Hepatitis B virus (HBV) and Diphtheria continue to impose substantial co-morbidity burdens in low- and middle-income countries where inadequate vaccination coverage and healthcare access create conditions for simultaneous infection. Understanding their coupled transmission dynamics is essential for designing effective integrated public health responses. This study presents, to our knowledge, the first application of a fractal-fractional derivative operator with Mittag-Leffler kernel to a fifteen-compartment Hepatitis BDiphtheria co-infection model incorporating explicit bacterial load dynamics. The model simultaneously represents sexual, vertical, and environmental HBV transmission routes alongside respiratory diphtheria transmission, extending beyond prior single-disease fractional models. The total human population and bacterial reservoir are partitioned into fifteen compartments (S, V, E D , I D , T D , R D , E HB , I aHB , I cHB , T HB , R HB , E C , I C , T C , R C ) and one bacterial compartment (B). The fractal-fractional derivative operator with Mittag- Leffler kernel (Atangana-Baleanu in Caputo sense) is applied to the governing system. Existence and uniqueness of solutions are established via the Banach and Krasnoselskii fixed-point theorems. The basic reproduction number R 0 is derived via the next-generation matrix method. Local and global asymptotic stability of the disease-free and endemic equilibria are analyzed. Model parameters are estimated by fitting to annual HBV case data from China (2004–2021, CDC) and weekly Diphtheria case data from Nigeria (2023 outbreak, NCDC). Elevated fractal dimension α and fractional order β jointly prolong infection peaks and delay epidemic resolution, with co-infected compartments IC reaching up to 2,500 individuals under high transmission scenarios. Increasing treatment rates and vaccination rates substantially reduce prevalence across all compartments. The basic reproduction numbers R 0H and R 0D (ranging from 0.001 to 0.005 across simulated parameter scenarios) confirm that integrated intervention strategies can drive the system below the epidemic threshold. These results demonstrate that fractal-fractional co-infection modelling, validated against real outbreak data, provides actionable insights for multi-disease intervention design that cannot be obtained from integer-order or single-disease models alone.
This study introduces and investigates the concept of the (α, β, γ, δ)-level graph for a Turiyam graph, a novel extension of fuzzy and neutrosophic graphs that incorporates a fourth independent membership degree representing the unknown or liberal component. We defined the (α, β, γ, δ)-level graph of a turiyam graph as a crisp graph derived by applying specific threshold values (α, β, γ, δ) to its vertex and edge membership values. In addition, we discuss its application in social networks. This study is supported by illustrative examples.
The paper proposes a system of Caputo-Katugampola fractional partial differential equations to model the spatiotemporal dynamics of the adoption of artificial intelligence (AI) in small and medium-sized enterprises (SMEs). The model embeds the Technology-Organization-Environment (TOE) framework into four coupled equations for AI readiness (R), technological infrastructure (T), organizational readiness (O) and cultural acceptance (C). The Caputo–Katugampola derivative of order α ∈ 2 (0, 1] with scaling parameter ρ > 0 encapsulates memory and non-local effects. The system is numerically solved using a space discretization based on the Legendre-Gauss-Lobatto spectral collocation method and a time discretization by a finite difference method. Lower α delays early adoption but speeds up later diffusion, AI readiness acts as a moderator for the TOE pathways, and the cultural acceptance of AI is moderated by the ethical interaction term η2, as revealed from numerical experiments applied to a representative domain of Indian SMEs. The findings provide quantitative managerial advice on the trade-off between the rate of adoption and ethical protections, and show that fractional models can be useful in application of the study of technology diffusion in management.
We study a class of conformable linear Volterra integro-dynamic equations (CVIDEs) on arbitrary time scales. Using the Anderson–Georgiev conformable dynamic calculus, we first derive an equivalent Volterra integral equation through the conformable variationof- constants formula. Explicit boundedness constants are then introduced on a time scale interval, and sufficient conditions for the existence of solutions are obtained by the Schauder and Krasnoselskii fixed point theorems. Under the same verifiable kernel smallness condition, the associated Volterra operator is a contraction, which yields uniqueness. We further establish Hyers–Ulam stability and Hyers–Ulam–Rassias stability with explicit stability constants. The latter result is formulated for positive nondecreasing Rassias weights, making all conditions used in the proof explicit. An example on the nonuniform time scale [Formula: see text] verifies the conformable regressivity, exponential factors, constants, contraction threshold, and exact solution values. Numerical plots illustrate the dependence of the solution and the stability bound on the Volterra-kernel amplitude.
This paper develops a fractional-order ecological model describing one prey and two predator species with harvesting, predator interference, and a Beddington-DeAngelis functional response. Caputo fractional derivatives are incorporated to capture memory effects in population dynamics. The model admits a biologically feasible equilibrium and preserves positivity and boundedness. Local stability is established using Matignon's criterion, revealing a memory-induced stability switch and a critical fractional order beyond which the coexistence equilibrium loses stability. For the integer-order case, harvesting-induced bifurcations, extinction thresholds, and Hopf bifurcation points are determined numerically using MATLAB. Numerical simulations demonstrate that fractional-order memory and predator interference jointly enhance system stability and delay harvesting-induced oscillations and population collapse. These findings highlight the important role of memory and interference mechanisms in promoting ecological resilience under sustained harvesting.
This study develops a deterministic compartmental model to investigate the effects of partial treatment on the transmission dynamics of tuberculosis (TB) and hepatitis C virus (HCV) co-infection and to identify effective intervention strategies. The basic reproduction number was derived using the next-generation matrix method, and sensitivity analysis was conducted to determine the parameters most influential in disease transmission. The results show that transmission-related parameters have the greatest influence on the reproduction number, while treatment-related parameters substantially reduce the burden of TB–HCV co-infection. Numerical simulations further indicate that partial treatment prolongs infectiousness, accelerates disease progression, and enhances interactions between TB and HCV, with TB identified as the dominant contributor to co-infection dynamics. An optimal control model incorporating awareness, detection, and treatment interventions was formulated using Pontryagin’s Maximum Principle. Numerical simulations demonstrate that the simultaneous implementation of these interventions can substantially reduce TB–HCV co-infection, with near elimination achievable within two years. Based on these findings, individuals diagnosed with either TB or HCV should be routinely screened for the other infection to facilitate early detection and timely concurrent treatment.
In this work, we investigate the invariance properties of nonlinear fractional partial differential equations (NLFPDEs) involving conformable fractional derivatives in both spatial and temporal variables by employing the framework of Lie symmetry analysis. Specifically, the study focuses on the generalized modified Burgers equation and the generalized modified Korteweg–de Vries (mKdV) equation with conformable fractional derivatives. The corresponding infinitesimal generators and symmetry vector fields for both the models are systematically derived. Then by using these Lie symmetries, the governing equations are transformed into ordinary differential equations. To obtain the exact analytical solutions, the ansatz methods and the [Formula: see text]-expansion technique are applied on the reduced equations. Furthermore, the generalized mKdV equation is transformed into an autonomous dynamical system, enabling a qualitative investigation through bifurcation theory.
In this paper, a new function class has been defined by involving a differential operator and Mittag-Leffler function This function class unifies the studies of well-known subclasses like convex, starlike and alpha-convex functions. Estimates of Taylor–Maclaurin coefficients [Formula: see text] and [Formula: see text] of the functions which belong to the defined function class are obtained. Further, the Fekete–Szegö inequality of this new function class is derived. Also, on taking some particular values of parameters, different corollaries have been obtained. The logarithmic coefficients and coefficient estimate of inverse function are also derived by using the defined class of function.
This paper is devoted to the study of the projective algebra of a certain class of [Formula: see text]-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We show that if an exponential [Formula: see text]-metric [Formula: see text], [Formula: see text], admits a projective vector field, then V is a conformal vector field with respect to the Riemannian metric [Formula: see text] or F has vanishing [Formula: see text]-curvature.
In this paper, we study bounded lattice homomorphisms from [Formula: see text]-spaces over complete [Formula: see text]-finite measure spaces into Banach lattices. For [Formula: see text], we work on the [Formula: see text]-ring of finite-measure sets and on simple functions with finite-measure support. We show that such operators are determined by their values on indicator functions of finite-measure sets, which yields a component-valued set function satisfying a local Boolean property. The operator is recovered as the unique bounded extension of the associated integral on simple functions. We obtain an exact operator-norm formula in terms of a natural size functional of the induced set function and prove a converse construction. In the AL-space case, we derive a Radon–Nikodým description and an explicit norm identity. Endpoint variants for [Formula: see text] on finite measure spaces and for [Formula: see text] under a [Formula: see text]-order continuity hypothesis are also included.
For a given positive integer n, the prime- [Formula: see text] graph denoted by [Formula: see text] is defined on the vertex set comprising all positive divisors of n greater than 1. An edge exists between two distinct vertices x and y if and only if their greatest common divisor, [Formula: see text], is a prime factor of n. This study explores the fundamental structural characteristics of [Formula: see text] systematically. Key findings establish that the graph is always connected for any integer n, with a diameter of at most 2 and a radius of 1. The paper provides characterizations and formulas for various graph invariants, including the clique number, chromatic number, girth, and vertex degrees, demonstrating their direct dependence on the prime factorization of n. It is shown that graphs [Formula: see text] and [Formula: see text] are isomorphic if n and m share the same prime factorization structure irrespective of the prime factors. Furthermore, conditions for planarity are determined. The analysis also covers properties such as the independence number, covering number, density of a graph establishing a relation between them and prime factorization of n. This research illuminates the deep interplay between the arithmetic properties of integers and the resulting topological features of their associated graphs.
In this paper, tuberculosis dynamics is studied using the fractal–fractional (FF) operator in the Caputo–Fabrizio (CF) sense. To understand the transmission dynamics of tuberculosis in India and to investigate the possibility of achieving tuberculosis eradication within the shortest possible time frame, a five-compartment mathematical model is presented. The entire tuberculosis-infected population is categorized into three groups, namely drug-sensitive (DS), multidrug-resistant (MDR), and isolated classes. The existence and uniqueness of the model governed by the CFFF derivative are established through the application of fixed-point theorems. The Ulam–Hyers stability of the model is examined using nonlinear analysis. Numerical solutions of the proposed model are obtained through the development of the Adams–Bashforth method. Based on the numerical simulations, it is suggested that tuberculosis could be eradicated in India if a high treatment success rate is achieved and at least 50% of MDR-TB cases are isolated.
Fractional calculus has emerged as a robust analytical framework with wide-ranging applications across mathematical analysis, including differential and integral equations, special functions, and series representations. Motivated by these developments, the present study focuses on analyzing and extending Saigo-type fractional integral operators. Building upon this foundation, we introduce a Saigo fractional integral operator characterized by an incomplete R-function kernel. Owing to the general structure of the proposed operator and the associated special functions, several known and new results are obtained as particular cases. These include image formulas involving the Riemann–Liouville and Erdléyi–Kober fractional integral operators, as well as representations connected with generalized Fox H-function, Wright hypergeometric, Mittag-Leffler, and Whittaker functions. The derived image formulas provide a unified framework for studying a broad class of special functions and may be useful in applications of fractional differential and integral equations. Furthermore, several earlier results reported in the literature arise naturally as special cases of the present findings, demonstrating the unifying and general nature of the proposed approach.
This work presents a symbolic algorithm for solving systems of equations, coupling Gröbner bases with integro-differential algebras to solve algebraic and differential systems. Building upon the basic work on boundary value problems, our method combines linear and nonlinear systems via noncommutative polynomial rings and an original reduction system. A rigorous confluence proof guarantees computation reliability, and its applications are illustrated through complete examples in robotics, mechanical, and electrical engineering. Through implementation in Python, taking advantage of recent improvements in symbolic computation libraries, showcases the practicality of the framework. This work contributes to symbolic computation by presenting a general-purpose, exact, and computationally efficient approach to solving intricate systems, which has potential applications in science and engineering.
Collatz conjecture states that each positive integer will return to 1 after two computations — either [Formula: see text] when x is odd or [Formula: see text] when x is even. We denote [Formula: see text] as ‘I’ and [Formula: see text] as ‘O’. Given a starting integer x, the computational sequence from x to 1 can be looked as a string consisting of ‘I’ and ‘O’. We call this sequence (string) as the dynamics of x. The key results are as follows: (1) To verify the Collatz conjecture, we randomly select an extremely large integer and verify whether it can return to 1. The largest one has been verified by us has 6,000,000 bits, which is overwhelmingly much larger than currently verified by others, e.g., 128 bits, and we only use a laptop. (2) To verify whether extremely large integers can return 1, we propose an dedicated algorithm that can compute [Formula: see text] for extremely large x whose length can be million bits, e.g., 5 million bits. The subtlety of our algorithm is replacing integer multiplication by bit addition, and further only by logical condition judgment (e.g., logic operation). (3) By observing dynamics of extremely large integers, we discover that the ratio — the count of ‘O’ over the count of ‘I’ in the dynamics goes to 1 asymptotically with the growth of starting integers. (4) We discover that once the length of starting integer is sufficiently large, e.g., 1 million bits, the corresponding dynamics presents sufficient randomness as a bit sequence (‘I’ is replaced with 1 and ‘O’ is replaced with 0). Thus, the computation of the dynamics of a sufficiently large integer can be looked as a pseudo-random bit sequence generator. We change the algorithm from outputting dynamics into outputting bit sequence, and then we compute randomly selected integers with L bit length, where L is 1, 2, 3, 4, 5, 6 million bits. We evaluate the randomness of the generated bit sequences by standards such as NIST SP 800-22 and GM/T 0005-2021. All sequences can pass the tests, and the larger the starting integer, the better. (5) We thus propose an algorithm for random bit sequence generator by only using logical judgment and less than 100 lines in ANSI C.
The novel fractional integral operator presented in this paper unifies and generalizes several existing fractional calculus operators. By applying this operator, we provide a significant extension of the classical results to the fractional setting by establishing a Chebyshev-type integral inequality for the synchronous functions. The primary inequality offers a flexible way to examine how functions behave when fractional integration is applied. The suggested approach is consistent with the existing results in the literature, as evidenced by the derivation of several corollaries and special cases as applications. The developed findings open up new directions for mathematical analysis study and their applications in practical sciences, while also adding to the expanding theory of fractional inequalities. The study discussed in this paper advances the inequality theory and fractional calculus.
Hepatitis B is one of the most severe viral infections, transmitted through contact with infected blood, sexual interaction, bodily fluids, and from mother to child during delivery. In this study, we develop a nonlinear fractional-order mathematical model of the Hepatitis B virus (HBV) using the Caputo fractional derivative operator. The use of the Caputo operator is particularly significant as it captures memory and hereditary properties of disease progression, providing a more realistic representation compared to classical integer-order models. For numerical simulations, we employ the Homotopy Analysis Transform Method (HATM), which combines the homotopy analysis method, homotopy polynomials, and the Laplace transform. Furthermore, the existence and uniqueness of the solution are established, and stability analysis is carried out using fixed-point theory. Sensitivity analysis is performed to identify the most influential parameters affecting HBV transmission. Numerical results, obtained using MATLAB, indicate that increasing media awareness significantly reduces the spread of HBV. Additionally, the inclusion of fractional-order dynamics enhances the models capability as an effective tool for controlling disease transmission. The findings of this study provide valuable insights for the healthcare sector and can assist in predicting disease outcomes and designing effective treatment and prevention strategies.
This paper introduces a generalized nonuniform multiresolution structure (GNMS) on the spectrum Λ=0,rN+2ℤ, where [Formula: see text] and r is an odd integer with [Formula: see text] such that [Formula: see text]. We characterize GNMS in reduced subspaces, develop a frame-like expansion and pyramid decomposition for signal representation, and validate the framework with numerical examples.
Integer factorization has raised concerns, particularly in the digitalization era, based on the fact that positive integers can be employed in the safety of information in cryptography. This study introduced a square table that is instrumental in the analysis of all positive integers concerning their prime factors. The systematic arrangement of triangular and nontriangular numbers in this table leads to formulas and instructions useful for factorization.
This study investigated the nonlinear fractional Van der Pol equation using the modified Atangana–Baleanu–Caputo (mABC) fractional derivative operator. The Van der Pol oscillator is a fundamental model in nonlinear dynamics, with applications spanning electrical circuits, cardiac rhythms, neural activity, mechanical systems, fluid dynamics, control engineering, robotics, and economic cycles. Unlike the classical ABC fractional derivative, the modified operator offers enhanced capability for modeling systems with long-term memory and hereditary properties. The main objective of this study is to test the applicability of the modified operator for solving this nonlinear oscillator problem, and to analyze how the fractional order and the damping term influence the system dynamics. A predictor–corrector numerical scheme based on Lagrange interpolation was developed and implemented for the coupled fractional system. The key findings demonstrate that the fractional order [Formula: see text] significantly influences the damping characteristics and oscillatory behavior. The proposed numerical scheme achieves good agreement with the reference methods, including the seventh-order Runge–Kutta (RK7) and three-step Adams–Bashforth (AB3) schemes, with observed quantitative differences attributable to the distinct kernel structures of the fractional operators. The numerical simulations performed in MATLAB R2017b yielded graphical and tabular results. The novelty of this study lies in the application of a mABC derivative to the Van der Pol system.
