Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

Abstract

This paper deals with the problem of global asymptotic stability for equilibria of a spatially diffusive SIR epidemic model with homogeneous Neumann boundary condition. By discretizing the model with respect to the space variable, we first construct Lyapunov functions for the corresponding ODEs model, and then broaden the construction method into the PDEs model in which either susceptible or infective populations are diffusive. In both cases, we obtain the standard threshold dynamical behaviors, that is, if , then the disease-free equilibrium is globally asymptotically stable and if , then the (strictly positive) endemic equilibrium is so. Numerical examples are given to illustrate the effectiveness of the theoretical results.

Keywords:

• SIR epidemic model
• Lyapunov function
• diffusion
• basic reproduction number

AMS Subject Classifications:

• 35Q92
• 37N25
• 92D30

Acknowledgements

We are deeply grateful to the editor and the anonymous reviewer for their helpful comments and suggestions to improve the previous version of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

TK was supported by Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science [No. 15K17585]; the program of the Japan Initiative for Global Research Network on Infectious Diseases (J-GRID); from Japan Agency for Medical Research and Development, AMED. JW 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 [No. 2014TD005]; Youth Innovation Talents of Heilongjiang Education Department.