An epidemiology model that includes a leaky vaccine with a general waning function

Abstract

Vaccination that gives partial protection for both newborns andsusceptibles is included in a transmission model for a diseasethat confers no immunity. A general form of the vaccine waningfunction is assumed, and the interplay of this together with thevaccine efficacy and vaccination rates is discussed. Theintegro-differential system describing the model is studied for aconstant vaccine waning rate, in which case it reduces to an ODEsystem, and for a constant waning period, in which case it reducesto a system of delay differential equations. For some parametervalues, the model is shown to exhibit a backward bifurcation,leading to the existence of subthreshold endemic equilibria.Numerical examples are presented that demonstrate the consequenceof this bifurcation in terms of epidemic control. The model canalternatively be interpreted as one consisting of two socialgroups, with education playing the role of vaccination.