A mathematical framework of HIV and TB co-infection dynamics.

The biological processes involved in diseases like human immunodeficiency virus (HIV) and tuberculosis (TB) require extensive research, particularly when both diseases occur together. This piece of research delves to explore a new fractional-order mathematical model that examines the co-dynamics of HIV and TB, taking into account the treatment effects. Although no definitive vaccine or cure for HIV exists, antiretroviral therapy (ART) can slow disease spread and prevent subsequent complications. The basic properties of the fractional model in the Caputo sense, including existence, uniqueness, positivity, and boundedness, are proved using crucial mathematical tools. The disease-free and endemic equilibria are determined for the co-infection model, along with the basic reproduction numbers [Formula: see text] for TB and [Formula: see text] for HIV, using the next-generation matrix technique. A comprehensive analysis is conducted to determine the local and global stability of the disease-free equilibrium point by applying the Routh-Hurwitz criteria and constructing a Lyapunov function, respectively. The stability of the disease-free state is also verified graphically by considering different initial conditions and observing the convergence of the curves to the disease-free equilibrium point. Furthermore, the model is examined under different scenarios by varying the reproduction numbers, specifically when [Formula: see text] and [Formula: see text], and when [Formula: see text] and [Formula: see text]. Using actual data from the USA from 1999 to 2022, crucial parameters are estimated. The final fitting of the model with real data demonstrates how effectively the model framework aligns with the data. Finally, computational simulations are performed for different cases to illustrate the behavior of the model solutions by varying the fractional order derivative, as well as examining the solution's behavior with respect to the stability points.