This paper is concerned with the oscillation of the second-order nonlinear functional dynamic equations on a time scale where γ is the quotient of odd positive integers, r(t), p(t), and are positive rd-continuous functions on , , and . We establish some new sufficient conditions for oscillation for the above equation. Several examples illustrating our results will be given.

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature.

This work explores Routh–Hurwitz stability and complex dynamics in models for awareness programs to mitigate the spread of epidemics. Here, the investigated models are the integer‐order model for awareness programs and their corresponding fractional form. A non‐negative solution is shown to exist inside the globally attracting set (GAS) of the fractional model. It is also shown that the diseasefree steady state is locally asymptotically stable (LAS) given that R₀ is less than one, where R₀ is the basic reproduction number. However, as R₀ > 1, an endemic steady state is created whose stability analysis is studied according to the extended fractional Routh–Hurwitz scheme, as the order lies in the interval (0,2]. Furthermore, the proposed awareness program models are numerically simulated based on the predictor‐corrector algorithm and some clinical data of the COVID‐19 pandemic in KSA. Besides, the model’s basic reproduction number in KSA is calculated using the selected data (R₀ = 1.977828168)_. In conclusion, the findings indicate the effectiveness of fractional‐order calculus to simulate, predict, and control the spread of epidemiological diseases.