Global stability analysis and optimal control of measles model with vaccination and treatment

A transmission and control model for measles infection is presented. The model incorporates vaccinated individuals and the role of treatment for both exposed and infected individuals. We present two main equilibrium points (disease-free and endemic) with the analysis of their stability. The basic reproduction number is calculated and we find that when it is less than unity, the disease-free equilibrium point is both locally and globally stable which means the disease can be eradicated under such condition. When it is greater than one, the infection is uniformly persistent and the endemic equilibrium is globally stable. The sensitivity index of basic reproduction number to the parameters within the model is also determined. Further, by using Pontryagin’s minimum principle, the optimal control problem is constructed with three controls i.e. vaccination, treatment of exposed individuals and treatment of infected individuals. Finally, the numerical simulations are established and our results show that a combination of all three controls gives the best result in reducing the number of measles infected individuals. These results indicate that being vaccinated followed by some treatments for both exposed and infected individuals would make measles eradication more efficient.