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

Abstract

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 R0 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∗ and R0>1. We further prove the existence of time-periodic traveling waves with wave speed c=c∗ by a delicate limitation argument. For du=dh, the nonexistence of traveling waves is proved by a contradiction argument for two cases involved with c∗ and R0, while the exponential decay of traveling waves with the critical speed is obtained by a dynamical system approach combined with Laplace transform.