A consistent discrete version of a non-autonomous SIRVS model
  A family of discrete non-autonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens non-standard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. The consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (respectively permanence) for the continuous model we also have extinction (respectively permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data. 
Joaquim Mateus, C\'esar Silva and Sandra Vaz
math.DS