Mathematical modelling and optimal control of typhoid fever

Abstract

Typhoid fever is a disease caused by a salmonella bacterium (Salmonella typhi) and transmitted by ingestion of water and/or food contaminated with faeces (stool). In this paper, we derive and analyse a model for the control of typhoid fever which takes into account an imperfect vaccine combined with some other control measures already studied in the literature. We begin by analysing the model without control. We compute the basic reproduction number R_0 and prove the local and global stability of the disease-free equilibrium whenever R_0 is less than one through Lyapunov's theory. When R_0 is greater than one, we prove the local asymptotic stability of the unique endemic equilibrium through the Centre Manifold Theory and we find that the model exhibits a forward bifurcation. Then, we extend the model by reformulating it as an optimal control problem, with the use of three time dependent controls, to assess the impact of vaccination combined with protection/environment sanitation and treatment on the spread of the disease in human population. By using optimal control theory, we establish conditions under which the spread of the disease can be stopped, and we examine the impact of combined control tools on the transmission dynamic of the disease. Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations and efficiency analysis show that, if we want to reduce significantly the spread of typhoid fever, treatment must be taken into account in all control strategies.