An unconditionally convergent discretization of the SEIR model

The SEIR (susceptible, exposed, infectious, recovered) model has been discussed by many authors, in particular with reference to the spread of measles in an epidemic. In this paper, the SEIR model with constant rate of infection is solved using a first-order, finite-difference method in the form of a system of one-point iteration functions. This discrete system is seen to have two fixed points which are identical to the critical points of the (continuous) equations of the SEIR model and it is shown that they have the same stability properties. It is shown further that the solution sequence is attracted from any set of initial conditions to the correct (stable) fixed point for an arbitrarily large time step. Simulations confirm this and results are compared with well-known numerical methods.