Modeling tuberculosis transmission in environmental ecology incorporating vaccination and treatment effects via a fractional-order approach and the Laplace–Adomian decomposition method

Tuberculosis (TB) is one of the leading diseases around the globe particularly in areas where the environmental and ecological conditions keep the disease persistent. Re-emergence of TB in the developing regions requires more holistic modeling efforts that would mirror real world complexities. It has been seen that the conventional models do not take into consideration the memory-dependent dynamics in the disease transmission. To this end, we would suggest a fractional order mathematical model that involves vaccination, treatment, and environmental interaction as it relates to the spread of TB. The memory effects are taken using fractional calculus and solutions obtained using La place-Adomian Decomposition Method (LADM) using semi-analytical solutions formula.The model emphasizes how human ecology reservoirs ensure transmissibility through simulating human ecology interactions with the environment. Stability analysis shows that TB will go extinct under the right conditions which in this case is the high rate of treatment and vaccination. The numerical simulations also show the model will be responding in a realistic way to different ecological conditions.The current investigation offers a powerful, newer scheme that integrates environmental microbiology, fractional calculus, and the methods of analysis. It also increases our knowledge into TB dynamics and gives a good idea in designing more competent public health interventions.