Stability and optimizing the treatment control of tuberculosis model via numerical approach

According to World Health Organization data, tuberculosis (TB) affects nearly one-third of the world’s population and causes several million deaths and new cases each year. Recent advances in fractal–fractional differential operators have proven effective in simulating complex real-world problems. In this study, we present a TB model with an emphasis on hospital treatment and public health education, using a fractal–fractional operator under the Mittag-Leffler function. The study focuses on biological feasibility elements such as unique solutions, existence, positivity, and feasible domains. The Lipschitz and growth conditions are used to demonstrate the existence and uniqueness of solutions to the proposed TB system. A next-generation matrix technique is used to calculate the effective reproductive number of tuberculosis to determine its spread. Suitable Lyapunov functionals are developed to demonstrate the global stability of both TB-free and endemic equilibria. Each model parameter’s impact on the effective reproductive number is assessed using a normalized sensitivity index calculation. A numerical iterative method with Newton polynomial interpolation is utilized to demonstrate the usefulness of the proposed model, and numerical simulations show that it is more efficient at various fractional orders. We looked at numerical data from a variety of factors and fractional order values, concentrating on their impact on disease eradication. The simulation results are compared between the Newton polynomial interpolation approach and the fractional Adams–Bashforth–Moulton predictor–corrector method for the model compartments. The fractal–fractional approach essentially combines the complex real-world dynamics of infectious diseases with theoretical mathematics. This approach offers deep insights that help improve public health decision-making and guide successful control measures.