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The monkeypox virus (MPXV) is what causes monkeypox (MPX) disease, which is comparable to both smallpox and cowpox. Using classical, fractional-order and complex order differential equations, we offer a deterministic mathematical model of the monkeypox virus in this study to research its possible breakouts in United States. The complex order derivative makes the fractional order derivative and the integer order derivative more common when the imaginary part of the complex order equals zero and when the real part in complex order derivatives is zero in this case, the new behaviour appears that doesn't appear in integer and fractional order derivatives. Eight nonlinear complex order differential equations make up this model consisting of humans and rodents population sizes. The population of humans Nh is divided into five different classes. The population of rodent Nr is divided into three classes, and the derivatives are described in the Atangana-Baleanu-Caputo sense and Mittag-Leffler kernels are employed. Numerical methods to simulate complex order systems such as The standard and nonstandard two-step Lagrange interpolation methods are utilised to fit the model. The basic reproduction number of the model is given. The stability of the suggested model's disease-free equilibrium point is shown. Finally, we study the duration and monkeypox outbreak's transmission pattern in the United States from June 13 to Sep 16, 2022, numerical simulations to illustrate our findings are presented. The results show that keeping diseased people apart from the general population decreases the spread of disease.

In this paper, optimal control for a variable-order diffusion-wave equation with a reaction term is numerically studied, where the variable-order operator is defined in the sense of Caputo proportional constant. Necessary optimality conditions for the control problem are derived. Existence and uniqueness for the solutions of fractional optimal control problem are derived. The nonstandard weighted average finite difference method and the nonstandard leap-frog method are developed to study numerically the proposed problem. Moreover, the stability analysis of the methods is proved. Finally, in order to characterise the memory property of the proposed model, three test examples are given. It is found that the nonstandard weighted average finite difference method can be applied to study such variable-order fractional optimal control problems simply and effectively.

AbstractAn optimal control problem for the variable-order fractional-integer mathematical model of vaccination Covid 19 is presented in this research, where the order of the derivatives varies during the course of the time interval, becoming fractional or classical when it is more favourable. The variable-order derivatives are defined here using both the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. The existence, uniqueness, boundedness and positivity of the solutions are given. In order to test the rate for the detection of symptomatic infected people, two control variables are introduced. The optimality conditions are derived. The fractional-integer operator is approximated using Grünwald–Letnikov. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.

Two novel crossover models for breast cancer that incorporate Ψ-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function Ψ(t) in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter ζ is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed Ψ-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The Ψ-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions.

Drug-induced liver injury (DILI) represents a significant clinical challenge characterized by hepatic dysfunction following exposure to diverse medications. Methotrexate (MTX) is a cornerstone in treating various cancers and autoimmune disorders. However, the clinical utility of MTX is overshadowed by its ability to induce hepatotoxicity. The current study aims to elucidate the hepatoprotective effect of the alcoholic extract of Egyptian Araucaria heterophylla resin (AHR) on MTX-induced liver injury in rats. AHR (100 and 200 mg/kg) significantly decreased hepatic markers (AST, ALT, and ALP), accompanied by an elevation in the antioxidant’s markers (SOD, HO-1, and NQO1). AHR extract also significantly inhibited the TGF-β/NF-κB signaling pathway as well as the downstream cascade (IL-6, JAK, STAT-3, and cyclin D). The extract significantly reduced the expression of VEGF and p38 with an elevation in the BCL2 levels, in addition to a significant decrease in the IL-1β and TNF-α levels, with a prominent effect at a high dose (200 mg/kg). Using LC-HRMS/MS analysis, a total of 43 metabolites were tentatively identified, and diterpenes were the major class. This study presents AHR as a promising hepatoprotective agent through inhibition of the TGF-β/NF-κB and JAK/STAT3 pathways, besides its antioxidant and anti-inflammatory effects.

Rumex vesicarius L. Polygonaceae is a wildly grown plant in Egypt, North Africa, and Asia with wide traditional uses. Several studies reported its biological activities and richness in phytochemicals. This research addresses a comprehensive metabolic profiling of the flowers, leaves, stems, and roots via RP-HPLC-QTOF-MS and MS/MS with chemometrics. A total of 60 metabolites were observed and grouped into phenolic acids, flavonoids, phenols, terpenes, amino acids, fatty acids, organic acids, and sugars. Principal component analysis and hierarchal cluster analysis showed the segregation of different parts. Moreover, the antioxidant capacity was determined via several methods and agreed with the previous results. Additionally, an in silico approach of molecular docking of the predominant bioactive metabolites was employed against two antioxidant targets, NADPH oxidase and human peroxiredoxin 5 enzyme (PDB ID: 2CDU and 1HD2) receptors, alongside ADME predictions. The molecular modelling revealed that most of the approached molecules were specifically binding with the tested enzymes, achieving high binding affinities. The results confirmed that R. vesicarius stems and roots are rich sources of bioactive antioxidant components. To our knowledge, this is the first comprehensive metabolic profiling of R. vesicarius giving a prospect of its relevance in the development of new naturally based antioxidants.

Abstract Background Sildenafil citrate (SC) attenuates endothelial dysfunction. However, its effects on diabetic retinopathy (DR), which is mainly a microvascular disease, remain unclear. Vascular endothelial growth factor (VEGF) is known to be a critical mediator of DR. Therefore, we investigated the effects of SC on diabetic retina by measuring VEGF levels. Methods In this study, twenty-eight rats were divided into the following groups: group I, the control group; group II, rats with streptozotocin-induced diabetes; group III, rats with streptozotocin-induced diabetes receiving daily oral sildenafil at 1 mg/kg; and group IV, rats with streptozotocin-induced diabetes receiving high-dose daily sildenafil at 2.5 mg/kg. After 3 months, VEGF was measured in the retina specimen in one eye and the vitreous body in the other eye by immunohistochemistry and enzyme-linked immunosorbent assay, respectively. Results We found that VEGF expression in the retina was low in all rats from groups I and IV and in 30% of rats from group III; 80% of rats in group II demonstrated high VEGF expression in the retinae (P

In this paper, two new climate change mathematical models are extended using the stochastic-deterministic piecewise hybrid fractional derivatives, where the hybrid fractional order operator is applied to extend the deterministic model and the fractional Brownian motion operator is applied to extend the stochastic model. The steady states analysis of the propsed models are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are Caputo proportional constant Adams-Bashfourth fifth step method for solving the hyprid fractional deterministic model and the modified nonstandard Euler Maruyama technique to study numerically the fractional Brownian motion stochastic model. Several numerical test examples are given to demonstrate the efficiency of the methods and to support the theoretical results.

In this paper, two different parallel approaches for a hybrid fractional order Coronavirus (2019-nCov) mathematical model are presented. Both parallel approaches are implemented using Julia high level language. Parallel algorithm implementations are developed for the HPC cluster using Message Passing Interface (MPI) technology and general-purpose computing on GPUs (GPGPU) using Compute Unified Device Architecture (CUDA) based on hardware environments. The algorithm implementation are used to solve the real-world problem of the hybrid fractional order Coronavirus (2019-nCov) mathematical model and to study the parallel efficiency. The introduced hybrid fractional order derivative is defined as a linear combination of the integral of Riemann-Liouville and the fractional order Caputo derivative. A parallel algorithm is designed based on the predictor-corrector method with the discretization of the Caputo proportional constant fractional hybrid operator for solving the model problem numerically. Simulation results show that, both the new parallel approaches achieve significant efficiency.

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study’s originality is the use of fresh data simulation techniques and different solution methodologies.

Alzheimer's disease (AD) has become a major public health concern and the fifth major cause of death among the aging population globally. In this study, the total phenols and flavonoids contents (TPC and TFC) and in vitro antioxidant actions of the methanol extract and the various fractions of Aframomum melegueta were evaluated using 2,2-diphenyl-2-picrylhydrazyl (DPPH) radical scavenging activity, nitric oxide scavenging activity (NO), lipid peroxidation (TBARS) activity and ferric reducing power assay (FRAP). Furthermore, acetylcholinesterase (AChE) and butyrylcholinesterase (BuChE) inhibitory activities of the two most potent fractions were investigated, and the phytochemicals identified in the ethyl acetate fraction, which had the best antioxidant and cholinesterase inhibitory effects were subjected to chemoinformatics studies. The extract and its fraction had high amounts of TPC and TFC. The ethyl acetate fraction exerted the best DPPH, NO, TBARS, and FRAP inhibition with IC50 values of 5.06, 6.58, 2.12, and 88.73 µg/mL, respectively. Interestingly, n-hexane and ethyl acetate fractions inhibited AChE (IC5016.83 and 11.67 µg/mL) and BuChE (IC50 7.54 and 5.21 µg/mL) enzymatic activities more than the standard inhibitor, rivastigmine which had 11.99 and 11.40 µg/mL IC50 values, respectively. A total of 18 compounds were identified in ethyl acetate fraction, and kaempferol was the major component, with 40.01 µg/g (30%). More strikingly, the top-scoring compounds (catechin, and kaempferol) exhibited good binding affinity, and interacted favorably with amino acids residues around and within the active sites of AChE and BuChE and also obeyed drug-likeness rules, and did not show a tendency towards toxicity when placed side by side with rivastigmine which is immunogenic. Thus, A. melegueta seeds contain safe bioactive chemicals, which could be a veritable remedy for managing Alzheimer's and other neurodegenerative diseases.

The objective of this article is to present the time-fractional nonlinear one-dimensional equations of damped heat wave propagation in a rigid thermal conductor slab and predicts the dependence of second sound velocity on temperature and heat flux. An efficient numerical method is presented for solving the proposed model. This method is a kind of non-standard finite difference weighted average. The numerical properties (stability, truncation error) of the proposed scheme are well investigated. The effectiveness of the numerical scheme is illustrated through test numerical examples.

In this artical, we present a novel complex order nonlinear mathematical model of HIV/AIDS and Malaria co-infection with multi-delay. The proposed model is formulated as a system of twelve complex order differential equations. The complex order derivative is defined in the Atangana-Baleanu-Caputo sense. Two numerical methods are utilized for simulating the determined system; standard two-step Lagrange interpolation method and the non-standard two step Lagrange interpolation method. In order to establish numerical simulations, the theoretical results and comparative studies are given.

Designing and implementing full-scale models by architecture students is one of the most difficult and uncommon experiences in design education. The study evaluates the impact of this experience on the development of creative design skills among first-year students. This is achieved by applying a methodology that combines quantitative analysis of learning outcomes, especially those associated with the skills of Basic Design courses, with qualitative analysis of sketches, scaled and full models, and video recordings of students during design and implementation stages. This study provides experimental support for the use of full-scale models in design education to develop students’ creative skills, in addition to discussing opportunities and challenges to help faculty and researchers in the field of design education.

Two novel crossover models for breast cancer that incorporate Ψ -Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function Ψ (t) in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter ζ is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed Ψ -Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The Ψ -nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods' efficacy and support the theoretical conclusions. [ABSTRACT FROM AUTHOR]

Researchers are actively exploring potential bioactive compounds to enhance the effectiveness of Lisuride (Lis) in treating Parkinson’s disease (PD) over the long term, aiming to mitigate the serious side effects associated with its extended use. A recent study found that combining the dietary flavonoid Tiliroside (Til) with Lis has potential anti-Parkinson’s benefits. The study showed significant improvements in PD symptoms induced by 1-Methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP) when Til and Lis were given together, based on various behavioral tests. This combined treatment significantly improved motor function and protected dopaminergic neurons in rats with PD induced by MPTP. It also activated important molecular pathways related to cell survival and apoptosis control, as indicated by the increased pAkt/Akt ratio. Til and Lis together increased B-cell lymphoma 2 (Bcl-2), decreased caspase 3 activity, and prevented brain cell decay. Co-administration also reduced tumor necrosis factor alpha (TNF-α) and Interleukin-1 (IL-1). Antioxidant markers such as superoxide dismutase (SOD), catalase, and reduced glutathione significantly improved compared to the MPTP-induced control group. This study shows that using Til and Lis together effectively treats MPTP-induced PD in rats, yielding results comparable to an 8 mg/kg dose of levodopa, highlighting their potential as promising Parkinson’s treatments.

Breast cancer is a deadly disease that affects countless women worldwide. The most conventional treatments for breast cancer, such as the administration of anticancer medications such as letrozole (LTZ), pose significant barriers due to the non-selective delivery and low bioavailability of cytotoxic drugs leading to serious adverse effects and multidrug resistance (MDR). Addressing these obstacles requires an innovative approach, and we propose a combined strategy that synergistically incorporates LTZ with berberine (BBR) into stabilised AuNPs coated with ascorbic acid (AA), known as LTZ-BBR@AA-AuNPs. The LTZ-BBR@AA-AuNPs, a novel combined drug delivery system, were carefully designed to maximise the entrapment of both LTZ and BBR. The resulting spherical nanoparticles exhibited remarkable efficiency in trapping these two compounds, with rates of 58% and 54%, respectively. In particular, the average hydrodynamic diameter of these nanoparticles was determined to be 81.23 ± 4.0 nm with a PDI value of only 0.286, indicating excellent uniformity between them. Furthermore, their zeta potential was observed to be −14.5 mV, suggesting high stability even under physiological conditions. The release profiles showed that after being incubated for about 24 h at pH levels ranging from acidic (pH = 5) to basic (pH = 7), the percentage released for both drugs ranged from 56–72%. This sustained and controlled drug release can reduce any negative side effects while improving therapeutic efficacy when administered directly to cancer. MDA-MB-231 cells treated with LTZ-BBR@AA-AuNPs for 48 h exhibited IC50 values of 2.04 ± 0.011 μg/mL, indicating potent cytotoxicity against cells. Furthermore, the nanoparticles demonstrated excellent stability throughout the duration of the treatment.

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge–Kutta method and the non-standard generalized fifth-order Runge–Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies’ simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.

In this work, we aim to propose a mathematical formalism for the human liver involving Caputo fractional derivative (CFD) with modified parameters. Regarding the proposed fractional order model (FOM), the positivity and boundedness theory of an explicit solution are established by using the fractional mean-value theorem in the sense of Caputo. In addition, the validation of the FOM is provided to ensure that it complies with the medical interpretation and we also discuss the stability of unique equilibrium point (EP) for the presented model. We apply the Generalized Mittag–Leffler function method (GMLFM) and Predictor–Corrector method (PCM) to solve this model. The GMLFM is used to give the proper results as a series of the time variable to obtain the desired solution at any time and the PCM is implemented to obtain the solution of the FOM over a long time period. The tabled results from this study are compared with the real clinical data and with other methods. The simulations approximate the real clinical data and support the obtained theoretical results. Precisely, the outcomes of the FOM by using the proposed methods coincide with most of the clinical data that means it more accurate. Ultimately, the graphical results indicate the superiority of the proposed FOM over the classical model.

Introduction: Dengue and Malaria are the most important mosquito-borne viral diseases affecting humans. Fever is transmitted between human hosts by infected female aedes mosquitoes. The modeling study of viral infections is very useful to show how the virus replicates in an infected individual and how the human antibody response acts to control that replication, which antibody playing a key role in controlling infection. Objectives: Optimal control of a novel variable-order nonlinear model of dengue virus is studied in the present work. Bang-bang control is suggested to minimize the viral infection as well as quick clearance of the virus from the host. Necessary conditions for the control problem are given. The variable-order derivatives are given in the sense of Caputo. Moreover, the parameters of the proposed model are dependent on the same variable-order fractional power. Two numerical schemes are constructed for solving the optimality systems. Comparative studies and numerical simulations are implemented. The variable-order fractional derivative can be describe the effects of long variable memory of time dependent systems than the integer order and fractional order derivatives. Methods: Both the nonstandard generalized fourth order Runge-Kutta and the nonstandard generalized Euler methods are presented. Results: We have successfully applied a kind of Pontryagin’s maximum principle with bang-bang control and were able to reduce the viraemia level by adding the dose of DI particles. The nonstandard generalized fourth order Runge-Kutta method has the best results than nonstandard generalized Euler method. Conclusion: The combination of the variable-order fractional derivative and bang-bang control in the Dengue mathematical model improves the dynamics of the model. The nonstandard generalized Euler method and the nonstandard generalized fourth order Runge-Kutta method can be used to study the variable order fractional optimal control problem simply.

Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo’s fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin’s maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists.

Abstract In this paper, we present a numerical technique for solving fractional optimal control problems with a fractional derivative called Caputo–Katugampola derivative. This derivative is a generalization of the Caputo fractional derivative. The proposed technique is based on a spectral method using shifted Chebyshev polynomials of the first kind. The Clenshaw and Curtis scheme for the numerical integration and the Rayleigh–Ritz method are used to estimate the state and control variables. Moreover, the error bound of the fractional derivative operator approximation of Caputo–Katugampola is derived. Illustrative examples are provided to show the validity and applicability of the presented technique.