Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior

The human immunodeficiency virus, which attacks the immune system and especially targets CD4 cells that are crucial for immunological defense against infections, is the cause of the severe illness known as acquired immunodeficiency syndrome (AIDS). This condition has the potential to take a patient’s life. Understanding the dynamics of AIDS and evaluating potential methods of prevention and treatment have both significantly benefited from the use of mathematical modeling. This research article proposes a unique technique that solves a model system of differential equations representing diverse populations, such as susceptible populations, acute populations, asymptomatic populations, and symptomatic populations or populations with AIDS. The method uses an artificial neural network (ANN) to do this. A specific Caputo–Fabrizio derivative is included in the suggested method to validate the system’s stability via the use of Krassnoselskii’s and Banach’s fixed-point approach in combination with the exponential kernel. In order to solve the differential equations and get the required data, the Laplace Adomian Decomposition (LAD) technique is used. Training the ANN involves obtaining simulated data from LAD and doing it within the context of a supervised learning framework. The performance of the ANN is assessed by comparing its predicted solutions to the LAD solutions. This allows for the calculation of the average error for each of the system’s functions. This study presents a potentially useful computational tool for understanding the dynamics of AIDS and delivering important insights into the design of new prevention and treatment methods.