Basic Reproduction Number $$R_0$$

Abstract

The basic reproduction number \(R_0\) plays a central role in structured population dynamics. Although some roots of \(R_0\) can be traced back to the nineteenth century, the specific concept was introduced to the demography literature in 1925. It took a further half century for this number to mature as a key concept in mathematical epidemiology, and it is only recently that the stable population theory has become a popular tool in the field. However, the progress of mathematical epidemiology over the past two decades has been remarkable, and the basic concept and applications of \(R_0\) are now better developed in epidemiology than in demography. In particular, the successful introduction of a general definition of the basic reproduction number for structured populations in the context of epidemic models gave rise to a new epoch in our understanding. Since then, the theory of the basic reproduction number has been developed as a central tenet of both infectious disease epidemiology and general population dynamics. Recently, this basic idea has evolved considerably to allow its application to time-heterogeneous environments. In this chapter, we sketch a general theory of \(R_0\). First, we formulate a general definition for the basic reproduction number \(R_0\) of structured populations in time-heterogeneous environments. Based on the generation evolution operator, we show that the basic reproduction number can be calculated as the spectral radius of the next-generation operator in a constant environment or in a periodic environment. Subsequently, we define the type-reproduction number in a time-heterogeneous environment and examine some applications in demography and epidemiology. Finally, we discuss some methods to estimate \(R_0\) from available data.