Global stability analysis of a viral infection model in a critical case

Recently, it has been proved that for the diffusive viral infection model with cell-to-cell infection, the virus-free steady state E₀ is globally attractive when the basic reproduction number R₀ < 1, and the virus is uniformly persistent if R₀ > 1. However, the global stability analysis in the critical case of R₀ = 1 is not given due to a technical difficulty. For the diffusive viral infection model including a single equation with diffusion term, global stability analysis in the critical case has been performed by constructing Lyapunov functions. Unfortunately, this method is not applicable for two or more equations with diffusion terms, which was left it as an open problem. The present study is devoted to solving this open problem and shows that E₀ is globally asymptotically stable when R₀ = 1 for three equations with diffusion terms by means of Gronwall's inequality, comparison theorem and the properties of semigroup.