Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 5 (2016), pp. 3895-3916 © Research India Publications http://www.ripublication.com/gjpam.htm Modelling Childhood Disease Outbreak in a Community with Inflow of Susceptible and Vaccinated New-born Timothy Kiprono Yano 1* , Oluwole Daniel Makinde 2 and David Mumo Malonza 3 1 Department of Mathematics, Pan African University Institute for Basic Science, Technology and Innovation, P.O. Box 62000, Nairobi, Kenya. 2 Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa. 3 Department of Mathematics, Kenyatta University, P.O. Box 43844 Nairobi, Kenya. *Correspondence Author: Abstract This paper investigates the transmission dynamics of a Childhood disease outbreak in a community with direct inflow of susceptible and vaccinated new-born. Qualitative analysis of the SEIR nonlinear model is performed for disease free and endemic equilibria using the stability theory of differential equations. The disease free state is found to be both locally and globally asymptotically stable when the vaccination reproductive number v R is less than unity. In addition, the model exhibits transcritical forward bifurcation phenomenon and the sensitivity indices of the vaccination reproductive number with respect to various model parameters is determined. Using the Adomian decomposition method (ADM) and the fourth order Runge-Kutta integration scheme (RK4), the semi-analytical and numerical solutions of the nonlinear model are obtained. Pertinent results are displayed graphically and in tabular form. A vaccination coverage threshold is obtained above which the disease will be effectively eliminated from the community. Keywords: Childhood diseases; Epidemiological model; Vaccination coverage; Forward bifurcation; Sensitivity indices; Adomian decomposition method; Runge-Kutta integration scheme.