Effect of Loss of Vaccine Induced Immunity in the Transmission Dynamics of Childhood Disease

Abstract

In this paper, a deterministic nonlinear model for the transmission dynamics of childhood disease is formulated and rigorously analyzed to explore the effect of loss of vaccine-induced immunity. The model was shown to exhibit two equilibria, namely, disease free equilibrium and an endemic equilibrium and their local stability was established using the computated effective reproduction number (Rv). The disease free equilibrium is globally asymptotically stable whenever Rv is less than unity by using an appropriate Lyapunov function. The global asymptotic stability of endemic equilibrium was established whenever Rv exceeds one by constructing a Lyapunov function using suitable combination of composite quadratic and logarithmic functions. Numerical simulation was done to validate its satisfactory agreement with the qualitative results, revealing that the loss of vaccine-induced immunity may be harmful to the spread of childhood disease provided it exceeds it critical threshold.