Dose–dependent infection rates of parasites produce the Allee effect in epidemiology

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Abstract
In many epidemiological models of microparasitic infections it is assumed that the infection process is governed by the mass–action principle, i.e. that the infection rate per host and per parasite is a constant. Furthermore, the parasite–induced host mortality (parasite virulence) and the reproduction rate of the parasite are often assumed to be independent of the infecting parasite dose. However, there is empirical evidence against those three assumptions: the infection rate per host is often found to be a sigmoidal rather than a linear function of the parasite dose to which it is exposed; and the lifespan of infected hosts as well as the reproduction rate of the parasite are often negatively correlated with the parasite dose. Here, we incorporate dose dependences into the standard modelling framework for microparasitic infections, and draw conclusions on the resulting dynamics. Our model displays an Allee effect that is characterized by an invasion threshold for the parasite. Furthermore, in contrast to standard epidemiological models a parasite strain needs to have a basic reproductive rate that is substantially greater than 1 to establish an infection. Thus, the conditions for successful invasion of the parasite are more restrictive than in mass–action infection models. The analysis further suggests that negative correlations of the parasite dose with host lifespan and the parasite reproduction rate helps the parasite to overcome the invasion constraints of the Allee–type dynamics.