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.

It is reasonable to use uncertain nonlinear differential equations to model systems that are subject to noise disturbances of unstable frequency, such as uncertain financial systems, biological population systems, infectious disease systems, and pharmacokinetic systems. Due to the coupling effect of nonlinearity and uncertainty, it is challenging to directly solve the responses of these equations. This paper addressed the challenge of studying the responses of these systems, especially the changes in steady-state behavior caused by parameter variations, known as "bifurcation" phenomena. We defined the concept of Hopf bifurcation in uncertain differential equations using cross-entropy and investigated the bifurcation phenomena in a class of second-order uncertain nonlinear differential equations. An efficient algorithm was designed to verify uncertain Hopf bifurcation and quantify the bifurcation threshold, with the validity of our definition confirmed through numerical simulations. This paper extended the classical Hopf bifurcation of ordinary differential equations to uncertain differential equations via the $\alpha$-path, thereby proposing a theoretical framework for uncertain bifurcation within uncertain dynamics.

In this paper, we study the higher-order uncertain differential equations (UDEs) as defined by Kaixi Zhang ^[11], mainly focus on the second-order case. We propose a pivotal condition (monotonicity in some sense, see more details in Section 3), introduce the concept of $\alpha$-paths of UDEs, and demonstrate its properties. Based on this, we derive the inverse uncertainty distribution of the solution. Finally, we present numerical examples to substantiate the rationality of the condition.