Optimal control for a tuberculosis model with undetected cases in Cameroon

• A deterministic model for the transmission dynamics of tuberculosis in the context weak diagnosis capacity is presented.
• The intervention strategy to control the disease and the cost functional are modeled.
• Using forward backward sweep method, the optimal schedule for tuberculosis control in Cameroon is discussed.
• Impact of parameters and nite time arising on the control are analyzed.
• Educational campaign and chemoprophylaxis may help to reduce by 80% the number of TB diagnosed cases.

This paper considers the optimal control of tuberculosis through education, diagnosis campaign and chemoprophylaxis of latently infected. A mathematical model which includes important components such as undiagnosed infectious, diagnosed infectious, latently infected and lost-sight infectious is formulated. The model combines a frequency dependent and a density dependent force of infection for TB transmission. Through optimal control theory and numerical simulations, a cost-effective balance of two different intervention methods is obtained. Seeking to minimize the amount of money the government spends when tuberculosis remain endemic in the Cameroonian population, Pontryagin’s maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically using the forward–backward sweep method (FBSM). Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods. It comes out that combining chemoprophylaxis and education, the burden of TB can be reduced by 80% in 10 years.