Dynamical analysis of a plateau pika disease model with time delay

In order to study the spread of plateau pika diseases, this paper proposes a delayed Filippov infectious disease model with a saturated incidence rate. This model adopts a dual-threshold control strategy and introduces a time delay to represent the virus’s incubation period, enhancing the model’s realism. First, we studied the stability of various equilibrium points and the existence of Hopf bifurcations. Then, we analyzed the conditions for the existence of sliding manifolds and their dynamical behavior using the Filippov convex method. Theoretical and numerical results indicate that, under varying thresholds and time delays, all solution trajectories of the system eventually converge to either regular equilibrium points, pseudo-equilibrium points, or a stable periodic solution. Additionally, we discussed boundary bifurcations, illustrating the transition from a stable regular equilibrium or periodic solution to a stable pseudo-equilibrium. Finally, we discussed global bifurcations, demonstrating the transition from grazing bifurcation to buckling bifurcation and then to crossing bifurcation as the time delay increases. The research results demonstrate that the Filippov control strategy can effectively manage the number of infected animals in many cases.