Dynamics of solutions of a diffusive time-delayed HIV/AIDS epidemic model: Traveling wave solutions and spreading speeds

We study a diffusive time-delayed HIV/AIDS epidemic model with information and education campaigns and investigate the dynamics of classical solutions of the model. In particular, we address the questions of disease's persistence-extinction, existence of epidemic waves, and spreading speeds. When the basic reproduction number is less than or equal to one, we show that the disease-free equilibrium solution is globally stable, hence there is no epidemic wave in this case. However, if it is bigger than one, we show that the disease will eventually persist. Furthermore, there is a minimum wave speed c_u^⁎, which decreases as a function of time-delay u, such that the system has an epidemic traveling wave solution with speed c for every c greater than c_u^⁎ and that there is no such traveling wave solution of speed less than c_u^⁎. Moreover the minimum wave speed c_u^⁎ converges to 0 as the time-delay approaches infinity. We also study the disease spreading speeds interval and show that in the absence of time-delay, there is a single disease spreading speed and this coincides with the minimal wave speed. We conclude with numerical simulations to illustrate our findings.