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Both mathematics and science exhibit a reciprocally advantageous and close relationship. The undeniable significance of mathematics in diverse scientific fields, including electrical engineering, physics, biology, and medicine, is beyond dispute. Due to the multitude of applications of mathematical biology in the present era, scholars have demonstrated a marked intrigue in this specific realm. In the past few years, there has been a discernible rise in the application of different fractional operators as a method of expressing the given problem. The utilization of this approach has gained significant acceptance as a customary technique for examining the propagation of epidemics. The current investigation explores innovative fractional operators in the framework of a model for an infectious disease. The mechanical characteristics of the model with fractional orders are established by employing diverse numerical methods and adjusting parameters related to time fractional components. To substantiate the theoretical discoveries, computational simulations are implemented for the suggested approach over a variety of fractional orders, displaying the outcomes of both fractional and fractal orders. We employed a highly efficient numerical methodology to obtain solutions for both the model and simulations. Additionally, we provide the criteria for the existence of a solution to the proposed epidemic model, and we determine the reproductive number under specific state conditions of the analyzed dynamic system. A fractional-order model of infectious disease is proposed to be analyzed through simulations, aiming to ascertain the possible efficacy of disease transmission within the community. These aspects are tackled through the application of the Banach contraction principle. Moreover, we have proposed future research directions incorporating the utilization of a novel hybrid fractional operator. Fractional calculus emerges as a significant area of investigation within applied mathematics, offering valuable tools for problem-solving across diverse fields such as medicine, science, and engineering. Recent scholars have emphasized the importance of mathematical techniques, especially those related to fractional and integer calculus modeling, as extremely advantageous when examining the dynamics of different disease models. The utilization of the fractional operator allows for the investigation of the dynamic impact of diseases on society, providing valuable insights for analysis, decision-making, and disease control. [ABSTRACT FROM AUTHOR]

In this manuscript, we developed a nonlinear fractional order Ebola virus with a novel piecewise hybrid technique to observe the dynamical transmission having eight compartments. The existence and uniqueness of a solution of piecewise derivative is treated for a system with Arzel'a-Ascoli and Schauder conditions. We investigate the effects of classical and modified fractional calculus operators, specifically the classical Caputo piecewise operator, on the behavior of the model. A model shows that a completely continuous operator is uniformly continuous, and bounded according to the equilibrium points. The reproductive number R0 is derived for the biological feasibility of the model with sensitivity analysis with different parameters impact on the model. Sensitivity analysis is an essential tool for comprehending how various model parameters affect the spread of illness. Through a methodical manipulation of important parameters and an assessment of their impact on Ro, we are able to learn more about the resiliency and susceptibility of the model. Local stability is established with next Matignon method and global stability is conducted with the Lyapunov function for a feasible solution of the proposed model. In the end, a numerical solution is derived with Newton's polynomial technique for a piecewise Caputo operator through simulations of the compartments at various fractional orders by using real data. Our findings highlight the importance of fractional operators in enhancing the accuracy of the model in capturing the intricate dynamics of the disease. This research contributes to a deeper understanding of Ebola virus dynamics and provides valuable insights for improving disease modeling and public health strategies. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra‐Fredholm integro‐differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

The focus of this work is on the absolute controllability of Hilfer impulsive non-instantaneous neutral derivative (HINND) with integral boundary condition of any order. Total controllability refers to the system's ability to be controlled during the impulse time. Kuratowski measure and semigroup theory in Banach space yield the results. Furthermore, we talked about optimal controllability in conjunction with appropriate limitations. Our established outcomes are described using an example. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order 1 < r < 2. By applying the results and facts belongs to the cosine function of operators, fractional calculus, the measure of noncompactness and fixed point approach, the main results are established. Initially, we focus the existence of mild solution, and later we establish the controllability of the considered fractional system. Finally, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our paper, we primarily concentrate on the existence and controllability results for fractional evolution inclusions of order 1 < r < 2 with Clarke's subdifferential type. By applying the facts related to the measure of noncompactness, fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our article, we primarily concentrate on the approximate controllability results for Sobolev type fractional stochastic evolution hemivariational inequalities. By applying the facts related to fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown‐like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS‐CoV‐2 virus into a system of equations to examine the existence and uniqueness results with the Atangana–Baleanu derivative by using a fixed‐point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India. [ABSTRACT FROM AUTHOR]