Mathematical analysis for a multi-group SEIR epidemic model with age-dependent relapse

Abstract

We consider a multi-group SEIR epidemic model in which recovered population relapse back to infectives depending on the time elapsed since the recovery. This leads to a hybrid system for which we can determine the basic reproduction number by the spectral radius of the next generation matrix and prove the threshold behaviors. The key idea to prove the global asymptotic stability of each equilibrium is the usage of the graph-theoretic approach to construct suitable Lyapunov functionals. The necessary arguments, including the existence of an endemic equilibrium, the asymptotic smoothness of the semiflow, the uniform persistence of the system, and the existence of a global attractor are also addressed.

Keywords:

• SEIR epidemic model
• multi-group model
• age-dependent relapse
• global asymptotic stability
• Lyapunov functional

AMS Subject Classifications:

• 35Q92
• 37N25
• 92D30

Acknowledgements

The authors would like to thank the anonymous referees and the editor for their helpful suggestions and comments which led to the improvement of our original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

J. Wang was supported by National Natural Science Foundation of China [grant number 11401182], [grant number 11471089]; Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province [grant number 2014TD005]; Youth Scientific Foundation of Heilongjiang University [grant number QL201203]; T. Kuniya was supported by Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science [grant number 15K17585]; the Japan Initiative for Global Research Network on Infectious Diseases (J-GRID) from Ministry of Education, Culture, Sport, Science & Technology in Japan, and Japan Agency for Medical Research and Development (AMED).