We propose a systematic Gagliardo-type formulation of fractional Sobolev spaces on arbitrary time scales, based on the Lebesgue Delta-measure and the off-diagonal interaction domain induced by the product measure. For fractional orders strictly between zero and one and for finite Lebesgue exponents, we define a nonlocal Gagliardo seminorm and the associated function space. This construction provides a notion of fractional regularity on time scales that is genuinely nonlocal and structurally distinct from the derivative-based approaches developed in the existing literature. We establish the basic functional properties of these spaces: they are Banach spaces in all admissible cases, reflexive in the strict range of exponents, and Hilbert in the quadratic case. On bounded time scales with finitely many connected components, we identify a sharp criterion for the construction to be nontrivial. We then compare the new framework with the derivative-based Riemann–Liouville fractional Sobolev spaces previously studied on time scales. On a continuous interval, in the supercritical regime, we obtain a norm equivalence with the bilateral Riemann–Liouville space on the subspace of functions with vanishing boundary trace. On hybrid time scales, we prove an explicit obstruction that rules out any analogous equivalence, due to the contribution of the mixed continuous–discrete interactions. On bounded hybrid time scales with finitely many connected components separated by a positive distance, we further establish a Poincaré-type inequality, a fractional Sobolev embedding, and fractional Hardy and Caffarelli–Kohn–Nirenberg-type inequalities for subcritical weights. Together, these results provide a complete functional and geometric framework, together with first geometric estimates, for the nonlocal Gagliardo-type approach to fractional Sobolev spaces on time scales.

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and thus the lack of vaccination has led to the widespread and clinical importance of monkeypox. Although mathematical epidemiology research on the disease is complementary to clinical studies, it has attracted attention in the last few years. The present study aims to discuss the indispensable effects of three control strategies such as vaccination, treatment, and quarantine to prevent the monkeypox epidemic modeled via the Atangana–Baleanu operator. The main purpose is to determine optimal control measures planned to reduce the rates of exposed and infected individuals at the minimum costs. For the controlled model, the existence-uniqueness of the solutions, stability, and sensitivity analysis, and numerical optimal solutions are exhibited. The optimal system is numerically solved using the Adams-type predictor–corrector method. In the numerical simulations, the efficacy of the vaccination, treatment, and quarantine controls is evaluated in separate analyzes as single-, double-, and triple-control strategies. The results demonstrate that the most effective strategy for achieving the aimed outcome is the simultaneous application of vaccination, treatment, and quarantine controls.

Modeling the dynamics of the flow rate system is challenged by the nonlinear behavior and the noisy measurement data. Accurate models require a comprehensive understanding of fluid mechanics, as well as knowledge of all instruments in the measurement chain. This study presents a black-box optimization approach to develop a nominal Fractional-Order (FO) model of a laboratory-scale flow system. The model was constructed by repeatedly solving an optimization problem using preprocessed experimental data and averaging the resulting optimal parameters. The nominal FO model was then validated against unseen, unprocessed measurement data to assess its robustness. The parameter sensitivity of the proposed model was analyzed by introducing +10% and +20% perturbations in each parameter individually. Error analysis evidences that root mean squared, mean absolute, and mean absolute percentage errors with the proposed model have reduced to 9.3%,5.1%, and 5.3%, respectively, compared to those integer-order models. Furthermore, residual-based distribution analysis confirms the robustness of the approach, with residuals tightly concentrated around the lowest values. Although the FO model incurs a higher computational cost during optimization, it was significantly reduced using an online optimizer. The proposed model demonstrates superior robustness and accuracy, making it a compelling choice for precise modeling.

In this paper, the homotopy perturbation method is utilized to derive analytical solutions for fractal linear and nonlinear Schrödinger equations. The effectiveness of the proposed technique is validated through several representative examples, demonstrating its capability to handle a broad class of fractal differential equations arising in mathematical physics.

Observability is a fundamental concept in control theory. Its primary purpose is to look into whether it is possible to reconstruct the system’s initial state using only the information from its outputs. This paper focuses on the regional reconstruction problem of the initial state for a semilinear time-fractional system, which refers to the possibility of recovering the value of the initial state on a desired boundary sub-region instead of the whole evolution domain or its boundary. To achieve this objective, we use the analytical method, where we suppose that the system’s dynamic generates an analytic semigroup. First, by establishing an internal sub-region, we establish a connection between the concepts of regional observability and regional boundary observability; we will later explain how to define the internal sub-region. Then, by imposing suitable assumptions on the analytic semigroup and the system’s non-linearity, we give the main theorems of this research from which we deduce a sequence that converges to the unknown initial state on the desired boundary sub-region. Moreover, we present an algorithm that produces some numerical simulations which align closely with our theoretical findings.

We consider the unexploited/exploited logistic equation and study the stability of equilibrium points through Lyapunov functions. Then, we apply first and second order optimality conditions for the optimal control of the total biomass yield. Finally, we note that the time-scale logistic equation present in the literature lacks biological significance and we propose a new version of a dynamic logistic equation, valid on an arbitrary time scale, for which any trajectory of the system, beginning with a positive initial condition, remains nonnegative.

Given a fractional-order linear equation $L^αu = f$, we define an appropriate symmetric bilinear form so that the fractional operator $L^α$ is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.

This paper presents a novel proportional-integral-derivative (PID) control framework for first alpha-order systems evolving in fractal time. The main contribution is the extension of classical control theory to systems exhibiting anomalous temporal scaling by employing local fractal derivatives. In contrast to fractional-order PID (FOPID) approaches, which primarily model memory effects, the proposed fractal PID framework captures time-scaling behavior arising in non-smooth environments, such as viscoelastic friction and irregular contact surfaces. The closed-loop dynamics are formulated as a second alpha-order fractal differential equation, from which a characteristic equation is derived to establish conditions for asymptotic stability. It is shown that, for a constant reference input and positive controller gains, the tracking error converges to zero as t ->infinity. In addition, a quantitative performance analysis demonstrates that the fractal-order alpha governs temporal stretching: smaller values of alpha lead to increased rise and settling times and reduced oscillation frequency. The effectiveness of the proposed approach is illustrated through applications to a thermal system with fractal heat input and robotic actuators operating in irregular environments. These results highlight the potential of fractal-time control as a systematic framework for modeling and controlling dynamical systems with non-integer temporal structure.

Recent global epidemics have triggered widespread negative emotions such as anxiety, fear, and anger. Understanding the dynamics of epidemic-related fear has therefore become a crucial aspect of public health research. This study proposes a new social mathematical model to investigate the epidemic-related fear propagation dynamics within a population. In the model, people are divided into five psychological states based on their fear status and awareness level: susceptible, aware, exposed, fearful, and recovered. Additionally, fear propagates among exposed and fearful people through terrible news in the media. The basic reproduction number is calculated in association with exposed and fearful people and terrible news. The sensitivity of this threshold parameter is analyzed with respect to the model parameters. The local and global stability of the proposed fear propagation model is discussed at the fear-free and endemic equilibrium points. The model parameters are hypothesized based on existing studies on COVID-19 fear in Turkey, and numerical simulations are presented accordingly. The results show that fear propagation behavior is exacerbated by loss of awareness and direct interpersonal interaction with exposed and fearful people, as well as indirect contact with terrible news in media. Under the influence of fear, people are concentrated in the fearful compartment and interact with susceptible and aware compartments, affecting the overall behavior of fear propagation dynamics. These findings highlight the critical role of psychological factors, particularly fear, in shaping behavioral responses during epidemics and underscore the need for targeted public health interventions that address both direct interpersonal interaction and indirect media-influenced fear propagation.

This paper studies second α -order dynamical systems within the framework of fractal calculus and examines their behavior under fractal proportional–derivative (FPD) and fractal proportional–integral–derivative (FPID) controllers. A brief overview of fractal calculus is provided, and fractal initial value and final value theorems are formulated. The structure of FPID control for second α -order systems is presented, and the stability conditions are analyzed within the fractal setting. The response of FPD controllers is also investigated through analytical expressions and numerical simulations for selected values of the fractal order α . As an illustrative example, a series RLC circuit is modeled using fractal derivatives to demonstrate how fractal-order dynamics influence the controlled system response. The results emphasize the role of the fractal order α in shaping transient behavior and steady-state characteristics.

This paper introduces a pioneer control methodology for servo-hydraulic systems, inspired by the dynamics of black holes, namely by designing a Black Hole inspired Controller (BHC) and a Fuzzy-BHC hybrid controller (FBHC). Drawing upon the astrophysical principles governing black hole attraction, a multifaceted study is provided with three core contributions: (i) the conceptualization and design of a disruptive controller rooted in phenomena found in black holes; (ii) identification of conditions required to maintain system stability; and (iii) an extensive set of simulations results employing a nonlinear hydraulic model to evaluate the controller’s performance in multiple scenarios, including those with optimized and non-optimized configurations, as well as under perturbations, model uncertainties and sensor noise. The findings underscore the effectiveness of this astrophysics-inspired controller in tracking step, sigmoid and sinusoidal trajectories, tuned via a black-hole-inspired metaheuristic optimizer, even disturbance scenarios, including those due to unmodeled friction. Performance evaluations indicate that, under the same optimization-based tuning procedure applied to all controllers, BHC achieves lower tracking-error indices than conventional PID and sliding-mode controllers, with a maximum reduction of approximately 60% across the evaluated scenarios and metrics. The FBHC can further improve performance, particularly under uncertainties and measurement noise, although in some nominal cases the additional adaptation may not yield further gains compared to a well-tuned BHC. These findings demonstrate the adaptability and robustness of the black hole-inspired control framework, supporting its potential for servo-hydraulic applications, or other control domains demanding robustness to disturbances and model uncertainties.

Rabies continues to pose a significant zoonotic threat, particularly in areas with high populations of domestic dogs that serve as viral reservoirs. This study conducts a comparative analysis of Stochastic Continuous-Time Markov Chain (CTMC) and deterministic models to gain insights into rabies persistence within human and canine populations. By employing a multitype branching process, the stochastic threshold for rabies persistence was determined, revealing important insights into how stochasticity influences extinction probabilities. The stochastic model utilized 10,000 sample paths to estimate the probabilities of rabies outbreaks, offering a rigorous assessment of the variability in disease occurrences. Additionally, the study introduces a novel mathematical formulation of rabies transmission dynamics, which includes environmental reservoirs, free-ranging dogs, and domestic dogs as essential transmission factors. The basic reproduction number (R0) was derived and analyzed within stochastic frameworks, effectively bridging the gap between these two modeling approaches. Numerical simulations confirmed that the results from the stochastic model closely aligned with those from the deterministic model, while also highlighting the importance of stochasticity in scenarios with low infection rates. Ultimately, the study advocates for a comprehensive approach to rabies control that integrates both the predictable trends identified through deterministic models and the impact of random events emphasized by stochastic models.

This paper deals with the gradient stability and the gradient stabilizability of Caputo time fractional diffusion linear systems. First, we give sufficient conditions that allow the gradient Mittag-Leffler and strong stability, where we use a direct method based essentially on the spectral properties of the system dynamic. Moreover, we consider a class of linear and distributed feedback controls that Mittag-Leffler and strongly stabilize the state gradient. The proposed results lead to an algorithm that allows us to gradient stabilize the state of the fractional systems under consideration. Finally, we illustrate the effectiveness of the developed algorithm by a numerical example and simulations.

We propose a nonstandard finite difference scheme for the Susceptible-Infected-Removed (SIR) continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with examples. In contrast with the SIR discrete-time model available in the literature, our new model is simultaneously mathematically and biologically sound.

This study presents a deterministic model to investigate rabies transmission dynamics, incorporating environmental effects and control strategies using optimal control theory. Qualitative and quantitative analyses reveal that the disease-free equilibrium is stable when the effective reproduction number Re<1 and unstable when Re>1. Mesh and contour plots illustrate an inverse relationship between Re and control strategies, including dog vaccination, health promotion, and postexposure treatment. Increased intervention reduces transmission, while higher contact rates among dogs raise Re. Numerical simulations with optimal control confirm the effectiveness of integrated strategies. Vaccination and treatment are identified as key interventions for achieving rabies elimination within 5 years.

We propose a new dynamic SIR model that, in contrast with the available model on time scales, is biological relevant. For the new SIR model we obtain an explicit solution, we prove the asymptotic stability of the extinction and disease-free equilibria, and deduce some necessary conditions for the monotonic behavior of the infected population. The new results are illustrated with several examples in the discrete, continuous, and quantum settings.

We conduct an analysis of a one-dimensional linear problem that describes the vibrations of a connected suspension bridge. In this model, the single-span roadbed is represented as a thermoelastic Shear beam without rotary inertia. We incorporate thermal dissipation into the transverse displacement equation, following Green and Naghdi's theory. Our work demonstrates the existence of a global solution by employing classical Faedo-Galerkin approximations and three a priori estimates. Furthermore, we establish exponential stability through the application of the energy method. For numerical study, we propose a spatial discretization using finite elements and a temporal discretization through an implicit Euler scheme. In doing so, we prove discrete stability properties and a priori error estimates for the discrete problem. To provide a practical dimension to our theoretical findings, we present a set of numerical simulations.

The COVID-19 pandemic has presented unprecedented challenges worldwide, necessitating effective modelling approaches to understand and control its transmission dynamics. In this study, we propose a novel approach that integrates asymptomatic and super-spreader individuals in a single compartmental model. We highlight the advantages of utilizing incommensurate fractional order derivatives in ordinary differential equations, including increased flexibility in capturing disease dynamics and refined memory effects in the transmission process. We conduct a qualitative analysis of our proposed model, which involves determining the basic reproduction number and analysing the disease-free equilibrium’s stability. By fitting the proposed model with real data from Portugal and comparing it with existing models, we demonstrate that the incorporation of supplementary population classes and fractional derivatives significantly improves the model’s goodness of fit. Sensitivity analysis further provides valuable insights for designing effective strategies to mitigate the spread of the virus.

In this research, we have derived a mathematical model for within human dynamics of COVID-19 infection using delay differential equations. The new model considers a ’latent period’ and ’the time for immune response’ as delay parameters, allowing us to study the effects of time delays in human COVID-19 infection. We have determined the equilibrium points and analyzed their stability. The disease-free equilibrium is stable when the basic reproduction number, R_0 , is below unity. Stability switch of the endemic equilibrium occurs through Hopf-bifurcation. This study shows that the effect of latent delay is stabilizing whereas immune response delay has a destabilizing nature.

Efforts to optimize therapeutic planning in cancer treatment are often constrained by the limited biological accuracy of predictive tumor growth models, hindering clinical applicability. This study presents an innovative oncology framework based on Black Hole-inspired control principles, wherein tumor masses are drawn toward remission or stable chronic disease states through attraction dynamics. This method operates independently of patient-specific factors, including tumor heterogeneity and uncertain dynamics, offering a significant departure from traditional survival probabilistic paradigms. Our research focused on a multifaceted approach to chemotherapy planning that included: (a) the design of a Black Hole-inspired control algorithm and (b) the simulation of tumor volume dynamics under Atezolizumab treatment. The simulations assessed tumor volumes of $200 \mathrm{~mm}^{3}$ and $12732 \mathrm{~mm}^{3}$, employing both Gompertzian and biphasic growth models, with and without accounting for chemotherapy resistance. Our results reveal that the Black Holeinspired control algorithm effectively managed chemotherapy outcomes across multiple scenarios, including tumor resistance, varying clinical parameters over time, and significant dynamic uncertainties. This evidence positions Black Hole-inspired control as a disruptive paradigm in oncology therapy, paving the way for high-impact research on integrating astrophysical-inspired algorithms for advanced personalized cancer treatments. -chemotherapy, drug delivery, cancer, natureinspired control algorithm, black hole