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Vaccination that gives partial protection for both newborns and susceptibles is included in a transmission model for a disease that confers no immunity. A general form of the vaccine waning function is assumed, and the interplay of this together with the vaccine efficacy and vaccination rates is discussed. The integro-differential system describing the model is studied for a constant vaccine waning rate, in which case it reduces to an ODE system, and for a constant waning period, in which case it reduces to a system of delay differential equations. For some parameter values, the model is shown to exhibit a backward bifurcation, leading to the existence of subthreshold endemic equilibria. Numerical examples are presented that demonstrate the consequence of this bifurcation in terms of epidemic control. The model can alternatively be interpreted as one consisting of two social groups, with education playing the role of vaccination.

1. Introduction. A classical SIS epidemic model for a disease that confers no immunity has only a disease free equilibrium (DFE) when parameters render the basic reproduction number R0< 1, and has one endemic stable equilibrium for parameters making R0> 1. Thus the disease dies out if R0< 1. By definition (see eg, Anderson and May [1]), R0 is “the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual”. A precise definition of R0 for an ODE model can be given as the spectral radius of the next generation matrix, see, eg,[4, 25]. In terms of stability, R0 is a threshold parameter, such that if R0< 1, then the DFE is locally asymptotically stable, and unstable if …