In this paper, we have constructed two muscular blood vessel systems influenced by external disturbances by using the non-autonomous delayed differential equation (DDE). As an important physiological structure of the human body, muscle blood vessels participate in many activities such as blood flow. Many diseases are associated with abnormal dynamics of muscle blood vessels. From a mathematical point of view, vasospasm is caused by a chaotic state of blood vessels. Vasospasm is the manifestation of this disease in the vascular system, which can cause blood vessel blockage and even harm human health when it is serious. We conducted the dynamical analysis of the S-type and N-type muscular blood vessel systems by utilizing bifurcation diagrams, time histories, and pseudo-phase portraits, and investigated the effects of different parameters on these systems. Specifically, when parameters change, rich dynamical phenomena occur, such as equilibria, periodic solutions, quasi-periodic solutions, Hopf bifurcation, and chaos, as well as the route of period-doubling bifurcation to chaos. Meanwhile, we analyzed approximate analytical solutions by using the method of multiple scales (MMS) and determined the stability of steady-state solutions through the Routh-Hurwitz criterion. The results indicate that the MMS can deduce better analytical results for some non-autonomous DDEs.

With the continuous evolution of the coronavirus, the Omicron variant has gradually replaced the Delta variant as the prevalent strain. Their inducing epidemics last longer, have a higher number of asymptomatic cases, and are more serious. In this article, we proposed a nonautonomous time-delayed susceptible-infected-removed (NATD-SIR) model to predict them in different regions of China. We obtained the maximum and its time of current infected persons, the final size, and the end time of COVID-19 epidemics from January 2022 in China. The method of the fifth-order moving average was used to preprocess the time series of the numbers of current infected and removed cases to obtain more accurate parameter estimations. We found that usually the transmission rate $\beta (t)$ was a piecewise exponential decay function, but due to multiple bounces in Shanghai City, $\beta (t)$ was approximately a piecewise quadratic function. In most regions, the removed rate $\gamma (t)$ was approximately equal to a piecewise linear increasing function of (a*t+b)*H(t-k), but in a few areas, $\gamma (t)$ displayed an exponential increasing trend. For cases where the removed rate cannot be obtained, we proposed a method for setting the removed rate, which has a good approximation. Using the numerical solution, we obtained the prediction results of the epidemics. By analyzing those important indicators of COVID-19, we provided valuable suggestions for epidemic prevention and control and the resumption of work and production.