In Vitro Fertilization (IVF) has been a major medical advancement in the field of fertility treatment. It has helped millions of individuals and couples overcome infertility by providing a workable option. It involves removing eggs from the ovaries of a female, fertilizing those eggs with male sperm in a monitored lab condition. In this work, we developed a new model to show the success of In Vitro Fertilization rates in women through a fractional-order compartmental modeling framework by using real data. The developed model is analyzed statistically, and the biological feasibility of the model. The Lipschitz condition, expressed by the Lipschitz inequality, suggests that the change in a function’s output is limited by a constant known as the Lipschitz constant in relation to the input changes between any two points in its domain. This characteristic ensures that solution routes remain consistent, resulting in well-behaved and singular solutions. The linear growth inequality describes the condition of linear growth by asserting that one variable can be limited by another that rises linearly. This inequality is crucial because it prevents differential equation solutions from rising too rapidly, ensuring that they remain within established boundaries. We used Volterra integral inequality with a Volterra-type Lyapunov function to analyze the fractional derivatives of Lyapunov functions in a fractional-order system, which is necessary to establish global asymptotic stability. This extends the standard Lyapunov stability theory to fractional-order systems, particularly in the analysis of complex models like the infectious disease model, by providing a method to constrain the behavior of the system without solving the differential equations explicitly. Additionally, a sensitivity analysis of several aspects is derived through mathematical simulations. Additionally, we use numerical simulations to validate our theoretical findings at different fractional orders.

A global health concern, neurodegenerative disorders like Parkinson’s and Alzheimer’s impact both mental and physical functioning. The complex interplay among immunological response, protein accumulation, and brain health necessitates sophisticated mathematical modeling. This study introduces a fractional-order mathematical model using the Mittag-Leffler derivative to describe the dynamics of neurodegeneration, incorporating key biological factors such as functioning and infected neurons, extracellular alpha-synuclein, microglia, and T-cells. A fundamental assumption of the model is that neuronal deterioration is influenced by memory effects, where past states impact current disease progression, making fractional-order calculus more suitable than traditional integer-order models. The model accounts for the secretion and clearance of alpha-synuclein, the activation of immune responses, and the role of microglia in mitigating or exacerbating neuronal damage. Sensitivity analysis emphasizes the crucial role of factors like neuronal cells production ${\mathrm{\Pi }}_N$, infection prevalence $\gamma$, and stimulation of microglial cells $\mathrm{\Theta }$. Numerical simulations support the long-run neuroinflammatory feedback mechanism, revealing that smaller values of fractional order $\eta <1$ reduce disease progression. This is based on the premise that increased memory ( $\eta$ values less than one) leads to slower transmission of pathological protein aggregation. The study demonstrates that building a surrogate machine learning model of the NARX-BRBNN type, calibrated using numerical solver output, not only decreases computing complexity but also accurately replicates the dynamics of the fractional equation. This comparison underscores the necessity of employing fractional-order numerical schemes for accurately modeling complex neurobiological systems. The study proposes focused treatment approaches and provides insightful information on the course of neurodegenerative diseases.

In this paper, the dynamical behavior of the fractional-order cancer model has been analyzed with the fractal-fractional operator, which discretized the conformable cancer model. The fractional-order model consists of the system of nonlinear fractional differential equations. Also, we discuss the fractional-order model to check the relationship between the immune system and cancer cells by mixing IL-12 cytokine and anti-PD-L1 inhibitor. The tumor-immune model has been studied qualitatively as well as quantitatively via Atangana-Baleanu fractal-fractional operator. The nonlinear analysis is used to check the Ulam-Hyres stability of the proposed model. Moreover, the dynamical behavior for the fractional-order model has been checked by using a fractal-fractional operator with a generalized Mittag-Leffler Kernel and verifying the effect of fractional parameters. Finally, the obtained solutions are interpreted biologically, and simulations are carried out to illustrate cancer disease and support theoretical results, which will be helpful for further analysis and to control the effect of cancer in the community.

A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices.

This paper investigates the existence of positive mild solutions and controllability for fractional differential evolution equations of order $\gamma \in (1,2)$ with nonlocal conditions in Banach spaces. Our approach is based on Schauder's fixed point theorem, Krasnoselskii's fixed point theorem, and the Arzelà-Ascoli theorem. Finally, we include an example to verify our theoretical results.

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $1<r<2$. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.

A repeatedly infected person is one of the most important barriers to malaria disease eradication in the population. In this article, the effects of recurring malaria re-infection and decline in the spread dynamics of the disease are investigated through a supervised learning based neural networks model for the system of non-linear ordinary differential equations that explains the mathematical form of the malaria disease model which representing malaria disease spread, is divided into two types of systems: Autonomous and non-autonomous, furthermore, it involves the parameters of interest in terms of Susceptible people, Infectious people, Pseudo recovered people, recovered people prone to re-infection, Susceptible mosquito, Infectious mosquito. The purpose of this work is to discuss the dynamics of malaria spread where the problem is solved with the help of Levenberg-Marquardt artificial neural networks (LMANNs). Moreover, the malaria model reference datasets are created by using the strength of the Adams numerical method to utilize the capability and worth of the solver LMANNs for better prediction and analysis. The generated datasets are arbitrarily used in the Levenberg-Marquardt back-propagation for the testing, training, and validation process for the numerical treatment of the malaria model to update each cycle. On the basis of an evaluation of the accuracy achieved in terms of regression analysis, error histograms, mean square error based merit functions, where the reliable performance, convergence and efficacy of design LMANNs is endorsed through fitness plot, auto-correlation and training state.

In this study, we proved existence results for nonlinear implicit fractional differential equations with the Caputo version of the Atangana-Baleanu derivative, subject to the boundary and nonlocal initial conditions. The Kuratowski's measure of non-compactness and its associated fixed point theorems–Darbo's fixed point theorem and Mönchh's fixed point theorem, are the foundation for the analysis in this paper. We support our results with examples of nonlinear implicit fractional differential equations involving the Caputo version of the Atangana-Baleanu derivative subject to both boundary and nonlocal initial conditions. In addition, we provide solutions to the problems we considered.

In this paper, a fractional food chain system consisting of a Holling type Ⅱ functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type Ⅱ functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations.

This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C $\mathrm{\nabla }$D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C $\mathrm{\nabla }$D and the Riemann-Liouville nabla derivative (RL $\mathrm{\nabla }$D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example.