Breast cancer is the most frequent cancer in the world, and it continues to have a significant impact on the total number of cancer deaths. Recently, oncology findings hint at the role of excessive glucose in cancer progression and immune cells' suppression. Sequel to this revelation is ongoing researches on possible inhibition of glucose flow into the tumor micro-environment as therapeutics for malignant treatment. In this study, the effect of glucose blockage therapeutics such as SGLT-2 inhibitors drug on the dynamics of normal, tumors and immune cells interaction is mathematically studied. The asymptomatic nature of the breast cancer is factored into the model using time delay. We first investigate the boundedness and non-negativity of the solution. The condition for existence of critical equilibrium point is determined, and its global stability conditions are derived using Lyapunov function. This revealed that a timely administration of the SGLT-2 inhibitors drug can eliminate tumor cells. Secondly, we determine the sufficient and necessary conditions for optimal control strategy of SGLT-2 inhibitors so as to avert side effects on normal cells using a Pontryagin's Minimum Principle. The results showed that if the ingestion rate of the inhibitor drug is equal to the digestion rate, the tumor cells can be completely eliminated within 9 months without side effects. The analytical results were numerically verified and the qualitative views of interacting cells dynamics is showcased.

In this paper, we examine static and spherically symmetric wormhole solutions in the context of Einstein–Cartan gravity, incorporating quantum corrections through the Casimir energy density as a matter source. To investigate the impact of quantum gravitational features, we apply the Generalized Uncertainty Principle (GUP), using the Dentournay, Gabriel, and Spindel (DGS) and Kempf, Mangano, and Mann (KMM) two GUP techniques. These GUP formalisms are implemented into the Casimir energy profile to investigate their effects on the wormhole geometry and matter content. We discuss the conservation equation, analyze the behavior of the null energy condition, and calculate the active gravitational mass to explore the physical acceptability of the wormhole solutions. In addition, we examine the complexity factor to describe the anisotropic character of the matter distribution filling the wormhole. The findings of this study demonstrate that GUP-induced corrections have a significant impact on the energy conditions, gravitational mass distribution, and structural complexity of the wormhole, indicating that Einstein–Cartan gravity, when incorporated with quantum vacuum effects, is capable of sustaining traversable wormhole geometries.

Understanding of the glucose risk factors-mediated mechanism in human breast cancer remains challenging. In this perception, for the first time, we proposed a complex nonlinear dynamical model that may provide a basic insight into the mechanism of breast cancer for the patient with existing glucose risk factors. The impact of glucose risk factors on the cancer cells’ population is evaluated using the formulated analytical model. The dynamical features of the cancer cells are described by a system of ordinary differential equations. Furthermore, the Routh–Hurwitz stability criterion is used to analyze the dynamical equilibrium of the cells’ population. The occurrence of zero bifurcation as well as two and three of the Jacobian matrix are obtained based on the sums of principal minors of order one. The glucose risk factors are exploited as the bifurcation parameters (acted as necessary and sufficient conditions) to detect the Hopf bifurcation. The presence of excess glucose in the body is found to affect negatively the breast cancer cells’ dynamics, stimulating chaos in the normal and tumor cells and thus drastically deteriorating the efficiency of the human immune system. The theoretical results are validated using the numerical simulations. It is concluded that the present findings may be beneficial for the future breast cancer therapeutic drug delivery and cure.

The findings of this study are connected with geometric function theory and were acquired by using subordination-based techniques in conjunction with Zeta-Riemann fractional differential operator information, we used the Zeta-Riemann fractional differential operator to investigate a certain classes of analytic functions. It is also shown that for particular choice of parameters for the new generalized classes, the classes of starlike, close-to-convex and α-convex functions emerges. Many classes of univalent functions were investigated with the use of convolution and subordination principle. Further, some classes are defined and developed by using the Zeta-Riemann fractional differential operator. The connections between the classes are given in the definition or in associated remarks and characterization properties are also proved, including combinations of functions belonging to those classes and inclusion relations.

In this paper, we propose a robust numerical scheme for solving a nonlinear fractional Keller–Segel system (KSDS) arising in chemotaxis theory. The KSDS describes the aggregation and movement of biological cells in response to chemical signaling. The system is formulated using the Erdélyi–Kober (EK) fractional operator, which incorporates memory and nonlocal temporal effects relevant to complex biological processes. To obtain approximate solutions, we develop an expansion homotopy perturbation method (EHPM). The proposed approach transforms the governing system into an equivalent integral formulation and employs fractional power‐series expansions and He’s polynomials to derive the solution components recursively. The theoretical validity of the model and the proposed scheme is established through a fixed‐point analysis, which provides sufficient conditions for the existence and uniqueness of solutions. Convergence of the constructed series is proved, and stability is investigated in the Hyers–Ulam sense. An error analysis is also presented to assess the reliability of the approximation. To provide an independent validation, the EHPM results are compared with those obtained by the Adomian decomposition method (ADM). The two methods generate identical solution components and consistent numerical results when the same truncation order is used, while EHPM avoids the separate construction of Adomian polynomials. Numerical results further illustrate the ability of the fractional KSDS to capture complex spatiotemporal aggregation patterns and demonstrate the important roles of chemotactic interactions and memory effects in the formation of structured biological patterns.