Asymptotic behaviour of the solutions to a virus dynamics model with diffusion

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the L₁ boundedness of the orbit. Then the precompactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number R₀ is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if R₀ > 1, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.