Analysis of a reaction–diffusion epidemic system with spatial heterogeneity and delay

This paper provides a thorough qualitative analysis of the dynamic behavior of a reaction–diffusion SIS epidemic system with delay in a heterogeneous environment subject to a homogeneous Neumann boundary condition. To begin with, the well-posedness including the existence and uniqueness of the solution was proved. Second, the basic reproduction number was defined and the threshold dynamics of the model were studied, and the asymptotic behavior and monotonicity of the basic reproduction number associated with the diffusion rate were discussed. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the existence of Hopf bifurcations at endemic equilibrium is given. In addition, the direction of Hopf bifurcation and stability of the bifurcating periodic solution were obtained by virtue of the normal form theory and center manifold reduction. Finally, some numerical simulations were provided to illustrate our theoretical results.