TRANSMISSION DYNAMICS AND OPTIMAL CONTROL OF AN AGE-STRUCTURED TUBERCULOSIS MODEL

Tuberculosis (TB) is still a serious threat to global public health, approximately 2 billion people worldwide are infected with TB. It is urgent to develop an optimal control strategy for TB. In this study, we propose an age-structured TB model taking into account vaccination, treatment, and relapse. We define the basic reproduction number $\mathcal{R}_{0}$ of the proposed model. Mathematical analyses show that the disease-free equilibrium state is globally asymptotically stable if $\mathcal{R}_{0} < 1$, and the endemic equilibrium state is globally asymptotically stable if $\mathcal{R}_{0}>1$. We combined TB data in China between 2007 and 2020 and the Markov-chain Monte-Carlo method to obtain the parameters and initial values of the model. Through the partial rank correlation coefficient method, we find the most sensitive parameters to $\mathcal{R}_{0}$. In light of the actual controllability, the transmission coefficient of TB and the treatment rate of the infectious population are chosen as controlled parameters to study the least cost-deviation problem. By using Pontryagin's maximum principle, we obtain the necessary conditions for optimal control. We also perform numerical simulations based on the forward-backward sweep method. Finally, we present optimal strategies that may help China achieve the End Tuberculosis Strategy by 2035 proposed by World Health Organization (WHO).