Mathematical modeling, forecasting, and optimal control of typhoid fever transmission dynamics

Abstract

In this paper, we derive and analyze a model for the control of typhoid fever which takes into account an imperfect vaccine combined with protection, environment sanitation, and treatment as control mechanisms. The analysis of the autonomous model passes through the computation of the control reproduction number Rc, the proof of the local and global stability of the disease-free equilibrium whenever Rc is less than one using Lyapunov’s theory. When Rc is greater than one, we prove the local asymptotic stability of the unique endemic equilibrium through the Center Manifold Theory and we find that the model exhibits a forward bifurcation. Using clinical data from Mbandjock, a town in the Centre Region of Cameroon, we calibrate the model by estimating model parameters. We find that the control reproduction number is approximatively equal to 2.4750, which means that we are in an endemic state (Rc>1). We also performed a sensitivity analysis by calculating the Partial Rank Correlation Coefficient (PRCC) of Rc and of infected compartments classes. Then, we extend the model by reformulating it as an optimal control problem, with the use of three time-dependent controls, namely vaccination, individual protection/environment sanitation, and treatment. Optimal control theory is used to analyze our optimal control model. Numerical simulations and efficiency analysis are performed to show the impact of each control strategy on the decrease of the disease burden.