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Öğe A new plentiful solutions for nanosolitons of ionic (NSIW) waves spread the length of microtubules in (MLC) living cells(Springer Science and Business Media LLC, 2025-02-20) Loubna Ouahid; M. A. Abdou; Jameelah S. Al Shahrani; A. Mohamed Abdel-Baset; Ali Akgül; Murad Khan HassaniThis article describes the developed Paul-Painlike method (PPM) to provide striking ODE of the nanosoliton of the ionic waves (NSIW) that spread the length of microtubules in live cells. Furthermore, Auxiliary Equation Approach (AEA) and Sardar Sub Equation Approach (SSEA) have been utilized similarly and concurrently to determine solutions for this particular model. In providing a physical explanation, various solitary wave structures are visually represented. These solutions include the anti-kink, kink shape, singular kink wave shape, and periodic bright, bright-dark and dark-singular soliton solution. Additionally, graphical illustrations (both 2-D and 3-D) demonstrate how the various parameters utilized affect the validity of analytical results. Furthermore, the uniqueness of the solutions we derived is highlighted by comparing the differences with earlier solutions of the model. The solutions produced may be beneficial in a number of significant investigations in medicine, as well as biology. The results demonstrate the effectiveness of the proposed techniques for determining many optical solitons of nonlinear evolution equations.Öğe A Standardized Numerical Methodology and Analysis for the Time Delayed Fractional Epidemic Model of Infectious Illnesses Spread by Lumpy Skin(L and H Scientific Publishing, LLC, 2025-06) Mudassar Rafique; Muhammad Aziz Ur Rehamn; Muhammad Rafiq; Zafar Iqbal; Nauman Ahmed; Ali AkgulThis study aims to investigate the solution of fractional order delayed lumpy skin infection model with Caputo operator numerically as well as analytically. This delay factor helps to control and slow down the spread of infection in individuals. In this study, existence and uniqueness of the underlying model is discussed. Equilibria of the lumpy skin model are computed along with reproductive number (R0), if R0 > 1 refers spread of illness and if R0 < 1 means control of disease. Local and global stability of fraction delayed model is also presented. Moreover, positive and bounded solution of proposed model are investigated. For the numerical solution of this model, we use Grunwald Letnikov non-standard finite difference scheme. The key properties of the numerical scheme are also investigated like positivity and boundedness. Numerical example is given to present the graphical solution of the fractional order delay epidemic model.Öğe A hybrid fractional model for cervical cancer due to human papillomavirus infection(Elsevier BV, 2025-03) Ali Akgül; Nauman Ahmed; Sadiya Ali Rano; Qasem Al-MdallalNumerous scientific and engineering applications exist for thermofluids. The primary cause of cervical cancer is the human papillomavirus (HPV), and thermos-fluid is crucial for identifying, treating, and understanding the cancerous phenomenon. In this work, a hybrid fractional order mathematical model of cervical cancer with modified parameters is studied. The proposed model consists of three fractional order nonlinear differential equations. The Grünwald Letnikov method is used to approximate the hybrid operator. A nonstandard finite difference scheme for the GL approximation is obtained to study the proposed model. Stability analysis of the used method is given, existence and uniqueness of solution to the hybrid model are given. Comparative studies between the two schemes is given. Convergence of the state variables is also shown to converge to true equilibrium points according to the stability conditions of the reproductive number. The effect of the order of fractional derivatives can also be observed in the simulations. In the end, concluding remarks are also given in the conclusion section that reflect the whole current research.Öğe ANALYSIS AND DYNAMICS OF CHOLERA EPIDEMIC SYSTEM IN SOCIETY VIA FRACTAL-FRACTIONAL OPERATOR(2025-01-01) Abbas, Fakhar; Ghaffar, Abdul; Akgül, Ali; Ahmad, Aqeel; Mustafa, Ghulam; Hendy A.S.; Abdallah, Suhad Ali Osman; El-Gawaad, N.S. AbdTo comprehend the dynamics of disease propagation within a society, mathematical formulation is a crucial tool to understand the complex dynamics. In order to transform the mathematical model with the objective of bolstering the immune system into a fractional-order model, we use the definition of Fractal-Fractional with Mittag-Leffler kernel. For an assessment of the stable position of a recently modified system, qualitative as well as quantitative assessments are carried out. We validate the property positivity and reliability of the developed system by evaluating its boundedness and uniqueness, which are important features of an epidemic model. The positive solutions with linear growth have been verified by the global derivative, and the level of effects of different parameters in each sub-section is determined through employing Lipschitz criteria. By employing Lyapunov’s first and second derivatives of the function, the framework is examined on a global scale to evaluate the overall effect with symptomatic and asymptomatic measures. Bifurcation analysis was performed to check the behavior of each sub-compartment under different parameters effects. The Mittag-Leffler kernel is used to obtain a robust solution via Fractal-Fractional operator for continuous monitoring of spread and control of cholera disease under different dimensions. Simulations are carried out to observe both the symptomatic and asymptomatic consequences of cholera globally, also to observe the actual behavior of cholera disease for control measures, and it has been confirmed that those with strong immune systems individuals recover early due to early detection measures. The actual state of cholera disease can be controlled by taking the following measures: early detection of disease for both individuals receiving medication and those who do not require medication because of their robust immune systems. This kind of research will be beneficial in determining how diseases spread and in developing effective control plans based on our validated findings.Öğe New exact soliton wave solutions appear in optical fibers with Sardar sub equation and new auxiliary equation techniques(Springer Science and Business Media LLC, 2025-02-05) Umair Asghar; Muhammad Imran Asjad; Yasser Salah Hamed; Ali Akgul; Murad Khan HassaniThis paper comprehensively analyzes exact solutions for the nonlinear long-short wave interaction system within the optical field. Consider two general techniques in this field, the Sardar sub-equation method, and a new auxiliary-equation technique. These methods are applied to derive a wide range of soliton solutions for nonlinear partial differential equations. By transforming the original partial differential equation into an ordinary differential equation using an appropriate transformation, various types of solitary wave solutions are obtained. The novelty of this work lies in the application of two powerful analytical methods. The study significantly broadens the scope of these techniques and their applications, providing a diverse set of exact solutions. To enhance comprehension, the obtained solutions are visualized through 3D, 2D, contour, and density plots, offering clear insights into the dynamics of solitary waves. Long-short-wave interaction model has many applications in different kinds of areas such as in optical fiber communication, to understand the interaction between different wave components that can influence the transmission of signals. This model is used to study the interaction between ion-acoustic waves and electron plasma waves. This helps in understanding energy transfer and wave stability in plasma, which is essential for applications like fusion energy research and space plasma. This is important in coastal engineering for predicting wave behaviors that affect coastal structures, sediment transport, and tsunami dynamics.Öğe Optical hausdorff quantum energy of spherical magnetic particles(Springer Science and Business Media LLC, 2025-02-04) Talat Körpinar; Zeliha Körpinar; Hatice özdemir; ALi Akgül; Murad Khan HassaniIn this article, a new approach for spherical magnetic curves under the spherical system in spherical space is given. Firstly, the Hausdorff derivative of the Lorentz spherical magnetic fields [Formula: see text] [Formula: see text] [Formula: see text] of spherical magnetic curves is constructed. On the other hand, the Lorentz spherical magnetic fields, by considering the Hausdorff derivative definition, are presented. Eventually, the Hausdorff energies of these spherical Lorentz fields according to the spherical system in [Formula: see text] spherical space are computed.Öğe Use of fractional calculus to avoid divergence in Newton-like solver for solving one-dimensional nonlinear polynomial-based models(Elsevier BV, 2025-04) Sania Qureshi; Amanullah Soomro; Ioannis K. Argyros; Krzysztof Gdawiec; Ali Akgül; Marwan AlquranThere are many different fields of study where nonlinear polynomial-based models arise and need to be solved, making the study of root-finding iterative solvers an important topic of research. Our goal was to use the two most significant fractional differential operators, Caputo and Riemann–Liouville, and an existing time-efficient three-step Newton-like iterative solver to address the growing interest in fractional calculus. The classical solver is preserved alongside a damping term created within it that tends to 1 as the fractional order α approaches 1. The solvers’ local and semi-local convergence are investigated, and the stability trade-off with convergence speed is discussed at length. The suggested fractional-order solvers are tested on a number of nonlinear one-dimensional polynomial-based problems that come up in image processing, mechanical design, and civil engineering, such as beam deflection; and many more.Öğe Second order slip micropolar MHD hybrid nanofluid flow over a stretching surface with uniform heat source and activation energy: Numerical computational approach(Elsevier BV, 2025-03) Syed Arshad Abas; Hakeem Ullah; Mehreen Fiza; Ali Akgul; Aasim Ullah Jan; Magda Abd El-Rahman; Seham M. Al-MekhlafiApplications: Micropolar fluids are extensively used in lubrication, polymer processing, and heat transfer applications to enhance performance in systems with suspended microstructures. These fluids find applications in industries such as medical, chemical, and microfluidics. Recent advancements have highlighted the potential of hybrid nanofluids in further improving thermal and flow characteristics. Novelty: Motivated by these developments, this study investigates the heat and mass transfer characteristics of a micropolar hybrid nanofluid comprising titanium dioxide (TiO2) and silver (Ag) nanoparticles suspended in water. The analysis focuses on the effects of slip boundary conditions, Joule heating, thermal radiation, heat sources, magnetohydrodynamic (MHD) effects, activation energy, and binary chemical reactions. Methodology: A mathematical model is formulated based on boundary-layer approximations, leading to a system of partial differential equations (PDEs) that describe the flow, thermal, and concentration fields. These PDEs are subsequently transformed into a set of ordinary differential equations (ODEs) using similarity transformations. The resulting higher-order nonlinear ODEs are solved numerically using the bvp4c technique in MATLAB. Findings: The results reveal that the inclusion of slip boundary conditions significantly influences the flow dynamics, reducing skin friction by 4.9 % and 10.4 % with increasing magnetic and material parameters, respectively, but enhancing it with a higher slip factor by 18.88 %. Additionally, an increased volume fraction of nanoparticles elevates the heat transfer rate by 6.3 % while diminishing the Sherwood number by 2.6 %, showcasing the thermal enhancement capabilities of the hybrid nanofluid. This study contributes to the field by providing new insights into the combined effects of Joule heating, activation energy, and chemical reactions on micropolar hybrid nanofluid flow. The result of bvp4c compared with previous literature and found to be closely aligned with published work. The findings have implications for the optimization of thermal management systems and processes in advanced engineering and industrial applications.Öğe Nonlinear normalized fractional electroosmotic spacelike fluid model(Elsevier BV, 2025-03) Talat Körpinar; Zeliha Körpinar; Ali Akgül; Qasem Al-MdallalIn this paper, we present optical recursively fractional SKμ−electroosmotic fractional recursivelySKμ−energy. Also, we have spacelike microfluidicsfractional SKμ− electroosmotic recursively tension energy. Moreover, we construct main Katugampola recursive-normal hyperbolic fractional KFα−tension field in hyperbolic space. Finally, we characterize spacelike radiative recursively fractional SKμ− phase in hyperbolic space.Öğe Thermal radiation effects of ternary hybrid nanofluid flow in the activation energy: Numerical computational approach(Elsevier BV, 2025-03) Hakeem Ullah; Syed Arshad Abas; Mehreen Fiza; Aasim Ullah Jan; Ali Akgul; Magda Abd El-Rahman; Seham M. Al-MekhlafiSignificance: The remarkable thermal conductivity and heat transfer characteristics of nanofluids make them extremely valuable in thermal engineering and other areas. Due to their increased effectiveness, nanofluids are incredibly useful for improving the efficiency of cooling systems, heating processes, and thermal management applications. Rotating machinery and gas turbine rotators are some industrial applications of hybrid nanofluids as heat transport fluids. Purpose: This study introduces a novel investigation into heat transport phenomena of ternary hybrid, hybrid and nanofluid containing copper, silver and alumina nanoparticles within two stretchy rotating disks maintaining a constant distance. The analysis incorporates the effects of thermal radiation, heat source, joule heating, and Arrhenius activation energy into the equations to stabilize the new composition's flow and thermal properties. Methodology: After utilizing von Karman similarity transformations to renovate the principal equations into the set of nonlinear differential equation systems, the resulting equations were solved using the bvp4c numerical approach with the assistance of MATLAB software. Findings: Graphs are used to explain the results in three different kinds of flows: hybrid fluid (Cu+Al2O3/H2O), nanofluid (Cu/H2O), and ternary hybrid fluid (Cu+Al2O3+Ag/H2O). Additionally, the outcomes of the variable parameters are presented and briefly discussed for different flow profiles. There is encouraging evidence that the numerical code for this study is compatible with previously published work. The skin friction improves 5 % due to the higher values of magnetic and stretching parameter at lower disk. The rate of the heat transfer improved 28 % for ternary nanoparticles as compared to hybrid and single nanofluids. Sherwood's number exhibits both growing and decreasing behaviors for Schmidt and Reynolds’ numbers. All the involved factors enhances the temperature profile. The radiation parameter boost the Nusselt number for ternary hybrid nanofluid up to 6 % and 3.4 % at lower and upper disk as compare to nanofluid.Öğe Comparative analysis of hall current impact on MHD laminar surface tension gradient 3D flow of propylene glycol based tetra hybrid nanofluid with generalized fick's and fourier's perspective(Elsevier BV, 2025-03) Munawar Abbas; Shirin Shomurotova; Qasem Al-Mdallal; Ali Akgül; Zuhair Jastaneyah; Hakim AL GarallehExamine the significance of the Cattaneo-Christov flux model on the Marangoni convection 3D flow of tetra hybrid nanofluid combined with Hall current in the present study. When exposed to a fluctuating magnetic flux, it demonstrates electrical conductivity over a stretchy sheet. Using the Cattaneo-Christov double diffusion (CCDD) model, the problem is simulated. In this work, the CCDD model is used to analyze the mass and heat transmission tetra hybrid nanofluid. Basic Fourier's and Fick's laws are generalized by their application. A tetra hybrid nanofluid consisting of Molybdenum disulfide (Mos2), copper (Cu), Silicon dioxide (SiO2) and cobalt ferrite (CoFe2o4), propylene glycol (C3H8O2) as the base fluid is used. This model is essential for precisely predicting the behaviors of heat transfer in nanofluid flows since it takes thermal relaxation time into consideration. Its uses include optimizing heat exchanger performance, enhancing cooling systems in electronics, and better thermal management in microfluidic devices. The basic set of equations is resolved employing the numerical technique (bvp4c). The nanofluid, hybrid, trihybrid, and tetra hybrid nanofluid graphs are all compared. The stretching ratio parameter indicates rising trends in the flow distributions, although the opposite performance is observed for thermal and concentration distributions. Rate of heat and mass transmission improve of tetra hybrid, trihybrid, hybrid nanofluids as increase the values of Marangoni convection.Öğe Characteristics of elastic deformation on Boger hybrid nanofluid using modified Hamilton–Crosser model: a local thermal nonequilibrium model(Springer Science and Business Media LLC, 2025-01-15) Mostafa Mohamed Okasha; Munawar Abbas; Muyassar Norberdiyeva; Dyana Aziz Bayz; Ibrahim Mahariq; Ansar Abbas; Ali Akgül; Ahmed M. GalalIn this investigation, elastic deformation characteristics on surface tension gradient flow of Boger hybrid fluid over a plate using modified Hamilton-Crosser Model are examined. The modeling takes into account the influence of local thermal nonequilibrium (LTNE). The expanded Cattaneo-Christov theory, which takes relaxation times into account, is the current theory for mass and heat transmission. Excellent heat transmission is offered by the energy equation-based LTNE model for both the liquid and solid phases. Therefore, in this work, two thermal distributions are used for both the liquid and solid phases. It can be applied to materials science to improve heat transmission procedures and nanotechnology, where accurate temperature control is essential for applications like electronic device cooling systems, microfluidic devices, and biomedical applications. Better modeling of complicated fluids in these systems is made possible by the addition of elastic deformation and LTNE, which enhances the systems' stability and efficiency, particularly under nonequilibrium heat conditions. The Bvp4c method is used to solve the model equation system numerically once the relevant similarity variables have condensed. To illustrate how different physical conditions affect the involved distributions, the findings are graphed. Results show that Boger fluid exhibits enhanced velocity at increasing solvent percent parameter values.Öğe Mathematical analysis and pattern formation in diffusive predator–prey system(Springer Science and Business Media LLC, 2025-01-07) Nauman Ahmed; Muhammad Waqas Yasin; Ali Akgül; Dumitru Baleanu; Ovidiu Tintareanu-MirceaPrey-predator interactions are modeled using various dynamical systems and these interactions are affected by several factors. The predation rate of prey, reproduction rate, and prey use various strategies to avoid predation, the movement of the prey and predator species, food, and secured shelter can lead to the emergence of various types of patterns. These patterns in the prey-predator dynamics explain the complicated ecosystem. A reaction-diffusion prey-predator model with harvesting in predator is numerically investigated. A conditionally positivity preserving scheme is used. The von Neumann technique is used for the stability analysis. The Taylor series is used for the consistency analysis and discrete approximation is consistent with the underlying model. Pattern formation is observed for the governing model. The spot, stripe, and spot-stripes patterns are successfully gained that describe the complicated dynamics of the prey-predator dynamics. 3D and 2D simulations are drawn for the underlying model. The underlying model has two equilibria, both are successfully gained. All the theoretical results are verified through the simulations.Öğe Modeling and analysis of dengue transmission in fuzzy-fractional framework: a hybrid residual power series approach(Springer Science and Business Media LLC, 2024-12-28) Mubashir Qayyum; Qursam Fatima; Ali Akgül; Murad Khan HassaniThe current manuscript presents a mathematical model of dengue fever transmission with an asymptomatic compartment to capture infection dynamics in the presence of uncertainty. The model is fuzzified using triangular fuzzy numbers (TFNs) approach. The obtained fuzzy-fractional dengue model is then solved and analyzed through fuzzy extension of modified residual power series algorithm, which utilizes residual power series along with Laplace transform. Numerical analysis has also been performed in this study and obtained results are shown as solutions and residual errors for each compartment to ensure the validity. Graphical analysis depict the model’s behavior under varying parameters, illustrating contrasting trends for different values of and examining the impacts of transmission and recovery rates on dengue model in uncertain environment. The current findings highlighted the effectiveness of proposed uncertainty in epidemic system dynamics, offering new insights with potential applications in other areas of engineering, science and medicine.Öğe Computational analysis of Yamada–Ota and Xue models for surface tension gradient impact on radiative 3D flow of trihybrid nanofluid with Soret–Dufour effects(Springer Science and Business Media LLC, 2024-12-23) Sayer Obaid Alharbi; Munawar Abbas; Ahmed Babeker Elhag; Abdullah A. Faqihi; Ali AkgülThis article discusses the significance of Soret and Dufour, non-uniform heat generation, activation energy on radiative 3D flow of trihybrid nanofluid over a sheet with Marangoni convection. The energy equation takes into consideration the impacts of the heat generation, while the concentration equation takes activation energy into account. This trihybrid nanofluid is based on ethylene glycol and contains nanoparticles of titanium dioxide (TiO2), cobalt ferrite (CoFe2O), and aluminum oxide (Al2O3). For the case of trihybrid nanoparticles, the Yamada–Ota and Xue nanofluid models have been modified. This model is helpful for optimizing heating and cooling systems in fields like energy systems, microelectronics, and aerospace engineering where exact control of thermal properties is essential. By adjusting the characteristics of nanofluids, it also enhances heat transfer rates, which is a critical component in the development of solar collectors and high-efficiency heat exchangers. By using the necessary similarity transformations, non-linear ODEs are obtained from the controlling PDEs. The shooting method (BVP4c) can be utilized to solve this system of highly nonlinear equations numerically. As the surface tension gradient parameter is increased, the velocity distribution, mass transfer, and heat transfer rates all increase but the performance of the thermal and solutal profiles is opposite.Öğe Comparison study of modified and classical Hamilton-Crosser models for electrophoretic and thermophoretic particle deposition in stagnation point flow of diamond -SiC-Co3O4/diathermic oil-based trihybrid nanofluid(Springer, 2024) Ahmed M. Galal; Sahar Ahmed Idris; Munawar Abbas; Shaxnoza Saydaxmetova; Ali Hasan Ali; Humaira Kanwal; Ali AkgülThe current work examines the impact of heat generation on the stagnation point flow of a magnetized trihybrid nanofluid around a rotating sphere with electrophoretic and thermophoretic particle deposition. The trihybrid (Diamond –SiC–Co3O4/Do) nanofluid flow model consists of nanoparticles of Cobalt oxide (Co3O4), diamond (ND), and silicon carbide (SiC) dissolved in diathermic oil (DO). By comparing the modified model with the classical Hamilton–Crosser model, this study aims to investigate the heat transfer rate of a trihybrid nanofluid based on diamond –SiC–Co3O4/ diathermic oil. Through the analysis of trihybrid nanofluids based on diamond –SiC–Co3O4/Do diathermic oil, this model can optimize heat transmission in systems that need effective thermal management, like chemical reactors, electronics cooling, and energy storage. Trihybrid nanofluids' special qualities improve thermal conductivity, stability, and deposition control, which raises operational efficiency and dependability. It also helps with the design of sophisticated cooling systems for automotive and aerospace applications. These governing equations were solved with MATLAB's bvp4c tool after being transformed into ordinary differential equations via similarity variables. Results imply that, when compared to the classical model, the modified model accurately predicts higher heat transfer rates. As a consequence, trihybrid nanofluid heat transfer properties are better understood and thermal conductivity models are more accurate. The study shows that the concentration profile improved for both classical and modified Hamilton–Crosser models to enhance the values of electrophoretic particle deposition; while, inverse behavior is observed for thermophoretic particle deposition.Öğe Scrutinization of local thermal non-equilibrium effects on stagnation point flow of hybrid nanofluid containing gyrotactic microorganisms: a bio-fuel cells and bio-microsystem technology application(Springer, 2024) Okasha, Mostafa Mohamed; Abbas, Munawar; Formanova, Shoira; Faiz, Zeshan; Ali, Ali Hasan; Akgül, Ali; Galal, Ahmed M.The impact of Stefan blowing on the stagnation point flow of HNF (hybrid nanofluid) across a sheet containing gyrotactic microorganisms under local thermal non-equilibrium conditions (LTNECs) is briefly discussed in this paper. The present work uses a simplified mathematical model to inspect the characteristics of heat transfer in the absence of LTNECs (local thermal equilibrium conditions). LTNECs, traditionally provide two distinct fundamental temperature gradients for the liquid and solid phases simultaneously. A hybrid nanofluid is a mixture of water as the base fluid and single-walled carbon nanotubes and multi-walled carbon nanotubes. Gyrotactic microorganisms are included into nanoparticles to increase their thermal efficiency in a variety of systems, including microbial fuel cells, enzyme biosensors, bacteria powered micromixers, chip-shaped microdevices like bio-microsystems, and micro-volumes like microfluidic devices. This model can also help environmental engineering by enhancing wastewater treatment procedures by allowing microorganisms to break down pollutants more effectively. It advances the development of more productive photo bioreactors, increasing the output of biofuels in the field of renewable energy. Material scientists can utilize this concept to develop controlled nanostructured materials with consistent composition and thermal properties. The considerable similarity transformation is used to build ordinary differential equations for the nonlinear dimensionless system. This problem is solved numerically by using the Bvp4c method. The results determine that when the Stefan blowing parameter increases, fluid flow increases but temperature, mass transfer rate, and heat transfer are decreased.Öğe Analyzing the dynamics of human papillomavirus transmission via fractal and fractional dimensions under Mittag-Leffler Law(Springer Heidelberg, 2024) Dasumani, Munkaila; Lassong, Binandam S.; Akgul, Ali; Osman, Shaibu; Moore, Stephen E.The human papillomavirus (HPV) is a common sexually transmitted infection and a leading cause of cervical cancer. The yearly hospitalization rate for diseases linked to HPV is alarming. However, the mathematical study of HPV disease using fractal-fractional derivatives has received less attention from researchers globally. In this study, we develop a compartmental model of HPV transmission dynamics that includes a hospitalized compartment. We investigate the dynamics of our model equations through fractal and fractional analysis using the Mittag-Leffler law. The fractional order enables us to capture the memory effects, and the fractal dimension helps to capture self-similarities in the HPV model. The fixed point theorem is employed to establish the existence and uniqueness of solutions for the proposed fractal-fractional model. We conduct a stability analysis utilizing Hyers-Ulam criteria to demonstrate that the model's equation exhibits robust and stable behavior. A novel numerical scheme is discussed and the numerical simulations are conducted using this proposed scheme. The simulation results reveal that the fractal dimension and fractional order significantly influence the dynamics of the HPV fractal-fractional model. Both factors substantially affect the model's trajectories, as demonstrated by numerical simulations. Moreover, higher fractional order and fractal dimension values lead to a decrease in individuals across the different compartments as the simulation progresses. The numerical simulations illustrate the need to employ fractal-fractional derivatives in studying infectious diseases like HPV. For researchers and other interested stakeholders, we recommend modeling the coinfection of HPV and other sexually transmitted diseases in future extensions.Öğe
