The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring $\Im /\wp$, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂ that satisfies certain criteria involving ℘, where $\alpha$ and $\beta$ are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results.

Capturing the factors influencing yellow fever (YF) outbreaks is essential for effective public health interventions, especially in regions like Nigeria, where the disease poses significant health risks. This study explores the synergistic effects of active case detection (ACD) and early hospitalization on controlling YF transmission dynamics. We develop a dynamic model that integrates vaccination, active case detection, and hospitalization to enhance our understanding of disease spread and inform prevention strategies. Our methodology encompasses mechanistic dynamic modeling, optimal control analysis, parameter estimation, model fitting, and sensitivity analyses to study YF transmission dynamics, ensuring the robustness of control measures. We employ advanced mathematical techniques, including next-generation matrix methods, to accurately compute the reproduction number and assess outbreak transmissibility. Rigorous qualitative analysis of the model reveals two equilibria: disease-free and endemic, demonstrating global asymptotic stability and its impact on overall YF transmission dynamics, significantly affecting control and prevention mechanisms. Furthermore, through sensitivity analysis, we identify crucial parameters of the model that require urgent attention for more effective YF control. Moreover, our results highlight the critical roles of ACD and early hospitalization in reducing YF transmission. These insights provide a foundation for informed decision making and resource allocation in epidemic control efforts, ultimately contributing to the enhancement of public health strategies aimed at mitigating the impact of YF outbreaks.

The purpose of this paper is to examine the behavior of a factor ring $R/P$ with a partial range I of a ring R, where P is a prime ideal of R and I is a non-zero ideal of R such that $P⊊I$. In order to accomplish this objective, we will utilize specific algebraic identities that involve P and are related to generalized $(\alpha ,\beta )$-derivations F and G, associated with $(\alpha ,\beta )$-derivations d and g, respectively, where $\alpha$ and $\beta$ are automorphisms of R. Additionally, we will list some important related consequences. Finally, we will provide an illustrative example to emphasize the importance of the hypotheses imposed in our theorems.

Let G be a finite group. A subgroup H of G is said to be weakly S-permutable in G if G has a subnormal subgroup T such that G = HT and T ∩ H ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. In this paper, we prove the following: For a Sylow p-subgroup P of G (p > 2), suppose that P has a subgroup D such that 1 < |D| < |P| holds and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then, the commutator subgroup G ′ is p-nilpotent. We certainly belive that this result will improve and extend a current and classical theories in the literature.

In this article, for a prime ideal $\rho$ of an arbitrary ring ℜ, we study the commutativity of the quotient ring $\Re /\rho$, whenever ℜ admits a generalized reverse derivation $\vartheta$ associated with a reverse derivation ∂ that satisfies certain identities in $\rho$. Additionally, we show that, for some cases, the range of the generalized reverse derivation $\vartheta$ lies in the prime ideal $\rho$. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our results are essential.