Analysis of the Diffusion SIR Epidemic Model With Networked Delay and Nonlinear Incidence Rate

This paper is concerned with the qualitative analysis of a diffusion SIR epidemic model with networked delay and nonlinear incidence rate. First, we prove the existence and uniqueness of the model by using the method of upper and lower solutions. Then, we prove that the trivial equilibrium (0, 0) is unstable; the disease‐free equilibrium ( N , 0) is locally asymptotically stable if the reproduction number and unstable if ; the endemic equilibrium ( S ∗ , I ∗ ) is locally asymptotically stable if or if and ; and ( S ∗ , I ∗ ) is unstable if and . Moreover, when , we show that hopf bifurcation occurs at ( S ∗ , I ∗ ) and . Numerical results are provided for theoretical discoveries.