The characteristics of epidemics and invasions with thresholds

In this paper we report the development of a highly efficient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensiohal reaction-diffusion models whose dynamics include a threshold or Allee effect. We prove that this methodology produces the correct results for single-component models which are generalizations of the Fisher model, and then demonstrate by numerical experimentation that analogous methods work for a wide class of epidemic and invasion models including the S-I and S-E-I epidemic models and the Rosenzweig-McArthur predator-prey model. As examplary application of this approach we consider the atto-fox effect in the classic reaction-diffusion model of rabies in the European fox population and show that the appropriate threshold for this model is within an order of magnitude of the peak disease incidence and thus has potentially significant effects on epidemic properties. We then make a careful re-parameterisation of the model and show that the velocities calculated with realistic thresholds differ surprisingly little from those calculated from threshold-free models. We conclude that an appropriately thresholded reaction-diffusion model provides a robust representation of the initial epidemic wave and thus provides a sound basis on which to begin a properly mechanistic modelling enterprise aimed at understanding the long-term persistence of the disease.