Five Classic Epidemic Models and Their Analysis
In this chapter, we present some standard mathematical methods for the analysis of compartmental epidemic models. We have chosen five classic epidemic models to demonstrate these methods. We start from the basic Kermack–McKendrick model and progressively expand it to a model with demography, and then introduce the Ross–MacDonald model for malaria. Each model is chosen to illustrate a specific mathematical approach for model analysis: the method of first integrals and level curves, the phase-line analysis, phase-plane analysis, reduction of dimension using homogeneity, and monotone dynamical systems. The general mathematical theories applied in this chapter are provided in Chapter 3 for reference and in-depth learning. Students in mathematics have a chance to learn these general theories in the setting of epidemic models and see how abstract theories of differential equations are applied to real-world problems. Students in public health and biological sciences will be able to learn the basic model analysis and gain exposure to some abstract mathematical concepts such as stability and bifurcations explained in the context of epidemiology, as well as to the theory of modern differential equations.