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

Abstract

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 R0≤1 and unstable if R0>1; the endemic equilibrium S∗,I∗ is locally asymptotically stable if R0≤R1 or if R1τ10∗. Moreover, when R1<R0, we show that hopf bifurcation occurs at S∗,I∗ and τ=τ10∗. Numerical results are provided for theoretical discoveries.