Modelling Tuberculosis Dynamics with Optimal Control Time-Dependent Interventions

Mahsa Hamidi (Departmant of Mathematics and Natural Sciences, Technical University of Berlin, Berlin)
Omid Solaymani Fard (Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad)
Olumuyiwa James Peter (Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai, Tamil Nadu 602105)
Zahra Dayheem (Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad)
Rantiola Famutimi (Department of Mathematical and Computer Sciences, University of Medical Sciences, Ondo City, Ondo State)
Hasan S Panigoro (Biomathematics Research Group, State University of Gorontalo, Bone Bolango)

Jambura Journal of Biomathematics (JJBM)
Vol. 7 No. 3: September 2026

Tuberculosis (TB) remains a major global health challenge, particularly in low- and middle-income countries. This study evaluates the impact of vaccination on TB dynamics using a mathematical model that incorporates two susceptible classes. The model examines the interplay between population characteristics, transmission pathways, treatment strategies, and intervention measures. The basic reproduction number, a key threshold parameter, is derived, and the global stability of both the disease-free and endemic equilibria is established. Furthermore, the model is extended to formulate an optimal control problem incorporating three time-dependent control functions: vaccination of susceptible individuals, public health education for the exposed population, and home-based treatment for infected individuals. Numerical simulations illustrate the theoretical results and show that, among the strategies considered, combining preventive measures with targeted vaccination is most effective in reducing TB infections. The findings provide insights into TB control and offer practical guidance for designing effective public health interventions.