Analysis of Fractional-Order Model for the Transmission Dynamics of Tuberculosis Incorporating Vaccination and Impact of Environmental Interventions

Tuberculosis (TB), caused by Mycobacterium tuberculosis, remains one of the leading causes of death from a single infectious agent, particularly in low-resource settings where health disparities persist despite available treatment and preventive strategies. Addressing the limitations of classical models, this study introduced a novel fractional-order mathematical model to better capture the complex transmission dynamics of TB and incorporated the epidemiological impact of key control measures, including vaccination, reinfection, and environmental interventions. The model applied Caputo fractional derivatives to captured memory-dependent and non-local characteristics inherent in TB progression. Using Schauder and Banach fixed-point theorems, the analysis established the existence and uniqueness of solutions for arbitrary-order fractional systems. Numerical simulations were performed to investigate the influence of fractional-order parameters on disease dynamics. Results showed that increasing the fractional order significantly reduces the rate of disease spread, lowers the peak incidence, and extends the duration of the epidemic curve outcomes that align with the hereditary and temporal properties of fractional calculus. Sensitivity analysis highlighted the critical roles of vaccination, timely treatment, and environmental hygiene, confirming their effectiveness in lowering the basic reproduction number and cumulative incidence. The findings of this study reveal that fractional-order models not only capture the memory and delay effects inherent in TB transmission, but also provided more realistic epidemic trajectories than classical integer-order systems. The novelty of this work lies in the integration of fractional calculus with environmental interventions, vaccination, and reinfection dynamics, offering a more comprehensive framework for analyzing the spread and control of tuberculosis. By demonstrating that higher fractional orders reduce transmission intensity, delay peak infection, and extend epidemic duration, the study provided valuable insights into how long-term memory effects influence disease behavior. Based on these results, the study, it was recommended that prioritizing early treatment, expansion of vaccination programs, and environmental health improvements (e.g., ventilation and sanitation) as effective strategies for reducing TB burden. The fractional framework also supports evidence-based policy planning by enhancing the predictive capacity of models used in public health, thereby offering a powerful tool for optimizing resource allocation and intervention timing in tuberculosis control efforts.