Mathematical modeling of infectious diseases: methods and some results

This paper is an overview of some techniques for analyzing mathematical models of infectious diseases, using dynamical systems and bifurcation theory. A non-standard finite-difference scheme for numerical integration of the model's differential equations is shown to preserve positivity and the asymptotic behavior of the model, unlike standard schemes such as Runge-Kutta. The methods are presented by discussing work on an epidemiological model in which vaccination or recovery from infection confers immunity, and a Gause-type predator-prey model with generalized functional response.