Optimal vaccination and treatment strategies for control of rubella with vertical transmission

Abstract

Rubella is a viral contagious disease associated with puffy cheeks, tender and swollen jaw. It spreads through direct contact with saliva or respiratory droplets from the mouth, nose or throat of infected persons. This work presents a system of nonlinear ordinary differential equations which models the transmission dynamics of the disease, with vertical transmission from infected
mothers to their offspring. The model also incorporates vaccination against the disease, and treatment of infected individuals as control measures for the disease. It is shown that the diseasefree equilibrium of the model is locally asymptotically stable when the control reproduction number, Rc is less than one. It is also shown that the model possesses a unique endemic equilibrium which exists when Rc > 1. This confirms the global stability of the disease-free equilibrium, and the absence of backward bifurcation in the model. Optimal control analysis is performed on the model to obtain the proportion of susceptible humans that will be vaccinated and infected humans that need to be isolated and treated for optimal control of the disease. Plots are presented to show the dynamics of the disease in the presence of the controls.