Insights from Price's equation into evolutionary epidemiology

The basic reproduction number, denoted by R0, is one of the most important quantities in epidemiological theory ([11], [23]). It is defined as the expected number of new infections generated by an infected individual in an otherwise wholly susceptible population ([2], [12], [23]). Part of the reason why R0 plays such a central role in this body of theory undoubtedly stems from its relatively simple and intuitively sensible interpretation as a measure of pathogen reproduction. If R0 is less than unity then we expect the pathogen to die out since each infected individual fails to generate at least one other infection during the lifetime of the infection. Given that R0 is a measure of pathogen reproductive success, it is not surprising that this quantity has also come to form the basis of most evolutionary considerations of host-pathogen interactions ([1], [18]). For example, mathematical models for numerous epidemiological settings have been used to demonstrate that natural selection is often expected to favour the pathogen strain that results in the largest value of R0 ([6], [18]). In more complex epidemiological settings such optimization criteria typically cannot be derived and instead a game-theoretic approach is taken ([5]). In this context a measure of the fitness of a rare mutant pathogen strain is used to characterize the evolutionarily stable strain (i.e., the strain that, if present within the population in sufficient numbers, cannot be displaced by any mutant strain that arises). Typically R0 again plays a central role as the measure of mutant fitness in such invasion analyses ([10], [18], [30]). In this chapter we consider an alternative approach for developing theory in evolutionary epidemiology. Rather than using the total number of new infections generated by an infected individual (i.e., R0) as a measure of pathogen fitness we use the instantaneous rate of change of the number of infected hosts instead (see also [3], [18]). This shifts the focus from a consideration of pathogen reproductive success per generation to pathogen reproductive success per unit time. One very useful result of this change in focus is that we can then model the time dynamics of evolutionary change in the pathogen population simultaneously with the epidemiological dynamics, rather than simply characterizing the evolutionary equilibria that are expected. Even more importantly, however, this seemingly slight change