Endemic Disease Models

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, the basic reproduction number [Formula: see text] determined a threshold. If [Formula: see text] the disease dies out, while if [Formula: see text] the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if [Formula: see text] and if [Formula: see text] there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when [Formula: see text] , and other situations in which there is an endemic equilibrium that is unstable for some values of [Formula: see text] .