Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment

Abstract

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction.

Keywords:

• SIS epidemic system
• reaction-diffusion
• delay
• Hopf bifurcation
• stability
• Lyapunov-Schmidt reduction
• heterogeneous environment
• endemic equilibria

2010 MSC Subject Classifications:

• 34K18
• 92D30

Acknowledgements

I am grateful to the editor and referees for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant Nos. 12071446, 11801089, and 11671123) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

I am grateful to the editor and referees for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China(Grant Nos. 12071446, 11801089, and 11671123) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan).