A comprehensive analysis of fractional-order model of tuberculosis with treatment intervention

Abstract

Tuberculosis (TB) remains one of the top infectious disease killers worldwide, with an estimated 10.6 million new cases and 1.3 million deaths reported in 2022 alone (WHO, 2023). The COVID-19 pandemic has further disrupted TB control efforts by limiting access to healthcare services, interrupting treatment regimens, and delaying diagnoses and leading to a resurgence in TB transmission. Tuberculosis is caused by Mycobacterium tuberculosis and spread through the air, TB posing a serious threat to vulnerable populations, especially those with weakened immune systems such as individuals living with HIV. These challenges emphasize the need for more robust and realistic modeling approaches to inform policy and intervention. In this study, We incorporated fractional-order derivatives and applied the Adams-Bashforth method to better understand how TB spreads and how it can be controlled. The model divides the population into six key groups: those susceptible to infection, exposed individuals, people with acute TB, those with chronic TB, individuals undergoing treatment, and those who have recovered. To capture the complexities of TB transmission, we incorporated fractional-order derivatives along with the Adams-Bashforth method, allowing us to account for memory effects and more accurately reflect real-world dynamics. Sensitivity analysis revealed that increasing treatment rates significantly improves recovery outcomes. In addition, we conducted a quantitative analysis of the model, deriving the basic reproduction number (R₀) using the next-generation matrix method. The results show that the endemic equilibrium is globally and asymptotically stable when R₀ > 1, indicating that TB will persist in the population under these conditions. Conversely, when R₀ < 1, the disease will die out over time, highlighting the critical role of reducing transmission and increasing treatment to achieve effective TB control. The simulations also explored various intervention strategies, including improved treatment access, faster diagnosis, and recognition of nonlinear transmission dynamicsThese results emphsize the importance of timely intervention and provide actionable insights for strengthening TB control policies.