Dynamical aspects of a tuberculosis transmission model incorporating vaccination and time delay

Abstract

To explore transmission dynamics of tuberculosis, a tuberculosis transmission model with vaccination and time delay is developed in current paper. Positivity and boundedness are analyzed. Local stability of tuberculosis-free equilibrium in respect of the time delay due to latent period of tuberculosis is analyzed and we have found threshold value of the time delay for the local stability of tuberculosis-free equilibrium. Then, local stability of tuberculosis-existence equilibrium following exhibition of Hopf bifurcation at the crucial value of the time delay due to latent period of tuberculosis is derived. It is shown that the developed model undergoes a Hopf bifurcation around the tuberculosis-existence when the time delay due to latent period of tuberculosis passes through the threshold value. Direction and stability of the Hopf bifurcation are investigated with the help of the normal form method and center manifold theory. Finally, numerical simulations are carried out in the justification of obtained analytical findings. The results obtained provide significant information for tuberculosis disease controlling.

Keywords

• Hopf bifurcation
• Stability
• Periodic solutions
• Time delay
• Tuberculosis transmission model
• Vaccination