In this article, the quasi-synchronization and Hopf bifurcation issues are investigated for the fractional-order quaternion-valued neural networks (QVNNs) with time delay in the presence of parameter mismatches. On the basis of noncommutativity property of quaternion multiplication results, the quaternion network has been split as four real-valued networks. A synchronization theorem for fractional-order QVNNs is derived by employing suitable Lyapunov functional candidate; furthermore, the bifurcation behavior of the hub-structured fractional-order QVNNs with time delay has been investigated. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical results.

The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real (R), complex (C), quaternion (Q), and octonion (O) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.

In this study, we provide a fractional-order mathematical model that considers the effect of vaccination on COVID-19 spread dynamics. The model accounts for the latent period of intervention strategies by incorporating a time delay $\tau$. A basic reproduction number, $R_0$, is determined for the model, and prerequisites for endemic equilibrium are discussed. The model’s endemic equilibrium point also exhibits local asymptotic stability (under certain conditions), and a Hopf bifurcation condition is established. Different scenarios of vaccination efficacy are simulated. As a result of the vaccination efforts, the number of deaths and those affected have decreased. COVID-19 may not be effectively controlled by vaccination alone. To control infections, several non-pharmacological interventions are necessary. Based on numerical simulations and fitting to real observations, the theoretical results are proven to be effective.

In this study, we investigate the problem of multiple Mittag-Leffler stability analysis for fractional-order quaternion-valued neural networks (QVNNs) with