Spatial and Temporal Dynamics of a Viral Infection Model with Two Nonlocal Effects.

We propose and study a viral infection model with two nonlocal effects and a general incidence rate. First, the semigroup theory and the classical renewal process are adopted to compute the basic reproduction number R-0 as the spectral radius of the next-generation operator. It is shown that R-0 equals the principal eigenvalue of a linear operator associated with a positive eigenfunction. Then we obtain the existence of endemic steady states by Shauder fixed point theorem. A threshold dynamics is established by the approach of Lyapunov functionals. Roughly speaking, if R-0 < 1, then the virus-free steady state is globally asymptotically stable; if R-0 > 1, then the endemic steady state is globally attractive under some additional conditions on the incidence rate. Finally, the theoretical results are illustrated by numerical simulations based on a backward Euler method.