This study explores the onset of food allergies in individuals using antacid medications, employing an innovative fractional mathematical model integrating fractional calculus and differential equations. Through this methodology, we analyze the intricate dynamics of food allergy epidemics and their interaction with antacid usage, offering essential perspectives for refining management strategies and implementing early intervention measures. Our research stems from a 2001 case, marking the initial occurrence of a severe anaphylactic reaction to manioc, a novel food allergy. Prompted by the prolonged acknowledgment of incidents involving a tuber ingrained in native Brazilians’ diets for over 500 years, we establish a mathematical framework addressing the persistence of the cassava allergen in the presence of proton pump inhibitors (PPIs). These medications are utilized to reduce gastric acidity in digestive disorders, elevate pH levels, foster the preservation of the allergen and subsequently enhance immune recognition. Our investigation hinges on employing a mathematical model to scrutinize how incidents related to food allergies impact the dynamics of scientific publication. The model, offering a distinctive perspective on food allergy characterization, relies on a novel set of differential equations incorporating fractional Caputo-Fabrizio (CF) operators. Insights into the model’s long-term behavior are derived from a comprehensive examination of its qualitative characteristics and subsequent computational simulations. We establish the existence and uniqueness of solutions within the proposed model, drawing on established results concerning fixed points. Utilizing the Ulam-Hyers method, we determine stability outcomes. For in-depth numerical analysis and simulation, our study employs Newton’s polynomial in conjunction with the three-step Adams-Bashforth numerical technique. The findings underscore the efficacy of fractional order techniques in accurately capturing the complex dynamics of food allergies and providing insightful information crucial for devising effective management plans.

This paper considers the novel fractional-order operator developed by Atangana-Baleanu for transmission dynamics of the SARS-CoV-2 epidemic. Assuming the importance of the non-local Atangana-Baleanu fractional-order approach, the transmission mechanism of SARS-CoV-2 has been investigated while taking into account different phases of infection and various transmission routes of the disease. To conduct the proposed study, first of all, we shall formulate the model by using the classical operator of ordinary derivatives. We utilize the fractional order derivative and the model will be extended to a model containing fractional order derivatives. The operator being used is the fractional differential operator and has fractional order ${\mathrm{\Phi }}_1$. The model is analyzed further and some basic aspects of the model are investigated besides calculating the basic reproduction number and the possible equilibria of the proposed model. The equilibria of the model are examined for stability purposes and necessary conditions for stability are obtained. Stability is also necessary in terms of numerical setup. The theory of non-linear functional analysis is employed and Ulam-Hyers’s stability of the model is presented. The approach of newton’s polynomial is considered and a new numerical scheme is developed which helped in presenting an iterative process for the proposed ABC system. Based on this scheme, sample curves are obtained for various values of ${\mathrm{\Phi }}_1$ and a pattern is derived between the dynamics of the infection and the order of the derivative. Further simulations are presented which show the cruciality and importance of various parameters and the impact of such parameters on the dynamics and control of the disease is presented. The findings of this study will also provide strong conceptual insights into the mechanisms of contagious diseases, assisting global professionals in developing control policies.

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

A mathematical epidemiological model for the transmission of Hepatitis B virus in the frame of fractional derivative with harmonic mean type incidence rate is proposed in this article. The proposed mathematical model is then fictionalized by utilizing the Atangana–Baleanu–Capotu ( $\mathcal{A}\mathcal{B}\mathcal{C}$) operator with vaccination effects. The threshold number R₀ is calculated by utilizing the next-generation matrix approach. The existence and uniqueness of solution of the proposed model are proved by utilizing the well-known fixed point theory. For the numerical solution of the proposed model with $\mathcal{A}\mathcal{B}\mathcal{C}$ derivative the well-known Adams–Bashforth–Molton (ABM) method is utilized. Likewise, stability is required in regard of the numerical arrangement. In this manner, Ulam–Hyers stability utilizing nonlinear functional analysis is utilized for the proposed model.