A noetic fractal-fractional dynamical formalism for history-dependent Seoul virus–Streptococcus pyogenes dual-pathogen interaction with physics-informed learning

Abstract

Seoul virus is an important rodent-borne hantavirus that can cause serious illness in humans, while Streptococcus pyogenes is a clinically significant bacterial pathogen associated with a wide range of infections and severe complications. The interaction between these two infections is not well understood from a mathematical modeling perspective. In this work, we develop a fractal–fractional model to investigate the transmission and co-infection dynamics of Seoul virus and Streptococcus pyogenes. The model is described using the Atangana–Baleanu–Caputo fractal–fractional derivative so that the effects of memory and the nonuniform nature of disease transmission can be represented more realistically than in conventional integer-order models. Several mathematical properties of the proposed system are studied, including the existence and uniqueness of solutions and the behavior of the disease-free equilibrium. The basic reproduction number is obtained to identify the threshold conditions associated with the spread of the infections, and the Ulam–Hyers stability of the model is also examined. In addition to the theoretical analysis, a physics-informed neural network is constructed to approximate the solutions of the fractal–fractional system. The governing equations and initial conditions are incorporated into the neural-network training process, allowing the model structure to guide the numerical approximation. The simulations illustrate the influence of the fractional and fractal parameters on the evolution of the co-infection system and show that the PINN approach can reproduce the model dynamics with good accuracy. The findings suggest that combining fractal–fractional operators with physics-informed learning offers a useful way to study co-infectious disease systems in which memory, heterogeneity, and nonlinear interactions play an important role.