The purpose of this work is to present the dynamics of a fractional SIR model with generalized incidence rate using two differential derivatives, that are the Caputo and the AtanganaBaleanu. Firstly, we formulate the proposed model in Caputo sense and carried out the basic mathematical analysis such as positivity, basic reproduction number, local and global dynamics of the disease free and endemic case. The disease free equilibrium is locally and globally asymptotically stable when the basic reproduction number is less than 1. We also show that the SIR fractional model is locally and globally asymptotically stable at endemic state when the basic reproduction number greater than unity. Further, to analyze the dynamics of Caputo model we provide some numerical results by considering various incidence rates and with different fractional order parameters. Then, we reformulate the same model using the Atangana-Baleanu operator having nonsingular and non-local kernel. We prove the existence and uniqueness of the Atangana-Baleanu SIR model using PicardLindelof approach. Furthermore, we present an iterative scheme of the model and provide graphical results for various values of the fractional order α. From the graphical interpretation we conclude that the Atangana-Baleanu derivative is more prominent and provides biologically more feasible results than Caputo operator.

The incidence rate function describes the mechanism of a disease transmission and has a key role in mathematical epidemiology. In the present paper, we develop a fractional SEIR epidemic model in the Caputo sense with generalized incidence function. Initially, we present the existence and positivity of the Caputo SEIR epidemic model and calculate the basic reproduction number. Further, we investigate the model equilibria and prove the detail stability analysis of the model. Finally, the numerical simulations are provided for various values of fractional order α and different incidence rates. From the numerical simulations we conclude that the order of the fractional derivative plays a significant role to provides more insights about the disease dynamics.

Buruli ulcer is an increasingly common tropical disease that has been even ignored in advanced nations like Australia and Japan. Some mammals, including possums, have shown symptoms of the disease. Fractional derivatives are applied for a better understanding of biological processes and their crossover behavior. Infectious disease can be controlled by predicting the future spread while using Mathematical Models. Recently, various mathematical models based on classical integer-order and non-integer-order models have been proposed to predict the dynamics of infectious disease outbreaks. As a consequence, it is investigated the dynamics of the possum model using non-integer order derivatives in order to gain a better understanding and deeper insight into several biological models. The model’s important properties, such as the positivity of the model solution, equilibrium points, and the invariant property of the proposed model have also been discussed. The next-generation approach is used to compute the basic reproductive number  0 . Both the stability i.e. the local and global are obtained when  0 is less or greater than 1. The global stability of the fractional Possum model is achieved using the Lyapunov function in a fractional environment. Furthermore, The existence and uniqueness of the fractional order models are demonstrated. Numerical simulations of the model are done, along with their graphical representations, to examine the effects of arbitrary order derivatives and depict the implications of our theoretical results. It can be seen from the graphical findings that the fractional model provides more clarity and a better understanding.

Viral infections have spread globally, profoundly affecting social and economic aspects of life and causing widespread suffering. Infection caused by the hepatitis B virus (HBV) is one of the significant global health challenges but can be effectively controlled with appropriate treatment and vaccination. In this study, we present a fractional modeling approach and novel computational technique to analyze the impact of treatment on HBV transmission dynamics using the Caputo derivative. The existence and uniqueness of solutions for the fractional model are established using fixed point theory. The local stability of the disease-free equilibrium is examined using the linearization technique, while global stability is confirmed through the Hyers–Ulam–Rassias approach. Numerical simulation of the Caputo HBV model is performed using the fractional Euler method for various fractional orders, validating the theoretical results. Furthermore, to enhance the accuracy of epidemiological modeling, we develop a novel computational technique using deep neural networks (DNNs) to solve the Caputo HBV model. To support our theoretical findings, we conduct a comprehensive evaluation of the DNN-generated solutions by benchmarking them against standard numerical results and assessing them through multiple phases of training, validation, testing, error distribution analysis, and regression evaluation. The neural network training state is visualized using the gradient value of Mu, along with validation checks and detailed curve-fitting analyses for each model equation. The precision of the proposed scheme is checked using the comparison of the outputs with best validation performances around \(10^{-06}\)–\(10^{-10}\), and minimum absolute error ranges from \(10^{-04}\) to \(10^{-06}\). We believe that the proposed hybrid architecture enhances accuracy and computational efficiency offering a novel and effective framework not previously explored in infectious disease modeling.