Theoretical evolutionary epidemiology analyzes the population dynamics of parasites and their hosts, together with their evolutionary dynamics (Day and Proulx 2004; Day and Gandon 2006, 2007). Classically, the analysis of evolutionary dynamics focuses on phenotypic traits like virulence, transmission, and recovery rates (Day and Proulx 2004, Gandon and Day 2007). In contrast, we focus here on the evolutionary dynamics of mean host and parasite fitness. Because host and parasite population growth rates are directly governed by mean population fitness, this analysis provides a direct link between evolutionary and epidemiological dynamics. In particular we show how this direct link helps to better understand the effect of recurrent mutations and environmental change in situations leading to population extinction.

This analysis also provides a way to rephrase classical evolutionary epidemiology models in the broader context of Fisher's fundamental theorem of natural selection. This theorem states that “the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time” (Fisher 1930). The validity of this theorem has been challenged in many situations in which evolution does not lead to the maximization of mean fitness (Moran 1964; Ewens 1969, 1989; Nagylaki 1992, 1993). Price (1972) showed this apparent lack of generality of Fisher's fundamental theorem comes from a misinterpretation of the theorem. The change in mean fitness , in the context of the environment *e*, between two points in time is (Price 1972; Frank and Slatkin 1992):

where the prime refers to the values of and *e* at the next time point. It is particularly useful to partition the change in mean fitness in the following way (Frank and Slatkin 1992):

where the first term, , is solely due to the effect of natural selection, whereas the second term, , refers to effect of the change of the environment, that is all the factors (biotic or abiotic) that may affect the fitness of genotypes (Fig. 1). This very general definition of the environment thus includes factors that may induce frequency-dependent or frequency-independent selection, and each factor may be viewed as one dimension of the environment (Mylius and Diekmann 1995). Price (1972) clarified Fisher's theorem by pointing out that it focuses only on , and thus only on the partial change of mean fitness due to the action of natural selection. Yet, the complexity introduced by the change of the environment has rarely been used to understand the dynamics of adaptation in particular case studies. The dynamics of mean fitness is often studied in simplified ecological scenarios in which genotypic fitness do not vary with time (Ewens 1989), or vary very slowly (Nagylaki 1979, 1993). Typically, only the change of the environment due to density-dependence has been analyzed in any detail (Fisher 1930; Kimura 1958; Frank and Slatkin 1992; Nagylaki 1992).

In this article we analyze the dynamics of adaptation in the broader context of interspecific interactions, where the environment of a species depends on the abundance and evolution of other species. In particular we demonstrate its potential relevance in the study of host-parasite interactions. We use a general epidemiological model to describe both the host and the parasite dynamics. We explore the dynamics of mean fitness in a diversity of situations. First, when the parasite evolves in a homogeneous host population, the degradation of the environment of the parasite corresponds to the decrease of the density of susceptible hosts. Second, allowing the host to coevolve adds yet another factor to the degradation of parasite's environment. Those different case studies help to illustrate the different factors governing the dynamics of parasite adaptation. In particular, following and extending Price (1972), we show that the dynamics of adaptation can always be partitioned among the effects of (1) natural selection, (2) recurrent mutations, and (3) changes in the environment.