Hopf Bifurcation and Sliding Mode Dynamics for an Anthrax Model with Time Delay and Spatial Diffusion

From the perspective of cost control, selecting the appropriate environmental purification threshold to suppress the anthrax spread remains challenging. In this paper, an environmental purification control model of anthrax with time delay and spatial diffusion is developed. By analyzing the relevant characteristic equations, we study the local stability of the regular equilibrium point and determine the sufficient conditions for the presence of Hopf bifurcation. Then, the direction of the Hopf bifurcation and stability of periodic solutions are studied using center manifold and normative theory. In addition, we identify the sliding domain and analyze its dynamical behavior by the Utkin equivalent control method. Finally, numerical simulations are conducted to validate the theoretical analysis.