Modeling the transmission dynamics of Two-strain TB with drug-sensitive and drug-resistant in Tanzania: A fractional order approach

Tuberculosis (TB) is a global infectious disease that causes significant human mortality annually. In this study, we develop a mathematical model to investigate TB transmission dynamics in Tanzania, considering drug-sensitive and drug-resistant strains from 2018 to 2023. The model incorporates treatment interventions along with nutritional supplements using the Caputo Fractional-Order Derivative approach. The analysis of model solutions and stability at both disease-free and endemic equilibrium points is conducted usingthe Jacobian matrix and Lyapunov functions. Analytically, the study proves that TB-infected populations exhibit R-0 > 1, confirming global stability, while disease-free states is locally and globally asymptotically stable when R-0 < 1 . Notably, variations in the fractional-order derivative ( 0 < alpha <= 1) significantly influence disease transmission dynamics under intervention scenarios with memory effects, affecting both the current infection rate and past states. The implementation of treatment interventions (0 <eta(s), eta(r) <= 1) significantly reduces the threshold index from R-e = 2.54385 to R-e =0.617896 . Moreover, with interventions, the integer-order derivative model reduces infection rates to 0.01 and 0.03 for drug-sensitive and drug-resistant individuals, respectively, but takes over 50 months. In contrast, the fractional-order derivative model successfully minimizes both infection rates to nearly zero within just 30 to 40 months after treatment intervention. This demonstrates that incorporating a fractional-order model provides a more effective strategy for tuberculosis infection clearance, offering flexible dynamics that capture long-term memory effects, an advantage that the integer-order model fails to achieve. Overall, the study suggests that the intervention efficacy enhanced by the fractional-order derivative technique accounting for memory effects and non-local solutions provides a flexible and accurate framework for understanding and predicting TB transmission, its progression, and effective control measures over time.