Modelling the transmission dynamics of infectious diseases with vaccination and temporary immunity

Abstract

In this dissertation, two non-linear mathematical models are proposed and analyzed to investigate the spread of infectious diseases in a variable size population through horizontal transmission in the presence of preventive or therapeutic vaccines which are capable of inducing temporary immunity and wane in time. In modeling the transmission dynamics, the population is divided into three subclasses namely; Susceptibles, Infectives and Vaccinated groups. It is assumed that both Vaccinated and Susceptible individuals are recruited into the community and can only become infected via contacts with the infectives group but the rate at which the vaccinated group may contract the diseases is extremely very low depending on the efficacy of the vaccine. All infectives are assumed to move at a constant rate to both Vaccinated and Susceptible groups. These models are analyzed by using the stability theory of differential equations and numerical simulation. The models exhibit two equilibria namely; the disease-free and the endemic equilibria. It is shown that if the vaccination reproduction number R0 < 1, the disease-free equilibrium is always globally asymptotically stable and in such a case the endemic equilibrium does not exist and the disease can be totally eliminated in the community. However, if R0 > 1, a unique endemic equilibrium exists that is locally asymptotically stable and consequently the equilibrium values of infective, vaccinated and susceptible population can be maintained at desired levels. Numerical simulations implemented on MAPLE using both Adomian decomposition technique and Runge-Kutta integration schemes, support our analytical conclusions and illustrate possible behaviour scenarios of the models.