Analysis of tuberculosis model with indirect environmental transmission and optimal control

In this paper, the dynamic behaviors of tuberculosis in the context of indirect environmental transmission are discussed by establishing the SEIRB epidemic model. The basic reproduction number is computed by employing the next-generation matrix approach. The global stability of disease-free equilibrium and endemic equilibrium is proved by constructing the Lyapunov function and the application of LaSalle's invariance principle. It shows that when the basic reproduction number is greater than 1, tuberculosis will spread among the population. When the basic reproduction number is less than 1, tuberculosis will disappear. Finally, an optimal control problem is constructed by using the extended model, which reveals the spread of tuberculosis can be effectively controlled by eliminating Mycobacterium tuberculosis in the environment and controlling tuberculosis patients at the same time. Numerical example results show the effectiveness of the optimization strategies.