Traveling waves of a delayed HIV/AIDS epidemic model with treatment and spatial diffusion

In this paper, a delayed HIV/AIDS epidemic model with treatment and spatial diffusion is introduced. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state is discussed. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper–lower solutions. By constructing a pair of upper–lower solutions, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state. It is shown that the existence of traveling waves of the proposed HIV/AIDS epidemic model is fully determined by the basic reproduction number and the minimal wave speed. Finally, numerical simulations are performed to show the feasibility and effectiveness of the theoretical results.