**1. Introduction**

In recent years, the study of epidemiology has been a vital problem in ecology. The research of population dynamics has developed rapidly, and many mathematical models have been used to analyze various infectious diseases. Many results have been established in the stability analysis of different epidemic models. The first susceptible–infective–recovered (SIR) epidemic model about disease transmission was established by Kermack and McKendrick in 1927 [1]. Since then, the population dynamics of infectious diseases have attracted the attention of scientists. In 2012, the field of mathematical biology was expanded, particularly in the context of the spread of infectious diseases by Fred Brauer et al. [2]. Nowadays, there are some research work devoted to study the stability of steady states of the SIR, SIRS, SEIR, etc. models [3–7]. It is well known, in the spread of infectious diseases, some infective individuals of population are immune after being recovered (e.g., measles, smallpox, mumps, and others). Meanwhile, some recovered individuals have no immunity (e.g., AIDS, hyperthyroidism, lupus erythematosus, and others), who will return to the susceptible population and continue to be infected. In fact, the probability of becoming infected is different among different individuals, which may depend on the type of infectious diseases and the status of individuals. Therefore, it is necessary to discuss the SEIRS model, which can more clearly describe the spread of infectious diseases in real life.

Time delay is ubiquitous and can be applied in many epidemiology related studies [8,9]. For example, measles has an incubation period of 8–13 days and the incubation period of canine madness is a few months or several years after infection. Sharma et al. [10] developed a five compartmental infection model to describe the spread of avian influenza A (H7N9) virus with two discrete time delays. In addition, Xu et al. [11] analyzed the stability of a SIRS model with time delay. Similarly, Shu [12] and

De la Sena [13] discussed the stability of the SEIR epidemic models with distributed delay respectively. Actually, many authors, such as Cooke [14], Gao [15] and Wang [16], have studied various SEIRS models with time delay.

Besides, age structure is also an important consideration in infectious diseases modeling such as rubella, poliomyelitis, and pertussis, which are transmitted only among children, and venereal diseases, which are transmitted only among adults. Besides, tuberculosis virus carriers in the early incubation period have a higher risk of becoming infective individuals than ones in the late incubation period [17]. Age-structured models have been applied in the epidemic dynamics for decades. In 1986, the dynamics of structured populations was discussed by Metz et al. [18]. Then, the mathematical theory of the age-structured population dynamics was proposed by Iannelli [19]. Afterwards, more and more epidemic models with age structure were studied in [20–28]. Recently, a new age-structured malaria model incorporating the age of latent period and the age of prevention period was formulated by Guo et al. [29]. A new SIRS epidemic model with relapse and infection age on a scale-free network was introduced Huo et al. [30]. However, as far as we can tell, there have been no results on an age-structured SEIRS model with time delay.

The main aim of this paper is to study the stability of an age-structured SEIRS model with time delay. The well-known method of characteristics [25–28] for first-order hyperbolic equations is used to solve this epidemic model. The explicit traveling wave solution is calculated at the preceding moment of time and is described in integral form. Under some hypotheses, the existence and uniqueness of the continuous traveling wave solution of the age-structured SEIRS model is investigated. Moreover, an age-structured SEIRS model with time delay is reduced to the nonlinear ordinary differential equation under some insignificant simplifications. After that, the dimensionless indexes are derived for the existence of the disease-free equilibrium point and the endemic equilibrium point. The local asymptotic stability of the disease-free equilibrium point is studied. By using Hurwitz's criterion and Descartes' rule of signs, the local asymptotic stability of the endemic equilibrium point of system is obtained.

The rest of the paper is organized as follows. In Section 2, an age-structured SEIRS model with time delay is proposed. In Section 3, the traveling wave solution is obtained and some sufficient conditions are established to guarantee the existence and uniqueness of the solution. In Section 4, the stability and Hopf bifurcation analysis of the proposed model are discussed. In Section 5, numerical simulations are provided to illustrate the effectiveness of our main results. Finally, some conclusions are given in Section 6.
