The SIR Model with Demography: General Properties of Planar Systems

Abstract

This chapter introduces a number of simple demographic models and fits them to population data. A demographic model is then integrated with the SIR model, which results in an SIR model with demography. The model is reduced to a 2 × 2 system and nondimensionalized. General tools for analysis of planar systems are presented and applied to the SIR model. The basic reproduction number is defined, and its mathematical and epidemiological significance is highlighted. Methods for establishing global stability of equilibria of planar systems are covered and applied to the SIR model. The concepts of Hopf bifurcation and periodic cycles are introduced and applied to the SIR model with more general incidence. Although much of the material presented in this chapter is basic for ODE books, its application to epidemic models that are characterized by multiple unknown parameters is nontrivial.