Periodic traveling waves for a diffusive influenza model with treatment and seasonality

This paper is concerned with the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By using the next generation operator theory, we first get basic reproduction number R-0 for the corresponding periodic ODEs. Then, by constructing sub-and super-solutions and using Schauder's fixed point theorem, we obtain the existence of time-periodic traveling wave solutions for the system with wave speed c >c(& lowast;) and R-0 > 1. We further prove the existence of time-periodic traveling waves with wave speed c = c(& lowast;) by a delicate limitation argument. For d(u) = d(h), the nonexistence of traveling waves is proved by a contradiction argument for two cases involved with c(& lowast;) and R-0, while the exponential decay of traveling waves with the critical speed is obtained by a dynamical system approach combined with Laplace transform.