Spatial propagation in a within-host viral infection model

Recent experimental evidence suggests that spatial heterogeneity plays an important role in within-host infections caused by different viruses including hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To examine the spatial effects of viral infections, in this paper we study the asymptotic spreading in a within-host viral infection model, which describes the spatial expansion speeds of viruses and infected cells within an infected host. We first establish the boundedness of solutions to the Cauchy problem via local (Formula presented.) -estimates and dual arguments. Then the spreading speed is estimated when the basic reproduction number of the corresponding kinetic system is larger than one. More precisely, the upper bounds of the spreading speed are given by constructing suitable upper solutions while the lower bounds of the spreading speed are obtained by introducing an auxiliary equation with nonlocal delay. When the basic reproduction number of the corresponding kinetic system is less than or equal to one, the virus dies out uniformly. Finally, we present some numerical simulations to illustrate our theoretical findings and discuss the biological relevance of these results.