Mathematical Modeling and Optimal Control Strategies for Reducing the Spread of the Tuberculosis Infection

Abstract

Tuberculosis is an infectious disease with transmission through the air and droplets of infected people. The problem of spreading and treating tuberculosis needs some development. The model used in this study is a model of development, in which infected individuals are divided into two subpopulations, namely the active infection subpopulation stage one and second stage infection is MDR (multi drug resistant). Four optimal control strategies are implemented with the aim of reducing the rate of spread of tuberculosis TB infection, prevention by increasing a clean and healthy lifestyle, and using masks when interacting between infected and susceptible, vaccination for family and close friends of infected individuals, implementing the DOTS strategy (Directly Observed Treatment Shortcourse) for infected individuals, and Intensive care in hospital. Then the optimal control of the model was analyzed using Pontriagin's Maximum Principle while for global stability using Lyapunov. The numerical results are obtained that the optimal control strategy given can accelerate in reducing the rate of spread of tuberculosis in Indonesia. this is evidenced by the graph of the population infected with tuberculosis every year after the control has decreased the number of infected, and the graph is heading to the equilibrium point.