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Dynamical behavior of a tuberculosis model with outcome of reinfection and optimal steering

T. K. Kar, Prasanta Kumar Mondal


In this paper, we propose and analyze a tuberculosis (TB) model and assume that treated individual may be again infected by infectious individual. The basic reproduction number is derived and it is observed that model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number and backward bifurcation which occurs due to sufficient level of reinfection. We extend our deterministic model to a stochastic version and study the dynamical behaviors of the deterministic model system in stochastic environment. We mainly use the Lyapunov functional to establish the global stability of endemic equilibrium point in stochastic sense. The effect of intensity of white noise on stability of the system is investigated using Euler’s method. We also apply optimal control theory to control the disease using vaccination and treatment as control variable. The objective function is to minimize the exposed and infectious individuals as well as the cost associated with it. Lastly, some simulations are given to verify our analytical results.


Tuberculosis, reproduction number, transcritical bifurcation, backward bifurcation, optimal control.

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