Mathematical Analysis Of An Age-Structured Vaccination Model For Measles

Abstract

An age-structured vaccination model for the transmission dynamics of measles in a population is designed and rigorously analysed. In the absence of vaccination, the model exhibits the phenomenon of backward bifurcation (where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium whenever the associated reproduction number is less than unity). This phenomenon is shown to arise due to the imperfect nature of children’s natural immunity against infection or measles induced mortality. For the case when the measles-induced mortality is negligible, it is shown, using a linear Lyapunov function, that the DFE of the model without vaccination is globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the vaccination free model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a non-linear Lyapunov function of Goh-Volterra type, to be globally-asymptotically stable for a special case. Numerical simulations of the vaccination model show that the use of an imperfect anti-measles vaccine can result in the effective control of measles in the community provided the vaccine efficacy and coverage rate are high enough.