**Zhe Yin, Yongguang Yu \* and Zhenzhen Lu**

Department of mathematics, Beijing Jiaotong University, Beijing 100044, China; 17121567@bjtu.edu.cn (Z.Y.); 18121571@bjtu.edu.cn (Z.L.)

**\*** Correspondence: ygyu@bjtu.edu.cn

Received: 20 February 2020; Accepted: 19 March 2020; Published: 23 March 2020

**Abstract:** This paper is concerned with the stability of an age-structured susceptible–exposed– infective–recovered–susceptible (SEIRS) model with time delay. Firstly, the traveling wave solution of system can be obtained by using the method of characteristic. The existence and uniqueness of the continuous traveling wave solution is investigated under some hypotheses. Moreover, the age-structured SEIRS system is reduced to the nonlinear autonomous system of delay ODE using some insignificant simplifications. It is studied that the dimensionless indexes for the existence of one disease-free equilibrium point and one endemic equilibrium point of the model. Furthermore, the local stability for the disease-free equilibrium point and the endemic equilibrium point of the infection-induced disease model is established. Finally, some numerical simulations were carried out to illustrate our theoretical results.

**Keywords:** SEIRS model; age structure; time delay; traveling wave solution; local asymptotic stability; Hopf bifurcation
