Modeling dual exposure risk in a delayed epidemic model with nonlinear thresholds

This paper proposes and studies a Filippov SIR model with dual delays, dual exposures and dual dynamic thresholds, aiming to reveal the complex epidemic dynamics under the combined effects of high exposure risk, delay effects and feedback control strategies. First, the dynamic characteristics of the three subsystems with delays are analyzed in depth. Then, through theoretical derivation, the existence conditions for the sliding segments and pseudo-equilibria of the Filippov SIR model without delays are presented. Numerical simulations further uncover a series of dynamic phenomena, including sliding bifurcation, touching bifurcation, pseudo-equilibrium bifurcation and the coexistence of a stable foci with a crossing cycle. Additionally, the impact of delays on the Filippov SIR model is explored, revealing that delay effects can lead to oscillatory periodic solutions, quasi-periodic solutions and chaos. The results indicate that reducing delay effects and exposure intensity is beneficial for controlling the spread of infectious diseases. Meanwhile, dynamically adjusting treatment strategies based on the status of susceptible and infected individuals not only facilitates the rational allocation of medical resources but also effectively reduces control costs.