then the CTL-inactivated steady state \(E_1\) $E_1$ is globally asymptotically stable; and if \(R_1>1\) $R_1>1$ , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state \(E_2\) $E_2$ 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.