FRACTIONAL-ORDER MODEL OF IMMUNIZATION AND MULTIDRUG RESISTANCE IN PEOPLE WITH TUBERCULOSIS: LYAPUNOV STABILITY, MODELING AND CHAOS CONTROL

This study presents a fractional-order mathematical model to investigate the transmission dynamics of tuberculosis (TB), including both pulmonary and multidrug-resistant cases. The model incorporates immunization as a control strategy, considering the potential future application of vaccines for infants and adults. Additionally, the model introduces a quarantine class to address the specific challenges of multidrug-resistant tuberculosis (TB). To capture the complex dynamics of the disease, fractal-fractional-order derivatives are employed, allowing for a more accurate depiction of the memory and nonlocal effects within the population. The model's equilibrium points, including the disease-free and endemic states, are derived and analyzed. A sensitivity analysis is conducted to assess the influence of key parameters on the basic reproduction number. Exploring the existence of solutions is achieved through fixed-point theory and its applications. We establish certain conditions for global asymptotical stability and use the next-generation matrix method to develop local stability analysis. The model's global stability is further investigated using the Lyapunov function method. Revisiting sensitivity analysis of the model parameters provides a comprehensive evaluation. Subsequently, the chaos of the TB epidemic model is investigated through the feedback control approach. In terms of computation, we generate a numerical scheme for the fractal-fractional tuberculosis (TB) model with the use of Newton polynomial. The graphical findings utilizing the numerical simulation (MATLAB version 18) are shown and briefly discussed. Utilizing available data, we graphically present the results, employing various fractal-fractional-orders to gain insight into the mechanisms underlying the dynamics with a chaotic behavior approach. The findings suggest that vaccination and quarantine interventions play significant roles in managing the spread of both drug-sensitive and drug-resistant tuberculosis (TB).