Sliding dynamics and codimension-2 bifurcations of an epidemic Filippov system with nonlinear threshold control

The implementation of preventive and control measures for major infectious diseases is often influenced by a multitude of factors, including the progression of infectious diseases, the current epidemic status, and the population size of various disease states. This paper introduces a threshold control strategy based on a non-smooth Filippov system, wherein the weighted sum of the susceptible population size and its change rate determines whether to enforce vaccination and isolation measures. We investigate the impact of this strategy on the dynamics of infectious disease transmission and analyze the effects of intermittent vaccination and isolation strategies with nonlinear recovery and threshold control functions. Based on the dynamics of subsystems, we analyze the sliding mode and the properties of the sliding regions, as well as the existence of the pseudo-equilibria. Additionally, we analyze the codimension-1 boundary equilibrium bifurcations of the proposed system, including boundary node bifurcation, boundary stable/unstable focus bifurcation, and boundary unstable-stable focus bifurcation. Leveraging the rich codimension-1 boundary equilibrium bifurcations, we explore two types of codimension-2 bifurcations and numerically illustrate the homoclinic boundary focus bifurcation and boundary Hopf bifurcation. Through an in-depth examination of boundary equilibrium bifurcations, we discover that the proposed system displays complex dynamical behaviors under different parameter values, including the emergence of new limit cycles, saddle–node bifurcations and grazing bifurcations of limit cycles. The main results indicate that under a specific control strategy, there exists a threshold value for the weighted sum of the size and change rate of the susceptible population that can effectively control the spread of infectious diseases. Moreover, whether the infected population remains low is contingent on the system’s initial state. Consequently, tailored and comprehensive control strategies must be devised to address the distinct characteristics of different population groups.