Turing–Hopf bifurcation and inhomogeneous pattern for a reaction–diffusion SIR epidemic model with chemotaxis and delay

In this paper, a reaction–diffusion SIR epidemic model with chemotaxis and delay is proposed to explore the completed dynamics, i.e., Turing and Hopf bifurcations and spatiotemporal inhomogeneous patterns. First, the basic reproduction number R0 is defined, and threshold criterion on the locally asymptotic stability of disease-free equilibrium is obtained. Second, the sufficient conditions on Turing bifurcation, Hopf bifurcation and Turing–Hopf bifurcation at the endemic equilibrium are obtained by taking delay and chemotaxis as bifurcation parameters respectively. It is proven that delay could induce Hopf bifurcation and chemotaxis could yield Turing bifurcation. Finally, the theoretical results and stable regions of endemic equilibrium with delay and chemotaxis are detailedly illustrated by numerical simulation. We find that complex spatiotemporal heterogeneous patterns could occur due to Hopf bifurcation-Turing instability, Hopf–Turing bifurcation or complex Hopf bifurcation, which could bring great challenges in disease prevention and control on each region with possible periodic outbreaks.