MATHEMATICAL ANALYSIS OF A TUBERCULOSIS CONTROL MODEL WITH VACCINATION, TREATMENT, AND PUBLIC AWARENESS VIA THE LAPLACE ADOMIAN METHOD

Abstract

Tuberculosis remains a major public health challenge in many endemic regions. Populations with incomplete treatment coverage and limited awareness require an integrated control approach involving vaccination, timely treatment, and sustained public awareness. This study develops and analyses a mathematical model to evaluate the combined effects of vaccination, treatment, and awareness campaigns on tuberculosis transmission and control. The transmission model incorporating vaccination, treatment, and public awareness was formulated and shown to be well-posed using the Lipschitz condition, which guarantees existence and uniqueness of nonnegative solutions. The effective reproduction number was derived using the next-generation matrix method, while local stability of equilibrium points was examined using the Routh-Hurwitz criterion. Sensitivity analysis was conducted to identify parameters that influence transmission. The Laplace-Adomian decomposition method was used to obtain approximate analytical solutions, and simulations were carried out in Maple 18 to assess intervention scenarios. Results indicate that public awareness can substantially lower susceptibility and exposure over the simulated period, while childhood vaccination, adult vaccination, and improved treatment coverage reduce infection when combined. The findings support the use of integrated awareness, vaccination, and treatment strategies for tuberculosis control in settings with suboptimal treatment coverage. The study's main contribution is the combined-intervention threshold and associated sensitivity structure; the LADM implementation provides a semi-analytical solution framework rather than a direct efficiency comparison with Runge-Kutta methods.