Curtailing the spread of typhoid fever: An optimal control approach

Abstract

In this paper, a deterministic compartmental model for the control of typhoid fever, which takes into account a partially effective vaccine and drug resistance effect was proposed. We incorporated five control strategies which consist of mass public health education, vaccination, treatment, second-line of treatment, and protection or environmental sanitation to curtail the spread of the disease in the population. The model consists of eight (8) compartments that include: Vaccinated population, susceptible population, exposed population, asymptomatic infected human population, symptomatic infected human population, resistant human population, treated human population, and the bacterial population. We developed a non-linear differential equation to study the dynamics of the model. We computed the basic reproduction number for a case of constant control which can be used to control the transmission dynamics of the disease and proved the local and global stability of the disease-free equilibrium, the result of stability analysis revealed that the disease-free equilibrium (DFE) is locally asymptotically stable if the basic reproduction number is less than one (1) using Routh–Hurwitz Criterion (RHC) and globally asymptotically stable if the basic reproduction number is less than one through the method of Castillo-Chavez. The stability analysis of the disease-free equilibrium result indicates that typhoid can be completely wiped out if the average number of secondary infection is kept at a number less than unity. We carry out sensitivity analyses on the reproduction number to ascertain the parameters that affect the reproduction number. Based on the results, an optimal control problem was formulated and analyzed using Pontryagin’s Maximum Principle (PMP) to determine the optimality system. We solved the optimality system using the forward and backward sweep method and the results revealed that the combination of vaccination, public health education, treatment, the second line of treatment, and environmental sanitation is the best strategy for controlling the spread of the disease.