Spatial dynamics of a viral infection model with immune response and nonlinear incidence

Incorporating humoral immunity, cell-to-cell transmission and degenerated diffusion into a virus infection model, we investigate a viral dynamics model in heterogenous environments. The model is assumed that the uninfected and infected cells do not diffuse and the virus and B cells have diffusion. Firstly, the well-posedness of the model is discussed. And then, we calculated the reproduction number $$ account for virus infection, and some useful properties of $$ are obtained by means of the Kuratowski measure of noncompactness and the principle eigenvalue. Further, when $$, the infection-free steady state is proved to be globally asymptotically stable. Moreover, to discuss the antibody response reproduction number $$ of the model and the global dynamics of virus infection, including the global stability infection steady state and the uniform persistence of infection, and to obtain the k-contraction of the model with the Kuratowski measure of noncompactness, a special case of the model is considered. At the same time, when $$ and $$ ($$), we obtained a sufficient condition on the global asymptotic stability of the antibody-free infection steady state (the uniform persistence and global asymptotic stability of infection with antibody response). Finally, the numerical examples are presented to illustrate the theoretical results and verify the conjectures.