Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission

In this paper, we focus on spreading speed of a reaction-diffusion SI epidemic model with vertical transmission, which is a non-monotone system. More specifically, we prove that the solution of the system converges to the disease-free equilibrium as t -> infinity if R-0 <= 1 and if R-0 > 1, there exists a critical speed c degrees > 0 such that if parallel to chi parallel to= ct with c is an element of (0, C degrees), the disease is persistent and if parallel to chi parallel to >= ct with c > c degrees, the infection dies out. Finally, we illustrate the asymptotic behaviour of the solution of the system via numerical simulations.