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This paper deals with the mathematical analysis of Tuberculosis by using fractal fractional operator. Mycobacterium TB is the bacteria that causes tuberculosis. This airborne illness mostly impacts the lungs but may extend to other body organs. When the infected individual coughs, sneezes or speaks, the bacterium gets released into the air and travels from one person to another. Five classes have been formulated to study the dynamics of this disease: susceptible class, infected of DS, infected of MDR, isolated class, and recovered class. To study the suggested fractal fractional model’s wellposedness associated with existence results, and boundedness of solutions. Further, the invariant region of the considered model, positive solutions, equilibrium point, and reproduction number. One would typically employ a fractional calculus approach to obtain numerical solutions for the fractional order Tuberculosis model using the Adams-Bashforth-Moulton method. The fractional order derivatives in the model can be approximated using appropriate numerical schemes designed for fractional order differential equations.

This manuscript is related to establishing appropriate results for the existence and uniqueness of solutions to a class of nonlinear impulsive implicit fractional-order differential equations (FODEs). It is remarkable that impulsive differential equations have attracted great popularity due to various important applications in the mathematical modeling of real-world phenomena/processes, particularly in biological or biomedical engineering domains as well as in control theory. The mentioned problem is considered under four-point nonlocal boundary conditions and the derivative is taken in the Caputo sense. Our results are based on fixed-point theorems due to Banach and Schaefer. To justify our results, two suitable examples are given.

Recent advances in electronics and microfluidics have enabled several research groups to develop fully integrated, sample-to-result isothermal nucleic acid amplification test (NAAT) platforms for the point of care. However, high component counts and costs have limited translation of these platforms beyond the clinic to low-resource settings—including homes. Many NAATs include complex, multi-component heater electronics based on flex circuits or multiple printed circuit boards (PCBs) to support essential NAAT steps such as lysis, sample deactivation, and nucleic acid amplification. In contrast, current commercial assays for home use, such as those for pregnancy or ovulation that include electronics, typically have just one onboard PCB. This work describes a generalizable strategy to integrate all heaters and the electronics needed to control them onto a single low-cost, USB-powered PCB. We built a multiplexable disposable NAAT (“MD NAAT”) platform that applies these principles, integrating small-area heaters that heat small regions to near-boiling (for pathogen lysis and deactivation) and large-area heaters (for amplification) on the same PCB. We show that both classes of heaters have high intra-board and inter-device reproducibility despite only heating a NAAT cartridge from below. We validated the small-area heaters by lysing methicillin-resistant Staphylococcus aureus (MRSA) cells and the large-area heaters by performing two types of isothermal NAATs (isothermal strand displacement amplification (iSDA) and loop-mediated isothermal amplification (LAMP)). These results demonstrate the merit of integrating NAAT heaters and control electronics onto a single printed circuit board and are a step toward translating NAATs to the home.

The simplest point-of-care assays are usually paper and plastic devices that detect proteins or nucleic acids at low cost and minimal user steps, albeit with poor limits of detection. Digital assays improve limits of detection and analyte quantification by splitting a sample across many wells (or droplets), preventing diffusion, and performing analyte amplification and detection in multiple small wells. However, truly digital nucleic acid amplification tests (NAATs) require costly consumable cartridges that are precisely manufactured, aligned, and operated to enable low detection limits. In this study, we demonstrate how to implement near-digital NAATs in low-cost porous media while approaching the low limits of detection of digital assays. The near-digital NAAT was enabled by a paper membrane containing lyophilized amplification reagents that automatically, passively meters and distributes a sample over a wide area. Performing a NAAT in the paper membrane while allowing diffusion captures many of the benefits of digital NAATs if the pad is imaged at a high spatial resolution during amplification. We show that the near-digital NAAT is compatible with a low-cost paper and plastic disposable cartridge coupled to a 2-layer rigid printed circuit board heater (the MD NAAT platform). We also demonstrate compatibility with biplexing and imaging with mobile phones with different camera sensors. We show that the near-digital NAAT increased signal-to-noise ratios by ~ 10×, improved limits of detection from above 10 3 copies of methicillin-resistant Staphylococcus aureus genomic DNA to between 100 and 316 copies in a biplexed reaction containing 10 5 copies of co-amplifying internal amplification control DNA, and reduced time-to-result from 45 min of amplification to 15–20 min for the positive samples.

Although reinforced concrete (RC) columns subjected to combined axial compression and flexural loads (i.e., eccentric load) are the most common structural members used in practice, research on FRP-confined circular RC columns subjected to eccentric axial compression has been very limited. More specifically, the available eccentric-loading models were mainly based on existing concentric stress–strain models of FRP-confined unreinforced concrete columns of small scale. The strength and ductility of FRP-strengthened slender circular RC columns predicted using these models showed significant errors. In light of such demand to date, this paper presents a stress–strain model for FRP-confined circular reinforced concrete (RC) columns under eccentric axial compression. The model is mainly based on observations of tests and results reported in the technical literature, in which 207 results of FRP-confined circular unreinforced and reinforced concrete columns were carefully studied and analyzed. A model for the axial-flexural interaction of FRP-confined concrete is also provided. Based on a full parametric analysis, a simple formula of the slenderness limit for FRP-strengthened RC columns is further provided. The proposed model considers the effects of key parameters such as longitudinal and hoop steel reinforcement, level of FRP hoop confinement, slenderness ratio, presence of longitudinal FRP wraps, and varying eccentricity ratio. The accuracy of the proposed model is finally validated through comparisons made between the predictions and the compiled test results.

Understanding the complex dynamics of HIV/AIDS transmission requires models that capture real-world progression and intervention impacts. This study introduces an innovative mathematical framework using fractal-fractional calculus to analyze HIV/AIDS dynamics, emphasizing memory effects and nonlocal interactions critical to disease spread. By dividing populations into four distinct compartments-susceptible individuals, infected individuals, those undergoing treatment, and individuals in advanced AIDS stages-the model reflects key phases of infection and therapeutic interventions. Unlike conventional approaches, the proposed nonlinear transmission function, , accounts for varying infectivity levels across stages (where is the total population and denotes the effective contact rate), offering a nuanced view of how treatment efficacy ( ) and progression to AIDS ( ) shape transmission. The analytical framework combines rigorous mathematical exploration with practical insights. We derive the basic reproduction number to assess outbreak potential and employ Lyapunov theory to establish global stability conditions. Using the Schauder fixed-point theorem, we prove the existence and uniqueness of solutions, while bifurcation analysis via center manifold theory reveals critical thresholds for disease persistence or elimination. We use a computational scheme that combines the Adams-Bashforth method with an interpolation-based correction technique to ensure numerical precision and confirm theoretical results. Sensitivity analysis highlights medication accessibility and delaying the spread of AIDS as a vital control strategy by identifying ( ) and ( ) as critical parameters. The numerical simulations illustrate the predictive ability of the model, which shows how fractal-fractional order affects outbreak trajectories and long-term disease burden. The framework outperforms conventional integer order models and produces more accurate epidemiological predictions by integrating memory-dependent transmission with fractional order flexibility. These findings demonstrate the model’s value in developing targeted public health initiatives, particularly in environments with limited resources where disease monitoring and balancing treatment allocation is essential. In the end, our work provides a tool to better predict and manage the evolving challenges of HIV/AIDS by bridging the gap between theoretical mathematics and actual disease control.

[This corrects the article DOI: 10.1371/journal.pone.0304375.].

Purpose>Augmented reality (AR) integrates the digital world with the real world and thus, provides a real-time experience to the users. With AR, the immediate surroundings become a learning platform for the users. The perception of the products has been enhanced many times with AR; thus, enriching user experience and responsiveness. The purpose of this paper is to bring forth the basics of AR and provide an overview of the research work carried out by researchers in the implementation of AR in different sectors.Design/methodology/approach>This paper summarizes the usefulness of AR in different industries. The authors have identified the peer-reviewed research publications from Web of Science, Scopus, Google Scholar, etc. The selection of literature has been made based upon the significance of AR in recent times. The industries/sectors where AR has been implemented successfully have been considered for this paper. The paper has been divided into various sections and subsections to bring more clarity to the readers.Findings>This paper presents a brief and a precise information on Industry 4.0 and AR. The basic working of AR system and its implications have also been discussed. The preference of AR over virtual reality (VR) has also been deliberated in this paper. The authors have presented the usefulness of AR in different sectors such as smart factories, ship yard building, online shopping, surgery and education. This paper discusses the AR-ready procedures being followed in these sectors.Originality/value>AR has been an add-on to VR systems. The processes in industries have become very handy and informative with AR. Because the application of AR in different sectors has not been discussed in a single paper; thus, this work presents a systematic literature review on the applications of AR in different sectors/industries.

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.