S i (t): number of susceptible individuals of the ith age group at time t; V i (t): number of vaccinated individuals of the ith age group at time t; E i (t): number of exposed (infected) individuals of the ith age group at time t; π i : age-specific vaccine efficacy for age group i. l i (t) and  i V t ( ) : disease incidence rates for the susceptible and vaccinated individuals of age group i. See [Additional file 1] for detailed formulae.