Reem Allogmany

In order to examine the dynamics of seasonal influenza transmission, this study suggests a fractional-order epidemiological model that was created using the Caputo fractional derivative. The model effectively replicates the observed pandemic progression over a order parameter, with an ideal value of [ ], offers better accuracy than traditional show that a non-integer order parameter, with an ideal value of [ ], offers better accuracy than traditional integer-order models. The idea of memory trace, which depicts the impact of previous infection levels on current disease dynamics, is a significant contribution of this work. This characteristic makes it possible to comprehend the impact of past immunity on present epidemiological results in greater detail. Furthermore, the model provides a more accurate depiction of disease spread by taking into consideration variation in infection severity among various population groups. The findings show that incorporating population heterogeneity and memory effects improves predictive performance and offers insightful information for designing public health interventions and policies. Overall, this study highlights fractional calculus’s potential as a potent instrument for enhancing the precision and usefulness of epidemiological models in the control of infectious diseases.

This paper proposes a numerical technique to study dynamical systems and uncover new behaviors in chaotic fractional-order models, a field that continues to attract significant research interest due to its broad applicability and the ongoing development of innovative methods. Through various types of simulations, this approach is able to uncover novel dynamic behaviors that were previously undiscovered. The results guarantee that initial conditions and fractional-order derivatives have a significant contribution to system dynamics, thus distinguishing fractional systems from traditional integer-order models. The approach demonstrated has excellent consistency with traditional approaches for integer-order systems while offering higher accuracy for fractional orders. Consequently, this approach serves as a powerful and efficient tool for studying complex chaotic models. Fractional-order dynamical systems (FDSs) are particularly noteworthy for their ability to model memory and hereditary characteristics. The method identifies new complex phenomena, including new chaos, unusual attractors, and complex time-series patterns, not documented in the existing literature. We use Lyapunov exponents, bifurcation analysis, and Poincaré sections to thoroughly investigate the system dynamics, with particular emphasis on the effect of fractional-order and initial conditions. Compared to traditional integer-order approaches, our approach is more accurate and gives a more efficient device for facilitating research on fractional-order chaos.

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations.

Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. Out of these, fractional jerk chaotic systems have gained attention regarding their applications in secure communication, signal processing, and control systems. This work develops a comparative analysis of a fractional jerk system that includes constant- and variable-order derivatives to contribute to chaos–stability analysis. Additionally, this study uncovers novel chaotic behaviors, further expanding our understanding of complex dynamical systems. The results yield new insights into using variable-order dynamics to enable chaotic systems to better adapt to real applications.

Viral infections pose a significant risk to public health all around the world. Knowledge of their transmission and epidemiology therefore forms the basis of prevention and control measures. Models are essential components to develop and quantify the characteristics of new pathogens, to evaluate the risks of emerging diseases, and to design strategies to control epidemics. This study investigates the validity of the fractional-order SVEIHR influenza model in predicting seasonal influenza outbreaks in Saudi Arabia using Atangana–Baleanu–Caputo (ABC) operators. The first part of our analysis was examining the positivity, boundedness, and global stability of the disease-free equilibrium of the model. The effective reproduction number of vaccination was calculated, indicating a decrease in the spread of the virus due to vaccination. Additionally, we determined that the reproduction number in the absence of vaccination suggests a higher rate of virus transmission. Therefore, we concluded that a reduction in [ ] occurs when vaccination rates increase. Our findings underscore the critical impact of vaccination on controlling the spread of influenza. Then raising coverage of protective vaccines becomes a logical goal for decreasing the basic reproduction number and thus mitigating the epidemic of influenza. We suggest that we extend this model by incorporating additional factors such as antiviral treatments and social behavior changes, which may further influence the dynamics of influenza transmission.

An in‐depth understanding of the mechanism concerning the parasitic disease called visceral leishmaniasis (VL) remains challenging. Thus, we modeled the dynamics of this illness using two fractional‐order operators, including Caputo–Fabrizio and Atangana–Baleanu. In the proposed dynamical model, the endemic and disease‐free equilibrium points were considered the symmetrical components. The fractional Euler method was applied to simulate the developed model, thus determining the equilibrium points' stability. The numerical simulation results were compared with the measured data to validate the model. The results obtained from the optimum fractional operator disclosed the minimum absolute and relative errors. The primary outcome of our study is the successful application of fractional‐order operators, specifically the Atangana–Baleanu operators with = 0.98, in modeling the dynamics of VL. Notably, the numerical simulation results, validated against real data from Sudan, demonstrated that the Atangana–Baleanu operators with = 0.98 yielded the best performance, with minimum absolute and relative errors. This underscores the precision of our fractional calculus‐based dynamical model in predicting VL dynamics compared to the classical framework, particularly for fractional‐order values of = 0.99 and = 0.98.

The objective of this research is to evaluate the heat and mass transfer in a water-based Darcy–Forchheimer hybrid nanofluid (HNF) flow across an expanding cylinder. The fluid flow has been studied under the influence of a magnetic field, viscous dissipation, heat source, thermal radiation, concentration stratification, and chemical reaction. Carbon nanotubes (CNTs) and iron ferrite (Fe₃O₄) nanoparticles (NPs) are added to the water, for the purpose of synthesizing the HNF. The fluid flow has been induced in the presence of gyrotactic microorganisms and the non-Fick’s model. Microorganisms are used to stabilize scattered nanoparticles through the hybrid nanofluid. The phenomena have been modeled in the form of a nonlinear system of partial differential equations (PDEs). The modeled equations are reduced to a dimensionless system of ODEs by using similarity substitution. The numerical solution of the derived sets of nonlinear differential equations is obtained by using the parametric continuation method. The impact of physical constraints on temperature, velocity, concentration, and microorganism profiles is presented through figures and tables. It has been observed that the heat and mass transport rates increase with the rising effect of the curvature parameter, while declining with the effect of the thermal stratification parameter.

Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.