Dynamics of an Epidemic Model with Vaccination and Saturated Incidence Rate

Measles and influenza are two major diseases-caused an epidemic in India. Therefore, in this paper, a $$ epidemic mathematical model for measles and influenza is proposed and analyzed, where pre and post vaccinations are considered as control strategies with waning natural, vaccine-induced immunity and saturation incidence rate. The dissection of the proposed model is conferred in terms of the associated reproduction number $$ , which is determined by the next-generation approach and obtained that if $$ , the disease-free equilibrium exists and it is locally as well as globally asymptotically stable. Further for $$ , a unique endemic equilibrium exists and it is also locally as well as globally asymptotically stable under certain conditions, which shows the prevalence and persistence of the disease in the population.