Spatiotemporal dynamics of a diffusive SI model in the regions of Turing-Hopf bifurcation point

The study of the spatiotemporal distribution of epidemics is one of the important topics in spatial epidemiology, and specifically, the spatiotemporal patterns of epidemics can effectively reveal the spread law and spatiotemporal distribution characteristics of epidemic. However, the spatiotemporal patterns of infected individuals are not well studied. To this end, this paper explores the spatiotemporal dynamics of a diffusive SI model with nonlinear incidence. Firstly, we theoretically analyze the existence of Hopf bifurcation, Turing bifurcation and Turing-Hopf bifurcation, and secondly, we accurately delineate the regions near the Turing-Hopf bifurcation point where different dynamical behaviors occur by analyzing its normal form. Then, we verify the theoretical analysis through numerical simulation. The results show that by slightly perturbing the control parameters, the system can switch between the four spatiotemporal patterns. In addition, we find that when the parameters are varied substantially, it may lead to the phenomenon that the system exhibits the coexistence of different steady states. This suggests that the diversity of spatiotemporal patterns of epidemics will increase significantly with the intervention of the Turing-Hopf bifurcation. These findings also provide theoretical support for epidemic prevention and control.