Optimal control problem from tuberculosis and multidrug resistant tuberculosis transmission model

Tuberculosis (TB) is one of the contagious and deadly disease in the world. TB can mutate into Multidrug Resistance Tuberculosis (MDR-TB) if the patient does not start with an appropriate treatments. These two diseases can be modeled using system of ten-dimensional ordinary differential equation which represents 10 groups of individuals. Analytic and numerical analysis are done to explain the existence of equilibrium points and the basic reproduction number (R₀) of the model. The analytic and numerical analysis results show that the disease free equilibrium point is locally asymptotically stable if (R₀₁ < 1) and unstable if (R₀₁ > 1). The level set of (R₀) respect to interventions parameters is discussed to understand the sensitivity of each parameters to control the spread of TB and MDR-TB. In this paper, the model is also constructed as an optimal control problem with involving three control variables, such as BCG vaccination, treatment with first-line anti-TB drug, and treatment with second-line anti-TB drug. The aim of this problem is to minimize the number of infected individuals and also minimize cost of the controls that given. Optimal control derived using Pontryagin Minimum Principle and then solved numerically using the gradient descent method. The effectiveness of optimal control is exhibited by comparing the number of total infected individuals with and without optimal control. It has been observed that the optimal control strategy gives better result in minimizing the number of total infected individuals.