Partly inhibiting of blow-up by chemotactic repulsion in a cross-diffusion model for virus infection

This work extends the May-Nowak model for virus dynamics to a cross-diffusion version, additionally illustrating diffusion in all components and chemotactically biased migration of healthy cells toward density gradients of infected cells as well as the repulsion of free viruses by infected cells, in which a quasi-steady-state approximation is applied to the equation describing the evolution of infected cells. It is firstly proven that for any given suitably regular initial data and for fixed other model parameters, one can find a number $ \chi_0>0 $ such that whenever the attractive chemotaxis coefficient $ \chi $ satisfying $ \chi\le \chi_0 $, the corresponding spatially two-dimensional no-flux initial-boundary problem admits a global classical solution that is uniform bounded. Moreover, our numerical simulations show that when the attractive chemotaxis coefficient $ \chi $ is sufficiently large, the solution blows up in a finite time; on the other hand, our numerical results also imply that the above-mentioned repulsion can partially suppress the blow-up in some sense.