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In this paper, a stochastic delay differential system driven by the square wave signal is presented. The sensitivity of the system to different types of signals is evaluated by the dynamic behaviors of the system. The stochastic dynamic behaviors of the system is analyzed theoretically, and the theoretical chaotic threshold under different external excitation signals is deduced. The numerical simulation results show that the stochastic delay differential system is more sensitive to the square wave signals with the same frequency than the harmonic signal and the square wave signal with the different frequency.

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<abstract><p>We present a stochastic time-delay susceptible-exposed-asymptomatic-symptom-vaccinated-recovered (SEAQVR) model with media publicity effect in this study. The model takes into account the impacts of noise, time delay and public sensitivity on infectious illness propagation. The stochastic dynamics of the system are analyzed at the Hopf bifurcation, using time delay and noise intensity as bifurcation parameters, and the theoretical conclusions are validated using numerical simulation. Increasing the time delay and sensitivity coefficient can effectively delay the occurrence of the peak number of infected individuals and mitigate the extent of infection. Additionally, time delay and noise intensity are shown to have specific thresholds, beyond which periodic infections occur. Notably, heightened public sensitivity reduces the threshold for time delay, and media publicity directly affects public sensitivity. The numerical simulation reveals that increasing media publicity intensity does not always yield better results, and that the sensitivity of the public at present is an important reference index for setting an appropriate publicity intensity.</p></abstract>