Basic Ideas in Epidemic Modeling

Abstract

In this chapter, we start by studying the early Kermack–McKendrick epidemic model, and introduce the basic ideas and ingredients of epidemic modeling. A crucial point is that we cannot precisely interpret the basic ideas and indices of infectious disease epidemiology without knowing the underlying nonlinear population dynamics. The early Kermack–McKendrick model is an infection-age-dependent outbreak model, and its extensions in the late 1970 s opened the door to the recent developments in mathematical epidemiology. The key idea of analyzing epidemic models is the basic reproduction number \(R_0\) and its well-known threshold principle: if \(R_0>1\), the final size of the epidemic is positive no matter how small the initial infected population, whereas if \(R_0 1\), there exists at least one endemic steady state, whereas if \(R_0<1\), there is no endemic steady state. This principle, however, does not hold under certain conditions. We provide examples in which subcritical endemic steady states exist even when \(R_0<1\) because of the reinfection of recovered individuals.