**6. Concluding Remarks**

In this article, the theory of an age-structured SEIRS model with time delay is analyzed. The model is based on the delayed nonlinear partial differential equation of initial-boundary value problems. The traveling wave solution of the system in Equation (1) is obtained using the method of characteristic and the recurrent algorithm. Then, we can obtain the existence and uniqueness of the continuous traveling wave solution of system according to hypotheses. The age-structured SEIRS model with time delay is reduced to a nonlinear ordinary differential equation under some insufficient simplifications. This allows us to obtain some sufficient conditions of existence of two equilibrium points of an age-structured SEIRS system: *R*<sup>1</sup> is a dimensionless index for the existence of the disease-free equilibrium point *H*0. *R*2, *R*3, and *R*<sup>4</sup> are dimensionless indexes for the existence of the endemic equilibrium point *H*∗. From the biological point of view, the endemic equilibrium point *H*∗ only exists in the case of high values of death and conversion rate of exposed and infected population. The disease-free equilibrium point *H*<sup>0</sup> and the endemic equilibrium point *H*∗ are given. The disease-free equilibrium point *H*<sup>0</sup> = (0, 0, 0, 0) is locally asymptotically stable if *R*<sup>1</sup> < 1, *R*<sup>2</sup> < 1, and *R*<sup>3</sup> < 1. The stability of the endemic equilibrium point *H*<sup>∗</sup> = (*N*<sup>∗</sup> *<sup>s</sup>* , *N*<sup>∗</sup> *<sup>e</sup>* , *N*<sup>∗</sup> *<sup>i</sup>* , *N*<sup>∗</sup> *<sup>r</sup>* ) with *τ* = 0 and *τ* > 0 are analyzed: for *R*<sup>0</sup> > 1, if the condition in Equation (36) holds, the endemic equilibrium point *H*∗ is locally asymptotically stable when *τ* = 0; if the conditions in Equations (43) and (44) hold, the endemic equilibrium point *H*<sup>∗</sup> is locally asymptotically stable when time delay 0 < *τ* < *τ*0; if the conditions in Equations (43)–(44) hold, the endemic equilibrium point *H*∗ is unstable when *τ* satisfies *τ* > *τ*0; Hopf bifurcation occurs at *τ* = *τk*(*k* = 0, 1, 2, ...). When time delay exceeds the critical value *τ*0, the system in Equation (20) loses its stability and Hopf bifurcation occurs. In this case, the susceptible, exposed, infected, and recovered population in the model will coexist in an oscillating mode and infectious diseases will get out of control. It can be seen that time delay has an important effect on the spread of infectious diseases. Therefore, we should shorten the time delay as much as possible in order to predict and eliminate infectious diseases. Our further research is using some bifurcation control strategies to control the occurrence of the Hopf bifurcation so as to control the occurrence of infectious diseases.

In general, this study provides the practical understanding of the different dynamic behaviors of an age-structured susceptible–exposed–infected–recovered–susceptible model with time delay, which is helpful for us to understand the application of epidemiology better in real life.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2227-7390/8/3/455/s1, Program S1: Software source code for Figures 1–4.

**Author Contributions:** Conceptualization, Z.Y.; Funding acquisition, Y.Y.; Methodology, Z.Y.; Software, Z.Y. and Z.L.; and Supervision, Y.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Nature Science Foundation of China (No. 61772063) and Beijing Natural Science Foundation (No. Z180005).

**Acknowledgments:** The authors would like to thank for the support funds.

**Conflicts of Interest:** The authors declare that there are no conflicts of interest regarding the publication of this paper.

## **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
