Spatial-Temporal Dynamics in Nonlocal Epidemiological Models

Throughout recorded history, nonindigenous vectors that arrive, establish, and spread in new areas have fomented epidemics of human diseases such as malaria, yellow fever, typhus, plague, and West Nile (Lounibos 2002). The spatial spread of newly introduced diseases is a subject of continuing interest to both theoreticians and empiricists. One strand of theoretical developments (e.g., Kendall 1965; Aronson and Weinberger 1975; Murray 1989) built on the pioneering work of Fisher (1937) and Kolmogorov et al. (1937) based on a logistic reaction-diffusion model to investigate the spread of an advantageous gene in a spatially extended population. With initial conditions corresponding to a spatially localized introduction, such models predict the eventual establishment of a well-defined invasion front which divides the invaded and uninvaded regions and moves into the uninvaded region with constant velocity. Provided that very small populations grow in the same way or faster than larger ones, the velocity at which an epidemic front moves is set by the rate of divergence from the (unstable) disease-free state, and can thus be determined by linear methods (e.g., Murray 1989). These techniques have been refined by Diekmann (1978, 1979), Thieme (1977a, 1977b, 1979), van den Bosch et al. (1990), etc. who used a closely related renewal equation formalism to facilitate the inclusion of latent periods and more general and realistic transport models. Behind the epidemic front, most epidemic models settle to a spatially homogeneous equilibrium state in which all populations co-exist at finite abundances. In many cases, the passage from the epidemic front to co-existence passes through conditions where the local abundances of some or all of the players drops to truly microscopic levels. Local rekindling of the disease usually takes place not only because of the immigration of infectives but also by in situ infections produced by the non-biological remnants of previous populations.