We investigate the initial behavior of a deterministic model of
parasitic infection, appropriate to transmission between homogeneously mixing
hosts, where the amount of infection which is transferred from one host to
another at a single contact depends on the number of parasites in the infecting
host. In this model, $R_0$ can be defined to be the lifetime expected number of
offspring of an adult parasite under ideal conditions, but it does not
necessarily contain the information needed to separate growth from extinction
of infection; nor need the growth rates of parasite numbers and numbers of
infected hosts be the same. Similar phenomena are observed if real time is
replaced by generation number, and the overlap of generations as time passes
need not correspond to that found, for instance, in the supercritical linear
birth and death process. The proofs involve martingale methods, applied to a
Markov chain associated with the deterministic differential equation
system.