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Background: Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal-fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population., Methods: Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model's equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel., Results: The stability analysis of the fractal-fractional model has been confirmed for both Ulam-Hyers and generalized Ulam-Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal-fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values., Conclusion: According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control., Competing Interests: Declaration of competing interest It is stated that we do not have any conflict of interest with any person institution or company., (Copyright © 2024 Elsevier Ltd. All rights reserved.)

The present study discussed a model to describe the SARS-CoV-2 viral kinetics in the presence of saturated antiviral responses. A discrete-time delay was introduced due to the time required for uninfected epithelial cells to activate a suitable antiviral response by generating immune cytokines and chemokines. We examined the system's stability at each equilibrium point. A threshold value was obtained for which the system switched from stability to instability via a Hopf bifurcation. The length of the time delay has been computed, for which the system has preserved its stability. Numerical results show that the system was stable for the faster antiviral responses of epithelial cells to the virus concentration, i.e., quick antiviral responses stabilized patients' bodies by neutralizing the virus. However, if the antiviral response of epithelial cells to the virus increased, the system became unstable, and the virus occupied the whole body, which caused patients' deaths. [ABSTRACT FROM AUTHOR]

In this paper, the higher nonlinear problems of fractional advection-diffusion equations and systems of nonlinear fractional Burger's equations are solved by using two sophisticated procedures, namely, the q-homotopy analysis transform method and the residual power series method. The proposed methods are implemented with the Caputo operator. The present techniques are utilised in a very comprehensive and effective manner to obtain the solutions to the suggested fractional-order problems. The nonlinearity of the problem was controlled tactfully. The numerical results of a few examples are calculated and analyzed. The tables and graphs are constructed to understand the higher accuracy and applicability of the current method. The obtained results that are in good contact with the actual dynamics of the given problem, which is verified by the graphs and tables. The present techniques require fewer calculations and are associated with a higher degree of accuracy, and therefore can be extended to solve other high nonlinear fractional problems. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of kidnapping incidents through the application of the Lotka-Volterra model with fractional order modeling. Drawing parallels between predator–prey interactions in ecological systems and the relationship between Kidnappers and Victims, this study aims to provide insights into the complex phenomenon of kidnapping. Theoretical foundations are established through key definitions and theorems, paving the way for the construction of the model. The existence and uniqueness of solutions to the model equations are examined, followed by a stability analysis to assess the long-term dynamics of kidnapping incidents. Numerical simulations are conducted to validate the theoretical framework and explore the sensitivity of the model to different parameters and initial conditions. By employing the Lotka-Volterra model with fractional order modeling, the study gives a deeper understanding of kidnapping incidents by drawing parallels with ecological systems, allowing for a more comprehensive analysis of the complex interactions between perpetrators and victims. [ABSTRACT FROM AUTHOR]