Global stability of the steady states of an SIS epidemic reaction–diffusion model

In this work, we investigate the SIS epidemic reaction–diffusion model under heterogeneous environment studied by Allen et al. in [L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model, Discrete Contin. Dyn. Syst. A 21 (1) (2008) 1–20]. In the two cases: (1) the diffusion rate dSdS of the susceptible individuals is equal to the diffusion rate dIdI of the infected individuals; (2) β(x)=rγ(x)β(x)=rγ(x) for any fixed constant r∈(0,∞)r∈(0,∞), where β(x)β(x) and γ(x)γ(x) respectively represent the rates of disease transmission and disease recovery, we completely determine the global stability of the disease-free equilibrium and the unique endemic equilibrium (if it exists). Our results partially answer the conjecture proposed by Allen, et al.