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

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.

Automated summary

This paper provides an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in a heterogeneous environment under homogeneous Neumann boundary condition. It explores the stability and effects of nonhomogeneous coefficients on disease-free equilibrium, as well as the existence, multiplicity, and structure of endemic equilibrium. The paper also analyzes the stability of endemic equilibrium, the existence of Hopf bifurcations, and the direction and stability of bifurcating periodic solutions.